Given a group $G$ and $S \subset G$, the Cayley (di)graph Cay$(G,S)$ is the graph whose vertex set is $G$ with an arc from $g$ to $h$ if and only if $hg^{-1}\in S$. Although the graph isomorphism problem in general is only known to be quasipolynomial, for some families of Cayley (di)graphs it reduces to understanding the automorphisms of the groups that are involved. The question of when this happens is known as the Cayley Isomorphism (CI) problem.
I will provide an overview of the current state of the CI problem for both finite and infinite groups.
On Calabi–Yau threefolds there are two types of integral invariants, quantum K-invariants and Gopakumar–Vafa invariants. In this talk, I will explain a joint project (with You-Cheng Chou) which aims to show that the quantum K-invariants and Gopakumar invariants are equivalent. At genus zero, this is a conjecture by Jockers–Mayr and Garoufalidis–Scheidegger (for the quintic), and a proof of the JMGS conjecture will be presented.
For A an abelian variety defined over a number field, a conjecture of Serre predicts that the set S of primes of ordinary reduction for A has positive density. If A is an elliptic curve without CM, Serre proved that S has density 1, and Elkies showed that, if L admits a real embedding, then the complement of S, that is the set of supersingular primes, is infinite.
In this talk, I will discuss generalizations of both Serre’s and Elkies’s theorems to abelian varieties of type IV, that is with multiplication by a CM field, which are parametrized by unitary Shimura curves.
This talk is based on joint work in progress with Victoria Cantoral-Farfan, Wanlin Li, Rachel Pries, and Yunqing Tang.
In 1955, Richard Brauer made what was arguably the first “local-global” conjecture in character theory. I’ll discuss the conjecture, often known as Brauer’s Height Zero Conjecture, and its recent proof due to joint work with G. Malle, G. Navarro, and P.H. Tiep.
A theorem of Furstenberg from 1967 states that if \Gamma is a lattice in a semisimple Lie group G, then there exists a measure on \Gamma with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary is absolutely continuous. We will discuss generalizations of this theorem in the setting of the Mapping class group and Gromov hyperbolic groups.
Consider the Monge-Kantorovich optimal transport problem where the cost function is given by a Bregman divergence. The associated transport cost, termed the Bregman-Wasserstein divergence here, presents a natural asymmetric extension of the (squared) 2-Wasserstein metric and has recently found applications in statistics and machine learning. On the other hand, Bregman divergence is a fundamental concept in information geometry and induces a dually flat geometry on the underlying manifold. Using the Bregman-Wasserstein divergence, we lift this dualistic geometry to the space of probability measures, yielding an extension of Otto’s weak Riemannian structure of the Wasserstein space to statistical manifolds. We do this by generalizing Lott’s formal geometric computations for the Wasserstein space. In particular, we define generalized displacement interpolations which are compatible with the Bregman geometry, and construct conjugate primal and dual connections on the space of distributions. We also discuss some potential applications. Ongoing joint work with Cale Rankin.
The celebrated Erdos-Ko-Rado theorem can be interpreted as a characterization of the size and structure of independent sets/co-cliques of maximum size in Kneser Graphs. Similar characterizations have been observed in a few other classes of Distance Regular Graphs. In this talk, we will consider a closely related problem of finding the size and structure of the biggest induced forest in a graph. This question is inspired by a graph process known as `Bootstrap Percolation’. In this process, given a fixed threshold $r$ and a set of ‘initially infected’ vertices, the states of vertices (either healthy or infected) are updated indiscrete time steps: any healthy vertex with at least $r$ infected neighbours becomes itself infected. An initially infected set of vertices percolates if the infection spreads to the entire graph. One of the extremal questions related to this process is to find the smallest percolating set. In the special case of $d$-regular graphs, when $r=d$, a set percolates exactly when its complement is independent. Meanwhile, in the case $r=d−1$, a set percolates if its complement is a set of vertices that induce a forest. We will characterize maximum induced forests in the Kneser graph and some other families of Distance Regular Graphs.
Modern advances in technology have led to the generation of ever-increasing amounts of quantitative data from biological systems, such as gene-expression snapshots of developing cell populations in a tissue, or geometric data of residue positions within a protein. Experimental observations are often limited to be partial, so information about the underlying process or structure must instead be inferred from data. Through its connection to the Schrödinger problem of large deviations for stochastic processes, we find that entropic optimal transport arises as a natural tool for reconstructing unobserved cellular trajectories under precise assumptions. We develop both a theoretical and computational framework for inferring cellular dynamics based on optimal transport, and demonstrate its potential to extract the genetic logic underlying biological dynamics. In another vein, we also discuss the utility of generalized notions of optimal transport for matching and summarizing topological features in geometric structures such as biomolecules.
Joint work with (the groups of) Prof. Geoffrey Schiebinger, Prof. Lénaïc Chizat and Prof. Michael Stumpf.
We prove that for each finite index subgroup H of the mapping class group of a closed hyperbolic surface, and for each real number r larger than one, there does not exist a faithful C-r-action (in Hölder’s sense) of H on a circle. For this, we partially determine the critical regularity of faithful actions by right-angled Artin groups on a circle. (Joint with Thomas Koberda and Cristobal Rivas)
Complex irreducible characters of finite groups have two main invariants that are used to measure their rationality: their character fields and their Schur indices. We will discuss recent work aiming to explicitly compute these invariants for certain families of finite reductive groups, in particular the symplectic, special orthogonal, and unitary groups.
We discuss about uniqueness of ancient mean curvature flow whose rescaled flow stays in a one-side of a shrinker. The rescaled flow locally and smoothly converges to the shrinker as time goes back. Hence, the flow is the graph of a positive function defined on the shrinker. Therefor, a parabolic Liouville’s theorem for positive entire ancient solutions gives us the uniqueness.
A linear space is a point-line incidence structure such that each pair of points is incident to exactly one line. It is regular if all lines are incident to the same number of points. (This is equivalent to a balanced block design with $\lambda=1$.)
I will discuss linear spaces which admit a very special class of groups of automorphisms, called extremely primitive groups. Together with Melissa Lee, we have almost finished the classification of these linear spaces. In the process, we discovered some new interesting constructions which I will describe.
Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We constructed an algebraic virtual cycle. A key step is a localisation of Edidin-Graham’s square root Euler class for $SO(2n,\mathbb{C})$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We also develop a theory of complex Kuranishi structures on projective schemes which are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the theory to the moduli spaces to prove the two virtual cycles coincide in homology after inverting 2 in the coefficients. In particular, when Borisov-Joyce’s real virtual dimension is odd, their virtual cycle is torsion. This is a joint work with Richard Thomas.
We study the dimensions of the eigenspaces for the Atkin-Lehner involutions acting on spaces of modular forms $M_k(\Gamma_0(Np))$, with the additional constraint imposed by fixing the mod $p$ Galois representation attached to the eigenforms. For this purpose we establish isomorphisms up to semisimplification between certain Hecke modules in characteristic $p$, generalizing Serre’s work relating modular forms mod $p$ to quaternion algebras. These isomorphisms are in turn obtained via a delicate study of congruences in the trace formulas for Hecke and Atkin-Lehner operators.
Adversarial training is a framework widely used by practitioners to enforce robustness of machine learning models. During the training process, the learner is pitted against an adversary who has the power to alter the input data. As a result, the learner is forced to build a model that is robust to data perturbations. Despite the importance and relative conceptual simplicity of adversarial training, there are many aspects that are still not well-understood (e.g. regularization effects, geometric/analytic interpretations, tradeoff between accuracy and robustness, etc…), particularly in the case of multiclass classification.
In this talk, I will show that in the non-parametric setting, the adversarial training problem is equivalent to a generalized version of the Wasserstein barycenter problem. The connection between these problems allows us to completely characterize the optimal adversarial strategy and to bring in tools from optimal transport to analyze and compute optimal classifiers. This also has implications for the parametric setting, as the value of the generalized barycenter problem gives a universal upper bound on the robustness/accuracy tradeoff inherent to adversarial training.
Joint work with Nicolas Garcia Trillos and Jakwang Kim.
This is based on joint work with Masoud Kamgarpour and GyeongHyeon Nam. Character varieties are geometric objects associated to maps from a finitely generated group to a reductive group. Seminal work by Hausel and Rodriguez-Villegas studied topological properties of character varieties by counting points over finite fields. They considered twisted GL(n) character varieties of the fundamental group of certain surfaces. In this talk, we will discuss character varieties associated to an orientable surface with punctures, and a reductive group G with connected centre. We give a sufficient condition for their smoothness. Assuming regular semisimple, and regular unipotent monodromy, we give an explicit, polynomial point count. This allows us to determine the Euler characteristic and the number of connected components.
We discuss applications of (a part of) the Iwasawa main conjecture to the non-triviality of Kato’s Kolyvagin systems and the structure of Selmer groups of elliptic curves over the rationals without any rank restriction.
Givental and Lee introduced quantum K-theory, a K-theoretic generalization of Gromov–Witten theory. It studies holomorphic Euler characteristics of coherent sheaves on moduli spaces of stable maps to given target spaces. In this talk, I will introduce the quantum K-theory for orbifold target spaces which generalizes the work of Tonita-Tseng. In genus zero, I will define a quantum K-ring which specializes to the full orbifold K-ring introduced by Jarvis-Kaufmann-Kimura. As an application, I will give a detailed description of the quantum K-ring of weighted projective spaces, which generalizes a result by Goldin-Harada-Holm-Kimura. This talk is based on joint work with Yang Zhou.
A set of permutations $\mathcal{F} \subset Sym(V)$ is said to be intersecting if any two of its elements agree on some element of $V$. The intersection density, $\rho(G)$, of a finite transitive permutation group $G \leq Syn(V)$, is the maximum ratio $\frac{|\mathcal{F}| |V|}{|G|}$ where $\mathcal{F}$ runs through all intersecting sets of $G$. If the intersection density of a group is equal to 1, we say that a group has the Erdos-Ko-Rado property. The intersection density for many groups has been considered, mostly considering which groups have the Erdos-Ko-Rado property. In this talk I will consider the intersection density of a vertex-transitive graph which is defined to be the maximum value of $\rho(G)$ where is a transitive subgroup of the automorphism group of $X$. I will focus on the intersection density of the Kneser graph $K(n,3)$, for $n\geq 7$.
Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it has been an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce two topological spaces that are natural analogs of the Gromov boundary for a larger class of metric spaces. First we construct the sublinearly Morse boundaries and show that it is a QI-invariant topological space that can be associated to all finitely generated groups. Furthermore, for many groups, the sublinear boundary can be identified with the Poisson boundaries of the associated group, thus providing a QI-invariant model for Poisson boundaries. This result answers the open problems regarding QI-invariant models of CAT(0) groups and the mapping class group. Lastly, for a subset of the metric spaces we define a compactification of the sublinearly Morse boundary and show that in these cases they are naturally identified with the Bowditch boundary. This talk is based on a series of joint work with Kasra Rafi and Giulio Tiozzo.
The fundamental gap is the difference of the first two eigenvalues of the Laplacian, which is important both in mathematics and physics. We will review many recent fantastic results for convex domains in $\mathbb R^n, \mathbb S^n, \mathbb H^n$ with Dirichlet boundary conditions. Then we will present a very recent estimate for the convex domain in surfaces with positive curvature. The last result is joint with G. Khan, H. Nguyen, M. Tuerkoen.
In 1975, Szemeredi proved that for every real number $\delta > 0 $ and every positive integer $k$, there exists a positive integer $N$ such that every subset $A$ of the set ${1, 2, \cdots, N }$ with $|A| \geq \delta N$ contains an arithmetic progression of length $k$. There has been a plethora of research related to Szemeredi’s theorem in many areas of
mathematics. In 1990, Cameron and Erdos proposed a conjecture about counting the number of subsets of the set ${1,2, \dots, N}$ which do not contain an arithmetic progression of length $k$. In the talk, we study a natural higher dimensional version of this conjecture by counting
the number of subsets of the $k$-dimensional grid ${1,2, \dots, N}^k$ which do not contain
a $k$-dimensional corner that is a
set of points of the
form ${ a } \cup { a+ de_i : 1 \leq i \leq k }$ for some $a \in {1,2, \dots, N}^k$ and $d > 0$, where $e_1,e_2, \cdots, e_k$ is the standard basis of $\mathbb{R}^k$.
Our main tools for proof are the hypergraph container method and the supersaturation result
for $k$-dimensional corners in sets of size $\Theta(c_k(N))$, where $c_k(N)$ is the maximum size of a $k$-dimensional corner-free subset of ${1,2, \dots, N}^k$.
The question of whether stable minimal surfaces are holomorphic under suitable geometric conditions has been much studied, beginning with work of Lawson-Simons in complex projective space, and the proof of the Frankel conjecture by Siu-Yau. A theorem of Micallef shows that a stable minimal immersion of a complete oriented parabolic surface into Euclidean 4-space is holomorphic with respect to an orthogonal complex structure, and the same result in all dimensions if additionally the minimal surface has finite total curvature and genus zero. However, Arezzo, Micallef and Pirola gave a counterexample in general. It is therefore necessary to strengthen the stability condition in the general question. We will discuss some recent progress on this question. This is joint work with R. Schoen.
We would like to know for which function f(k) it is true that any oriented graph of minimum semidegree at least f(k) necessarily contains a given oriented path with k edges. For the directed path, f(k)=k/2 works, and perhaps this is true for other orientations as well. We show that this is approximately the case for large antidirected paths, and more generally, for large antidirected trees of bounded maximum degree. This is joint work with Camila Zárate.
I will report on the joint works with J. Brum, N. Matte-Bon and M. Triestino about R-focal actions of groups on the real line by homeomorphisms. I will describe and motivate this notion, provide some examples and give some applications to the problem of understanding the space of actions of a given (class of) group(s).
In this talk, I will explain quantum spectrum and asymptotic expansions in FJRW theory of invertible singularities of general type. Inspired by Galkin-Golyshev-Iritani’s Gamma conjectures for quantum cohomology of Fano manifolds, we propose Gamma conjectures for FJRW theory of general type. Here the Gamma structures are essential to understand the connection between algebraic structures of the singularities (such as Orlov’s semiorthogonal decompositions of matrix factorizations) and the analytic structures, such as asymptotic expansions in FJRW theory. The talk is based on the work joint with Ming Zhang.
The Rapoport-Zink space for $\mathrm{GSp}(4)$ is a local counterpart of the Siegel threefold. Its $\ell$-adic etale cohomology is equipped with an action of the product of three groups: $\mathrm{GSp}_4(\mathbb{Q}_p)$, its non-trivial inner form $J(\mathbb{Q}_p)$, and the Weil group of $\mathbb{Q}_p$. This action is expected to be strongly related with the local Langlands correspondence for $\mathrm{GSp}_4$ and $J$. In this talk, I will explain how the $\mathrm{GSp}_4(\mathbb{Q}_p)$-supercuspidal part of the cohomology is described by the local Langlands correspondence. If time permits, I will also give some observations on the $\mathrm{GSp}(6)$ case.
This talk will be about the interface of representation theory and machine learning. In machine learning, one sometimes wants to learn quantities which are invariant or equivariant with respect to a group. For example, the decision as to whether there is a tiger nearby should not depend on the precise position of your head and thus this decision should be rotation invariant. Another example: quantities that appear in the analysis of point clouds often do not depend on the labelling of the points, and are therefore invariant under a large symmetric group. I will explain how to build networks which are equivariant with respect to a group action. What ensues is a fascinating interplay between group theory, representation theory and deep learning. Examples based on translations or rotations recover familiar convolutional neural nets, however the theory gives a blueprint for learning in the presence of complicated symmetry. These architectures appear very useful to mathematicians, but I am not aware of any major applications in mathematics as yet. Most of this talk will be a review of ideas and techniques well-known in to the geometric deep learning community. New material is joint work with Joel Gibson (Sydney) and Sebastien Racaniere (DeepMind).
It is a classical problem to determine the minimum and maximum genus of a $2$-cell embedding of a graph $G$. However there will be many other embeddings with a genus between these two values: we will study the average genus across all of these embeddings.
We give an overview of recent work on the average genus for fixed graphs. We also discuss extensions of the problem to random graphs and random maps, and links with representation theory.
I will discuss the irrationality at a prime $p$ of the values of irreducible characters of a finite group. Among $p$’-degree characters, this irrationality behaves very nicely due to its conection with Navarro’s refinement of the celebrated McKay conjecture. I will present some recent work on the continuity of $p$-irrationality of $p$’-degree characters, bounding almost $p$-rational characters, and bounding odd-degree characters. Some results are joint with G. Malle and A. Maroti.
The Springer theory relates nilpotent orbits in the Lie algebra of a reductive algebraic group to irreducible representations of Coxeter groups. We discuss a Springer theory for graded Lie algebras and the character sheaves arising in this setting, concentrating on the construction of cuspidal character sheaves. Irreducible representations of Hecke algebras of complex reflection groups at roots of unity enter the description of character sheaves. We will explain the connection between our work and the recent work of Lusztig and Yun, where irreducible representations of trigonometric double affine Hecke algebras appear in the picture. This is based on joint work with Kari Vilonen and partly with Misha Grinberg.
The DT/PT correspondence is a formula which relates Donaldson-Thomas invariants counting closed subschemes in Calabi-Yau 3-folds, and Pandharipnade-Thomas invariants counting stable pairs on them. It gives an economical way of formulating GW/DT correspondence by Maulik-Nekrasov-Okounkov-Pandharipande. The DT/PT correspondence was proved by wall-crossing in the derived category, where on the wall we have a special type of Ext-quivers called DT/PT quivers. In this talk, I will give a categorical wall-crossing formula for categories of matrix factorizations associated with DT/PT quivers. The resulting formula is a semiorthogonal decomposition which involves quasi-BPS categories, and is a categorical analogue of numerical DT/PT correspondence. This is a joint work with Tudor Padurariu.
I will discuss geometric aspects of hyperbolic manifolds related to Dehn filling.
In this talk we establish two surprising types of Weyl’s laws for some compact $\mathrm{RCD}(K, N)$/Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for $\mathrm{RCD}(K, N)$ spaces. Our results depends crucially on analyzing and developing important properties of the examples constructed by Pan-Wei. Of independent interest, this also allows us to provide a counterexample to a conjecture by Cheeger-Colding. This is a joint work with Xianzhe Dai (University of California, Santa Barbara), Jiayin Pan (University of California, Santa Cruz) and Guofang Wei (University of California, Santa Barbara).
Interest from the machine learning community in optimal transport has surged over the past five years. A key reason for this is that the Wasserstein metric provides a unique way to measure the distance between data distributions—one that respects the geometry of the underlying space and behaves well even when distributions lack overlapping support.
In today’s talk, I will present two recent works that leverage the benefits of the Wasserstein metric in vastly different contexts. First, I will describe how the Wasserstein metric can be used to define a novel notion of archetypal analysis — in which one approximates a data distribution by a uniform probability measure on a convex polygon, so that the vertices provide exemplars of extreme points of the data. Next, I will discuss an application of optimal transport to collider physics, in which comparing collider events using the Wasserstein metric allowed us to achieve state of the art accuracy with vastly improved computational efficiency. In both cases, I will discuss both the theoretical benefits and the computational challenges of optimal transport in the machine learning context.
The symmetry of a discrete object (such as a graph, map or polytope, and even a Riemann surface or 3-manifold) can be measured by its automorphism group – the group of all structure-preserving bijections from the object to itself. (For example, an automorphism of a map takes vertices to vertices, edges to edges, faces to faces, and preserves incidence among these elements.) But also/conversely, objects with specified symmetry can often be constructed from groups.
In this talk I will give some examples showing how this is possible, and some methods which help, and describe some significant outcomes of this approach across a range of topics. Included will be some very recent developments, including a few unexpected discoveries.
Langlands’ reciprocity conjecture parameterizes smooth irreducible representations of a reductive group $G$ over a local field, or automorphic representations of a reductive group $G$ over a global field, in terms of the $L$-group of $G$, which is a split extension of the dual group of $G$ and the Galois group. Langlands’ functoriality conjecture predicts relationship between such representations given a homomorphism between the $L$-groups. In practice, one is often confronted with extensions of the dual group by the $L$-group that are not split. We will explore this phenomenon based on the following construction. To a connected reductive group $G$ over a local field $F$ we define a compact topological group $\tilde\pi_1(G)$ and an extension $G(F)_\infty$ of $G(F)$ by $\tilde\pi_1(G)$. From any character $x$ of $\tilde\pi_1(G)$ of order $n$ we obtain an $n$-fold cover $G(F)_x$ of the topological group $G(F)$. We also define an $L$-group for $G(F)_x$, which is a usually non-split extension of the Galois group by the dual group of $G$, and deduce (from the linear case) a refined local Langlands correspondence between genuine representations of $G(F)_x$ and $L$-parameters valued in this $L$-group. We will present one concrete application: a characterization of the local Langlands correspondence for semi-simple discrete $L$-parameters, that is uniform for all local fields.
We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property in terms of the Wasserstein distance, a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.
Among the groups of orientation-preserving homeomorphisms of the circle, the one which preserves a veering pair of laminations are 3-manifold groups. We explain motivation, previous results, and main ingredients of the proof. This is based on the joint work with KyeongRo Kim and Hongtaek Jung.
Tilting modules for a reductive algebraic group G in prime characteristic were formally introduced (in this setting) by Donkin in the early 1990’s. They have since come to play an increasingly central role in the study of G-modules, featuring prominently in modular character formulas and also shedding light on the “Humphreys-Verma Problem.”
In the early days, Donkin made a series of conjectures that predicted properties of the direct summands of tensor products involving the Steinberg module. Some of these conjectures remain open, while others have been resolved only in the last few years. In this talk I will discuss what we presently know and do not know, and why it is important. This is based on joint work with Chris Bendel, Dan Nakano, and Cornelius Pillen.
If $\Sigma_{i}$ is a stable submanifold of $M_{i}$, for $i=1,2$, then $\Sigma_{1}\times \Sigma_{2} $ is a stable submanifold in $M_{1}\times M_{2}$. Are all the stable submanifolds in $M_{1}\times M_{2}$ like that? We will show that is the case for specific dimensions and codimensions in the product of a complex or quaternionic projective space with any other Riemannian manifold. We will also talk about the behaviour of stable submanifolds under a complex structure of the product of two complex projective spaces. We will describe how our proofs were motivated by work that has be done by Simons, Lawson, Ohnita, Torralbo and Urbano. Part of this work is joint with Shuli Chen (Stanford).
The Ichino-Ikeda conjecture is an explicit relation between the central L-value and squares of a certain period of automorphic forms.
This conjecture has been established by Beuzart-Plessis, Yifeng Liu, Wei Zhang, Xinwen Zhu, Chaudouard and Zydor for unitary groups.
I will report on a joint work in progress with Michael Harris and Ming-Lun Hsieh on the construction of $p$-adic L-functions for $U(3)\times U(2)$ via the Ichino-Ikeda conjecture.
Caldararu-Costello-Tu defined Categorical Enumerative Invariants (CEI) as a set of invariants associated to a cyclic A-infinity category (with some extra conditions/data), that resemble the Gromov-Witten invariants in symplectic geometry. In this talk I will explain how one can define these invariants for Calabi-Yau A-infinity categories - a homotopy invariant version of cyclic - and then show the CEI are Morita invariant. This has applications to Mirror Symmetry and Algebraic Geometry.
The game of Cops and Robber is traditionally played on a finite graph. But one can define the game that is played on an arbitrary geodesic space (a compact, path-connected space endowed with intrinsic metric). It is shown that the game played on metric graphs is essentially the same as the discrete game played on abstract graphs and that for every compact geodesic surface there is an integer $c$ such that $c$ cops can win the game against one robber, and $c$ only depends on the genus $g$ of the surface. It is shown that $c=3$ for orientable surfaces of genus $0$ or $1$ and nonorientable surfaces of crosscap number $1$ or $2$ (with any number of boundary components) and that $c=O(g)$ and that $c=\Omega(\sqrt{g})$ when the genus $g$ is larger. The main motivation for discussing this game is to view the cop number (the minimum number of cops needed to catch the robber) as a new geometric invariant describing how complex is the geodesic space.
The collection of parking functions under a natural Sn-action (which has Catalan-many orbits) has been a central object in Algebraic Combinatorics since the work of Haiman more than 30 years ago. One of the lines of research spawned around it was towards defining and studying analogous objects for real and complex reflection groups W; the main candidates are known as the W-non-nesting and W-non-crossing parking functions.
The W-non-nesting parking functions are relatively well understood; they form the so called algebraic W-parking-space which has a concrete interpretation as a quotient ring (of the ambient polynomial algebra over a system of homogeneous parameters that carry the reflection representation of W). The W-non-crossing parking functions on the other hand have defied unified explanations while simultaneously proving themselves central in the study of Coxeter and Artin groups (their geometric group theory and combinatorics) and in the representation theory of Dynkin quivers. One of the main open problems in the field since the early 2000’s has been to give a type-independent proof of the W-isomorphism between the algebraic and the non-crossing W-parking spaces. In this talk, I will present the first such proof, solving the more general Fuss version of the problem, that proceeds by comparing a collection of recursions that are shown to be satisfied by both objects. This relies on a variety of recent techniques we introduced, in particular a parabolic decomposition of the algebraic parking space and its relation with the spectrum of Laplacian matrices for reflection arrangements.
For $r\geq 1$, a graph has $r$-friendship property if every pair of vertices has exactly $r$ common neighbours. The motivation for this definition is from the Friendship theorem, which is on the graphs with $1$-friendship property. The Friendship theorem, first proved by Erdős, Rényi, and Sós in 1966, states that if $G$ is a graph in which every pair of vertices has exactly one common neighbour, then $G$ has a universal vertex $v$ adjacent to all others, and the graph induced by $V(G)\setminus {v }$ is a matching. In this presentation, we study graphs with $r$-friendship property, where $r\geq 2$. We show all such graphs are strongly regular. Furthermore, we prove that for any $r\geq 2$, there are only finitely many graphs with $r$-friendship property. This is an ongoing joint work with Karen Gunderson.
In this talk, we will present our recent results on the local multiplicity formula of strongly tempered spherical subgroups. In particular, we formulate the epsilon dichotomy conjecture for those spherical subgroups and prove this conjecture for the tempered representations of endoscopy type. Globally, the corresponding automorphic periods are related to the central values of certain L-functions of symplectic type. The conjectural Ikeda–Ichino type formulas are given. This is a joint project with Chen Wan at Rutgers University, Newark.
We explain how to generalize a classical construction of Thurston for pseudo-Anosov maps to obtain loxodromic elements for the action of a big mapping class group on a ray graph. Our work is based on ideas of Hooper, Thurston and Veech. Joint work with Israel Morales.
$Q$-valued maps minimizing a suitably defined Dirichlet energy were introduce by Almgren in his proof of the optimal regularity of area minimizing currents in any dimension and codimension. In this talk I will discuss the extension of Almgren’s result to stationary $Q$-valued maps in dimension $2$.
This is joint work with Jonas Hirsch (Leipzig).
Graphs having three distinct eigenvalues are a fundamental object of study in spectral graph theory. Strongly regular graphs are the most well-studied examples. In 1995, at the 15th British Combinatorial Conference, Willem Haemers asked do there exist any connected graphs having three distinct eigenvalues apart from strongly regular graphs and complete bipartite graphs. Haemer’s question prompted responses from Muzychuk-Klin and Van Dam who found new families of nonregular graphs having three distinct eigenvalues.
Muzychuk and Klin initiated the study of a graph with three distinct eigenvalues via its Weisfeiler-Leman closure. They classified such graphs whose Weisfeiler-Leman closure has rank at most 7. In this talk, we extend this classification up to rank 9. Our results include a correction of the literature (where the rank 8 case was erroneously claimed to be impossible) and discussion of further study.
We define integer valued invariants of an orbifold Calabi-Yau threefold $X$ with transverse ADE orbifold points. These invariants contain equivalent information to the Gromov-Witten invariants of $X$ and are related by a Gopakumar-Vafa like formula which may be regarded as a universal multiple cover / degenerate contribution formula for orbifold Gromov-Witten invariants. We also give sheaf theoretic definitions of our invariants. As examples, we give formulas for our invariants in the case of a (local) orbifold K3 surface. These new formulas generalize the classical Yau-Zaslow and Katz-Klemm-Vafa formulas. This is joint work with S. Pietromonaco.
When considering proper colourings of oriented graphs, one possible approach is to insist that the colouring be consistent with the orientation: all edges between two colour classes should be oriented in the same way. The oriented chromatic number of an oriented graph is the least $k$ so that there is a homomorphism from the oriented graph to some tournament of order $k$. In this talk, I will present new work with Nir on the oriented chromatic number of random models of oriented graphs of bounded degree. Previous extremal results can be used to bound the oriented chromatic number of a random $d$-regular oriented graph between $\Omega(\sqrt{2}^d)$ and $O(d^2 2^d)$. Using colourings by doubly-regular tournaments, we improve the upper bound to $O(\sqrt{e}^d)$. I will present these results and discuss an optimization result for functions over doubly stochastic matrices that extends the optimization result of Achlioptas and Naor that was central to results on chromatic numbers of unoriented random graphs.
Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class in the sense of Behrend-Fantechi on the moduli stack of lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class. This can be taken as a generalization of the tautological invariants on the moduli space of stable curves to the moduli space of stable surfaces. The method presented can also be applied to the moduli space of stable map spaces from semi-log-canonical surfaces to projective varieties.
One of the central objects of interest in additive combinatorics is the sumset $A+B= {a+b:a \in A, b \in B}$ of two sets $A,B$ of integers. Our main theorem, which improves results of Green and Morris, and of Mazur, implies that the following holds for every fixed $\lambda > 2$ and every $k>(\log n)^4$: if $\omega$ goes to infinity as $n$ goes to infinity (arbitrarily slowly), then almost all sets $A \subseteq [n]$ with $|A| = k$ and $ |A + A| < \lambda k$ are contained in an arithmetic progression of length $\lambda k/2 + \omega$. This is joint work with Marcelo Campos, Mauricio Collares, Rob Morris and Victor Souza.
Hassett–Tschinkel and Benoist–Wittenberg recently introduced a new rationality obstruction that refines the classical the Clemens–Griffiths intermediate Jacobian obstruction to rationality, and exhibited its strength by showing that this new obstruction characterizes rationality for intersections of two quadrics. We show that this phenomenon does not extend to all geometrically rational threefolds. We construct examples of conic bundle threefolds over $\mathbb{P}^2$ that have no refined intermediate Jacobian obstruction to rationality, yet fail to be rational. This is joint work with S. Frei, L. Ji, S. Sankar, and I. Vogt.
In this talk, I will explain how to use mean curvature flow to obtain explicit lower bound for the density of topologically nontrivial minimal cones. This is joint with Jacob Bernstein.
I’ll discuss a proposed notion of geometric finiteness in the mapping class group, motivated by both the analogy with Kleinian groups and the geometry of extension groups, following Farb and Mosher’s work on convex cocompactness. This is joint work with Spencer Dowdall, Matthew Durham, and Alessandro Sisto.
The irreducible representations of the Symmetric group are a classical subject, seemingly well understood. Yet, the multiplicities in the irreducible decomposition of the tensor product of two irreducibles, the Kronecker coefficients, present an 80+ long standing open problem. The problem has seen its revival in the context of Geometric Complexity Theory for the separation of complexity classes, the algebraic analogues of P vs NP.
In this talk we will review some recent results on the asymptotics and complexity of Kronecker coefficients and the underlying characters of the Symmetric group. Based on a series of joint works with Christian Ikenmeyer and Igor Pak.
In classical homotopy theory, two spaces are considered homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory that recognizes the difference between the vertices and edges of a graph was invented. This theory is called A-homotopy theory in honor of Ron Aktin, who created the foundations of this theory in Q-analysis in the 1970s. A-homotopy theory allows us to compare graphs and to compute invariants for graphs. The intended use of these invariants is to find areas of low connectivity in a large network where information might be missing or where the network might be made more efficient.
In algebraic topology, each space $X$ has associated spaces $\widetilde{X}$ and continous maps $p: \widetilde{X} \to X$, and each pair together $(\widetilde{X}, p)$ is called a covering space of $X$. Under certain conditions, a space has a covering space with special properties, called a universal cover. Among other things, universal covers allow us to factor maps between topological spaces, which can be quite useful. In this talk, I will give a brief introduction to A-homotopy theory and discuss the universal covers I developed for graphs with no 3 or 4-cycles as well as the covering graphs obtained from quotienting these universal covers. I will end by mentioning some of the useful properties of these universal covers.
A Kronecker coefficient $g(\lambda, \mu, \nu)$ is a non-negative integer that depends on three partitions $\lambda$, $\mu$, $\nu$ of a natural number $n$. It is the multiplicity of an irreducible representation $V^\nu$ of the symmetric group of degree n in the tensor (or Kronecker) product $V^\lambda \otimes V^\mu$ of two other irreducible representations of the same group.
The study of ways of computing Kronecker coefficients is an important topic on algebraic combinatorics. Several tools have been used to try to understand them, notably from representation theory, symmetric functions theory and Borel-Weil theory. These numbers generalize the well-known Littlewood-Richardson coefficients but are still very far to be fully grasped.
It is known that each Kronecker coefficient can be described as an alternating sum of numbers of integer points in convex polytopes. In this talk we present a new family of polytopes that permit efficient computations on Kronecker coefficients associated to partitions with few parts and provides insight in the behavior of Murnaghan stability.
The dual Specht module of the symmetric group algebra over $\mathbb{Q}$ has two distinguished bases, namely the standard basis and Young’s seminormal basis. We study how the Young’s seminormal basis vectors are expressed in terms of the standard basis, as well as the denominators of the coefficients in these expressions. We obtain closed formula for some Young’s seminormal basis vectors, as well as partial results for the denominators in general.
This is a joint work with Ming Fang (Chinese Academy of Sciences) and Kay Jin Lim (Nanyang Technological University).
The PRIMA Early Career Researcher’s Showcase is a short session for early career participants (undergraduate, graduate and postdoctoral scholars) to highlight their areas of research.
We find new polynomial upper bounds for the size of nodal sets of eigenfunctions when the Riemannian manifold has a Gevrey or quasianalytic regularity.
Among infinitely many factorizations A=VV* of a psd matrix A, we seek the factor V that has the smallest (1,2) norm. In this talk we review the origin of this problem as well as existing results regarding the optimal value. We discuss also the conjecture that the squared (1,2) norm of V is equivalent to the (1,1) norm of A.
Harder-Narasimhan (HN) theory gives a structure theorem for principal G bundles on a smooth projective curve. A bundle is either semistable, or it admits a canonical filtration whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary algebraic stacks, I will discuss work in progress with Andres Fernandez Herrero to apply this general machinery to the stack of “gauged” maps from a curve C to a G-scheme X, where G is a reductive group and X is projective over an affine scheme. Our main application is to use HN theory for gauged maps to compute generating functions for K-theoretic enumerative invariants known as gauged Gromov-Witten invariants. This problem is interesting more broadly because it can be formulated as an example of an infinite dimensional analog of the usual set up of geometric invariant theory, which has applications to other moduli problems.
Traditional Indigenous marriage rules have been studied extensively since the mid-1800s. Despite this, they have historically been cast aside as having very little utility. Here, I will walk through some of the interesting mathematics of the Gamilaraay system and show that, instead, they are in fact a very clever construction. Indeed, the Gamilaraay system dynamically trades off kin avoidance – to minimise incidence of recessive diseases – against pairwise cooperation, as understood through Hamilton’s rule.
Let $d\geq 2$ and let $H$ denote the absolute multiplicative Weil height on $\bar{\mathbb{Q}}$. Let $f(z)\in \bar{\mathbb{Q}}[z]$ of degree $d$ and let $a\in\bar{\mathbb{Q}}$, the multiplicative and logarithmic canonical heights of $a$ with respect to $f$ are defined as
$$ \hat{H}_f (a) =\lim H(f^n(a))^{1/d^n} \quad \text{and}\quad \hat{h}_f(a)=\log \hat{H}_f(a). $$
Let $n$ be a positive integer. For $1\leq i\leq n$, let $f_i\in\bar{\mathbb{Q}}[z]$ of degree $d$ and let $a_i\in\bar{\mathbb{Q}}$. In this talk, we provide a complete characterization of when the $\hat{H}_{f_i}(a_i)$’s are multiplicatively dependent modulo constant meaning there exist integers $m_1,\ldots,m_n$ not all of which are $0$ and $a\in \bar{\mathbb{Q}}$ such that:
$$ \hat{H}_{f_1}(a_1)^{m_1} \cdots \hat{H}_{f_n}(a_n)^{m_n}=a. $$
As an immediate consequence, we characterize all the pairs $(f,a)$ such that $\hat{H}_f(a)$ is an algebraic number and proves the existence of $(f,a)$ such that $\hat{h}_f(a)$ is an irrational number. The proof uses the Medvedev-Scanlon classification of preperiodic subvarieties under the dynamics of a split polynomial map and the construction of a certain auxiliary polynomial. This is joint work with Jason Bell.
Coxeter groups are groups generated by involutions $s_i$, with the relations of form $(s_is_j)^m=$ id. For each Coxeter group, we will be discussing a particular system of “voracious” paths between any two vertices of the Cayley graph. This system turns out to have the “fellow traveller” property and is generated by a finite state automaton. This is joint work with Damian Osajda.
The Allen-Cahn equation is a semi-linear elliptic equation arising in the van der Waals-Cahn-Hilliard theory of phase transitions. Earlier fundamental work by De Giorgi, Modica, Sternberg etc. revealed intriguing relationship between the Allen-Cahn equation and the theory minimal surfaces. Based on the deep regularity theory by Hutchinson, Tonegawa and Wickramasekera, Guaraco recently introduced a new PDE approach to the existence of minimal surfaces via the Allen-Cahn equation and there have been substantial progress along this direction in the past few years. In this talk, we will consider the Allen-Cahn equation on bounded domains and describe some geometric and analytic aspects of the boundary behaviour of the limit-interfaces. This work is substantially supported by research grants from Hong Kong Research Grants Council and National Science Foundation China.
Hurwitz numbers count covers of Riemann surfaces with given ramification data. The case of covers of $\mathbb{P}^1$ with one variable ramifications and a uniform ramification elsewhere is particularly interesting, as these numbers are related to the KP integrable hierarchy. A large family of such Hurwitz problems can be computed using topological recursion: a universal procedure, recursive on the Euler characteristic of the surfaces involved, which requires a spectral curve as input. The case of Atlantes Hurwitz numbers evaded this approach up to now, as it seemed to have the same spectral curve as completed cycles Hurwitz numbers. I will introduce all these notions, and explain how Atlantes Hurwitz numbers do satisfy a variant of topological recursion, with a transalgebraic spectral curve, and this distinguishes it from the completed cycles case. This is joint work with Vincent Bouchard and Quinten Weller.
I’ve been introduced to the concepts of ethnomathematics, decolonization and Indigenization throughout my career as a mathematics educator and researcher. I will share the ways in which these concepts shape research and teaching conversations that I now have.
Over the last few decades, the (unconstrained) LASSO $$ \min_{x\in \mathbb{R}^n} \frac12 \lVert A x-b\rVert_2^2 + \lambda \lVert x\rVert_1, $$ has become an indispensable tool in statistical learning, data science, and signal processing, thanks to its ability to efficiently recover sparse approximate solutions to (underdetermined) linear systems.
In this talk, we will present a novel variational analysis of this popular optimization program. First, we will establish smoothness results as well as Lipschitz properties for the optimal value and optimal solution maps of the LASSO as functions of the measurement vector $b$, the sampling matrix $A$, and, most notably, the tuning parameter $\lambda$. Then, we will illustrate how to apply the proposed variational analysis in the context of compressed sensing, validating our theoretical findings with numerical experiments. Finally, we will discuss extensions to related optimization problems such as the the square-root LASSO.
In 1900, Hilbert posed the following problem: “Given a Diophantine equation with integer coefficients: to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in (rational) integers.”
Building on the work of several mathematicians, in 1970, Matiyasevich proved that this problem has a negative answer, i.e., such a general `process’ (algorithm) does not exist.
In the late 1970’s, Denef–Lipshitz formulated an analogue of Hilbert’s 10th problem for rings of integers of number fields.
In recent years, techniques from arithmetic geometry have been used extensively to attack this problem. One such instance is the work of García-Fritz and Pasten (from 2019) which showed that the analogue of Hilbert’s 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. In joint work with Lei and Sprung, we improve their proportions and extend their results in several directions.
I this talk, I will give semiorthogonal decompositions of derived categories of several classical moduli spaces, e.g. symmetric products of curves, Brill-Noether loci, (relative) Quot schemes, Hilbert schemes of points. In particular, they contain a proof of Jiang’s conjecture for semiorthogonal decompositions of Quot schemes of locally free quotients. They are by-products of my research on categorifications of wall-crossing in Donaldson-Thomas theory, and the proofs involve techniques of derived algebraic geometry, categorical Hall algebras, matrix factorizations and Koszul duality.
The study of eigenfunctions of Laplacians lies at the interface of several areas of mathematics, including analysis, geometry, mathematical physics and number theory. These special functions arise in physics and in partial differential equations as modes of periodic vibration of drums and membranes.
A fundamental question surrounding Laplace-Beltrami eigenfunctions targets their concentration phenomena, via high-energy asymptotics or high-frequency behaviour. One popular approach to this question involves studying the growth of the $L^p$ norms of these eigenfunctions on the ambient manifold, or on submanifolds thereof. What can one say about the behaviour of eigenfunctions on rougher sets? We will discuss answers to this question, based on joint work with Suresh Eswarathasan.
Previous work has shown that a neural network with the rectified linear unit (ReLU) activation function leads to a convex polyhedral decomposition of the input space. In this talk, we will see how one can utilize this structure to detect and analyze adversarial attacks in the context of digital images.
When an image passes through a network containing ReLU nodes, the firing or non-firing at a node can be encoded as a bit ($1$ for ReLU activation, $0$ for ReLU non-activation). The sequence of all bit activations identifies the image with a bit vector, which identifies it with a polyhedron in the decomposition. We identify ReLU bits that are discriminators between non-adversarial and adversarial images and examine how well collections of these discriminators can ensemble vote to build an adversarial image detector and also present further applications of this induced geometry.
We show that for every non-elementary hyperbolic group, an associated topological flow space admits a coding based on a transitive subshift of finite type. Applications include regularity results for Manhattan curves, the uniqueness of measures of maximal Hausdorff dimension with potentials, and the real analyticity of intersection numbers for families of dominated (Anosov) representation, thus providing direct proof of a result established by Bridgeman et al. in 2015. Joint work with Steve Cantrell (Chicago).
A pangram is an expression which contains all letters of the alphabet in the expression’s language. An autogram is an expression which describes itself (correctly) by (for example) describing how many copies of each letter appear in the expression. The first pangrammatic autogram found in English was the result of an 18-month long project in the 1980’s resulting is the sentence, “Only the fool would take trouble to verify that his sentence was composed of ten a’s, three b’s, four c’s, four d’s, forty-six e’s, sixteen f’s, four g’s, thirteen h’s, fifteen i’s, two k’s, nine l’s, four m’s, twenty-five n’s, twenty-four o’s, five p’s, sixteen r’s, forty-one s’s, thirty-seven t’s, ten u’s, eight v’s, eight w’s, four x’s, eleven y’s, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least, a single !” In this presentation, Edward Doolittle, Associate Professor of Mathematics at First Nations University of Canada, will describe his effort to “translate” such sentences into the Plains Cree language. Joint work with Arok Wolvengrey, Professor of Linguistics at First Nations University.
BPS structures were introduced by Bridgeland to describe certain outputs of both Donaldson-Thomas theory and four-dimensional N=2 supersymmetric quantum field theory. To solve a totally different problem, the so-called loop equations in the theory of matrix models, Chekhov and Eynard-Orantin introduced the topological recursion, which takes initial data called a spectral curve and recursively produces an infinite tower of geometric invariants, often with an enumerative interpretation. $\newline$
We will describe recent work and ambitions connecting these two different theories. In the simplest cases, we describe an explicit formula for the “free energies” of the topological recursion in terms of a corresponding BPS structure constructed from the same initial data. We will sketch the relation to the WKB analysis of quantum curves, and gesture towards a wide range of generalizations.
The remarkable successes of neural networks in a huge variety of inverse problems have fueled their adoption in disciplines ranging from medical imaging to seismic analysis over the past decade. However, the high dimensionality of such inverse problems has simultaneously left current theory, which predicts that networks should scale exponentially in the dimension of the problem, unable to explain why the seemingly small networks used in these settings work as well as they do in practice. To reduce this gap between theory and practice, we provide a general method for bounding the complexity required for a neural network to approximate a Lipschitz function on a high-dimensional set with a low-complexity structure. The approach is based on the fact that many sets of interest in high dimensions have low-distortion linear embeddings into lower dimensional spaces. We can exploit this fact to show that the size of a neural network needed to approximate a Lipschitz function on a low-complexity set in a high dimensional space grows exponentially with the dimension of its low-distortion embedding, not the dimension of the space it lies in.
The use of infinitesimal methods (nonstandard analysis) in calculus can simplify computations, including the determination of convergence or divergence of a series. The level comparison test for series with nonnegative terms is an example. Featuring a computation that is similar in difficulty to the test for divergence, this test hinges on whether the reciprocal of the “omegath” term of the series lies in the “convergence zone” or in the “divergence zone.” In this talk the test is described, justified, and demonstrated. (The level comparison test is introduced in the textbook Calculus Set Free: Infinitesimals to the Rescue, Oxford University Press, 2022.)
The goal of Optimal Recovery is to uncover procedures that are worst-case optimal for the task of learning / recovering unknown functions from a priori information (the model) and a posteriori information (the observations). Optimal recovery procedures can sometimes be chosen as linear maps, at least when the observations are accurate. When they are corrupted, however, the problem becomes more complicated, especially if the focus is put on the practical construction of the (near) optimal procedure. This talk showcases positive results in three situations: deterministic observation errors bounded in $\ell_2$; deterministic observation errors bounded in $\ell_1$; and stochastic log-concave observation errors.
A recent trend in geometric group theory is to understand the large scale geometry of a group with respect to a collection of subgroups. The objects of study are pairs consisting of a finitely generated group and a finite collection of subgroups, and there is a notion of quasi-isometry of pairs inducing an equivalence. This relation captures classical phenomena in the study of quasi-isometric rigidity and brings up natural questions. The talk will introduce some of these topics and discuss recent results obtained in joint work with Sam Hughes and Luis Sánchez Saldaña.
One guiding principle is the idea that Laplacian eigenfunctions corresponding to larger frequencies should oscillate more. The same principle should then also be true, in some form, for linear combinations of high-frequency eigenfunctions (in one dimension, this is Sturm-Liouville theory). Recent progress on this question is based on the notion of optimal transport and a very simple idea which we formalize: if it’s easy to buy milk, then there are must be many supermarkets (and, conversely, if there are only few supermarkets at least some people have to travel a large distance to buy milk). This turns into a geometric inequality that is interesting in its own right.
In this talk we give survey what is currently known for Chen’s flow, and discuss some very recent results. Chen’s flow is the biharmonic heat flow for immersions, where the velocity is given by the rough Laplacian of the mean curvature vector. This operator is known as Chen’s biharmonic operator and the solutions to the elliptic problem are called biharmonic submanifolds. The flow itself is very similar to the mean curvature flow (this is essentially the content of Chen’s conjecture), however proving this requires quite different strategies compared to the mean curvature flow. We focus on results available in low dimensions – curves, surfaces, and 4-manifolds. We provide characterisations of finite-time singularities and global analysis. The case of curves is particularly challenging. Here we identify a new shrinker (the Lemniscate of Bernoulli) and use some new observations to push through the analysis. Some numerics is also presented. The work reported on in the talk is in collaboration with Yann Bernard, Matthew Cooper, Philip Schrader and Glen Wheeler.
We study non-commutative projective varieties in the sense of Artin-Zhang, which are given by non-commutative homogeneous coordinate rings, which are finite over their centre. We construct moduli spaces of stable modules for these, and construct a symmetric obstruction theory in the CY3-case. This gives deformation invariants of Donaldson-Thomas type. The simplest example is the Fermat quintic in quantum projective space, where the coordinates commute up to carefully chosen 5th roots of unity. We explore the moduli theory of finite length modules, which mixes features of the Hilbert scheme of commutative 3-folds, and the representation theory of quivers with potential. This is joint work with Yu-Hsiang Liu, with contributions by Atsushi Kanazawa.
Tribal colleges and universities have and continue to seek out connections between the local heritage and culture and the mainstream education content. In math, calls for culture to be more integrated into the classroom have been met with epistemological challenges as well as a dearth of math and local culture resources. This research project addresses both of these challenges. This presentation will specifically share on the collaborative development, evaluation, and confirmation of an epistemological framework for curriculum development at Sitting Bull College. Following an Indigenous Research Methodology, a group of tribal college math instructors, Lakota language immersion teachers, and fluent elders gathered to discuss D/Lakota math connections. Specific examples from these recorded discussions will be shared. The framework and the results demonstrate that math fluency and Dakota/Lakota language fluency can grow together.
We study Campana’s orbifold conjecture for finite ramified covers of $\mathbb P^2$ with three components admitting sufficiently large multiplicities. We also prove a truncated second main theorem of level one for analytic maps into $\mathbb P^2$ intersecting the coordinate lines in sufficiently high multiplicities. In particular, the exceptional set for the later result can be described explicitly.
This is joint work with Ji Guo.
This is the first of two talks about a “negative” spin analogue of the Witten $r$-spin theory. In the first talk, we will present the construction and properties of a cohomological field theory (without a flat unit) that parallels the famous Witten r-spin class for negative spin. The class for $r=2$ was constructed and studied by Norbury in 2017. By studying certain deformations of this class, we use Teleman’s reconstruction theorem to prove relations in the tautological ring, and in the special case of $r=2$ they reduce to relations involving only kappa classes which were recently conjectured by Norbury–Kazarian.
This is based on the following joint work with Elba Garcia-Failde (who will give the second talk) and Alessandro Giacchetto: https://arxiv.org/pdf/2205.15621.pdf
When the data is large, or comes in a streaming way, randomized iterative methods provide an efficient way to solve a variety of problems, including solving linear systems, finding least square solutions, optimizing with gradient descent and Newton methods, solving feasibility problems and beyond. However, if the data is corrupted with arbitrarily large sparse errors, one can expect the iterates are diverted far from the solution with each corrupted equation they encounter. I am going to talk about our recently proposed robust versions of Randomized Kaczmarz and Stochastic Gradient Descent methods for solving linear systems that avoid harmful corrupted equations by exploring the space as they proceed with the iterates. We show that the convergence rates in expectation are comparable with the convergence of the vanilla methods on uncorrupted systems. I will also touch on how to use the information obtained on the exploration phase efficiently, and what structural characteristics of the data matrix are crucial for such methods. Based on the joint work with Deanna Needell, Jamie Haddock, Will Swartworth, Ben Jarman, and Lu Cheng.
In this talk I will introduce you to the wonderful world of 2-dimensional topology! We will mainly focus on how we can use combinatorial constructions of curves (1-dimensional objects) on surfaces (2-dimensional objects) to answer questions about the group of symmetries of a surface, which we call the mapping class group. I intend for this talk to be accessible to folks with a wide variety of backgrounds including undergraduate and graduate students.
The Steklov eigenvalue problem has been one of the central topics in spectral geometry over the last decade. In particular, a lot of research has been focused on the asymptotic distribution of Steklov eigenvalues. In this talk, we investigate asymptotics for Steklov eigenvalues on surfaces with a boundary that is only smooth to finite order. In particular, we obtain remainder estimates in Weyl’s law with a rate of decay depending on the order of smoothness.
Quantization is one of the compression techniques to reduce computation cost, memory, and power consumption of deep neural networks (DNNs). In this talk, we will focus on a post-training quantization algorithm, GPFQ, that is based on a deterministic greedy path-following mechanism, and its stochastic variant SGPFQ. In both cases, we rigorously analyze the associated error bounds for quantization and show that for quantizing a single-layer network, the relative square error essentially decays linearly in the number of weights – i.e., level of over-parametrization. To empirically evaluate the method, we quantize several common DNN architectures with few bits per weight, and test them on ImageNet, showing only minor loss of accuracy compared to unquantized models.
I will report on joint work with Daniel Bragg. There are many ways to express the universality properties of moduli spaces. For example, Vakil established years ago that any singularity type defined over the integers appears in natural moduli spaces, proving what he called “Murphy’s Law”. We have discovered another, similar, phenomenon: many natural moduli spaces contain all finite gerbes. In particular, any finite gerbe over any field appears as the residual gerbe of some point of the stack of curves.
An important heuristic principle in the study of eigenfunctions of the Laplace-Beltrami operator is that their properties should resemble those of polynomials. In this light, I will discuss oscillations and zeros for linear combinations of Laplace eigenfunctions on Riemannian manifolds. In particular, I will prove that zeros become dense in the manifold if not too many eigenfunctions are summed. Time permitting, I will mention related open questions on eigenfunctions sums.
In this talk, I will explain how the Batyrev-Manin conjecture on rational points can be generalized to Deligne-Mumford stacks by using twisted sectors. In the original conjecture, the so-called a- and b-invariants are determined by positions of the ample line bundle in question and the canonical divisor in the Néron-Severi space relative to the pseudo-effective cone. In generalization to stacks, we introduce orbifold versions of these algebro-geometric notions. Once we define them suitably, the generalized conjecture is formulated more or less in the same way as the original conjecture was. The Malle conjecture on Galois extensions of a number field is then regarded as a special case of it. This is a joint work with Ratko Darda.
For a balanced bistable reaction-diffusion equation, an axisymmetric traveling front has been well studied. We prove that an axially asymmetric traveling front with any positive speed does exist in a balanced bistable reaction-diffusion equation. Our method is as follows. We use a pyramidal traveling front for an imbalanced reaction-diffusion equation whose cross section has a major axis and a minor axis. Preserving the major axis and the minor axis to be given constants and taking the balanced limit, we obtain an axially asymmetric traveling front in a balanced bistable reaction-diffusion equation. This traveling front is monotone decreasing with respect to the traveling axis, and its cross section is a compact set with a major axis and a minor axis.
This is the second talk of the series started by Nitin Chidambaram on a “negative” spin analogue of the Witten r-spin theory. In this second talk, we will exploit the relation between cohomological field theories and topological recursion to prove W-algebra constraints satisfied by the descendant potential of the class (constructed in Nitin’s talk). Furthermore, we conjecture that this descendant potential is the r-BGW tau function of the r-KdV hierarchy, and prove it for r=2 (confirming a conjecture of Norbury) and r=3.
This is based on the following joint work with Nitin Chidambaram (who gave the first talk) and Alessandro Giacchetto: https://arxiv.org/pdf/2205.15621.pdf
Given a graph, we define a new stability condition for the algebraic and tropical moduli spaces of rational curves. Using the theory of geometric tropicalization, we characterize when the tropical compactification of the compact moduli space agrees with the theory of geometric tropicalization. In this talk, I will give a gentle introduction to the moduli spaces of graphically stable curves and their tropical counterparts, and briefly describe that such a tropicalization occurs only when the graph is complete multipartite.
In this talk, I will discuss joint work with M. Engelstein, L. Li, and S. Mayboroda, where we introduce the notion of Dahlberg-Kenig-Pipher operators in the context of domains $\Omega$ with low dimensional boundaries. We show that when the boundary of the domain is uniformly rectifiable with small constant, then elliptic measure $\omega$ associated to this domain is an $A_\infty(d\sigma)$ weight with small constant. As consequence, we show that for $C^1$ domains, $\log (d\omega/d\sigma) \in \mathrm{VMO}$. One of the main difficulties in this context is the lack of outer graphical approximations to $\Omega$, since the exterior of $\Omega$ can be empty.
Shannon’s coding theorem establishes conditions under which a given stochastic source can be encoded by a given stochastic channel with zero probability of error. One can also consider a deterministic channel, given by a function from one space of sequences to another, and ask which sources can be transmitted by that channel not just with zero error probability but in fact injectively. Working with an existing formalization of this idea in symbolic dynamics, I will present a characterization of the subshifts that can be transmitted injectively by a given sliding block code on a mixing shift of finite type (i.e. the space of sequences realizable by an ergodic Markov chain). The result generalizes a classical embedding theorem of Krieger and answers a question posed to me by Tom Meyerovitch.
The moduli stack of complexes of vector bundles over a Gorenstein Calabi-Yau curve, considered up to chain isomorphisms, admits a 0-shifted Poisson structure in the sense of Calaque-Pantev-Toen-Vaquie-Vezzosi. We will give several classical examples of Poisson varieties appearing in representation theory and integrable system, that are naturally Poisson substacks of the above mentioned stack. Using derive algebraic geometry, we are able to prove some new results on these Poisson varieties. This is a joint work with Alexander Polishchuk.
Given a bounded domain $\Omega \subset \mathbb R^n$, one says that the $L^p$-regularity problem is solvable for the Laplace equation in $\Omega$ if, given any continuous function $f$ defined in $\partial \Omega$ and the harmonic extension $u$ of $f$ to $\Omega$, the non-tangential maximal function of the gradient of $u$ can be controlled in $L^p$ norm by the tangential derivative of $f$ in $\partial\Omega$. In my talk I will review a joint result with Mourgoglou where we proved the $L^p$-regularity for more general domains, and a more recent result (with the additional collaboration of Poggi) where we extend this result to elliptic PDE’s in divergence form.
The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem. In this talk, I will review some results and techniques related to this problem. I will specifically concentrate on the case of Hilbert schemes of points on K3 surfaces and on generic abelian varieties of any dimension. This is joint work in progress with C. Li, E. Macrì and P. Stellari.
Sparse polynomial methods have, over the last decade, become widely used tools for approximating smooth, high-dimensional functions from limited data. Notable applications include parametric modelling and computational uncertainty quantification, discovering governing equations of dynamical systems, and high-dimensional PDEs. A standard sparse polynomial approximation scheme involves solving a convex optimization problem involving a $\ell^1$- or weighted $\ell^1$-norm. This is typically done using first-order methods. However, such methods generally converge very slowly, making them undesirable for the many applications that require high accuracy. Motivated by this problem, in this talk I will describe a general scheme for accelerating first-order optimization algorithms. This scheme applies to any convex optimization problem possessing a certain $\textit{approximate sharpness}$ condition and essentially any first-order optimization algorithm. For a wide range of problems, i.e., not just $\ell^1$-minimization-type problems, this scheme achieves either optimal rates of convergence or improves on previously established rates. To emphasize its usefulness and generality, I will showcase applications in compressive imaging and machine learning.
In 1987, physicists Bak, Tang, and Wiesenfeld created an idealized version of a sandpile in which grains of sand are piled on the vertices of a (combinatorial) graph and are subjected to certain avalanching rules. In the last three decades, there has been a broad effort in the statistical physics community to understand the dynamics of this model, which has a natural underlying algebraic structure. The sandpile monoid and the sandpile group encode the short and long-term dynamics of the system. Disguised under different names, these algebraic structures have been widely studied in diverse contexts including algebraic combinatorics, arithmetic geometry, and algebraic geometry.
In this talk we give an introduction to the sandpile model and the algebraic structures attached to it. We provide a broad overview of the theory and discuss some of the more celebrated results. We end the talk with a discussion about an ongoing project that showcases some of the unresolved fundamental questions accessible to undergraduates.
Recent work has been done to relate ideas between topological recursion and Higgs bundles, such as studying the relationship between topological recursion and the geometry of the moduli space of Higgs bundles, and the relationship between quantum curves and quantization. In this talk I will discuss my current work, exploring these ideas in in a more general, twisted setting where the coefficient line bundle of the Higgs field is arbitrary.
A common approach for compressing large-scale data is through matrix sketching. In this talk, we consider the problem of recovering low-rank matrices or tensors from noisy sketches. We provide theoretical guarantees characterizing the error between the output of the sketching algorithm and the ground truth low-rank matrix or tensor. Applications of this approach to synthetic data and medical imaging data will be presented.
In this joint talk, we’ll discuss some pedagogical methods we have developed and used in Applied Precalculus, a course for pre-Business majors at TCU to prepare them for Applied Calculus. The population of students who take this course frequently provides certain challenges to the instructor, so we implemented a new course structure in several sections of the course a few years ago, and it has been evolving ever since. In this new course structure, the course is flipped, with exposure to material happening outside of class, and the course is also somewhat individually-paced, incorporating mastery-based grading in a series of module quizzes. We will discuss our motivation for this course structure, details on its implementation, how it has evolved over time, and what challenges still remain.
This presentation will share our work, through the Aboriginal and Torres Strait Islander Mathematics Alliance (ATSIMA), in teaching mathematics in connection to the culture of our Communities. I will give an overview of my personal journey in mathematics and how this led to the development of ATSIMA. I will introduce the Goompi Model, which is built on the premises that mathematics is an intrinsic part of culture and can guide the development of connected, culturally responsive pedagogies in mathematics. I will then share some stories from the classroom to show what is possible from this approach and have an open dialogue across the gathering to share ideas.
I will discuss local-global principles for two notions of semi-integral points, termed Campana points and Darmon points. In particular, I will introduce a semi-integral version of the Brauer-Manin obstruction interpolating between Manin’s classical version for rational points and the integral version developed by Colliot-Thélène and Xu. Lastly, we will apply these tools to study semi-integral points on quadric orbifolds.
We consider tautological bundles and their exterior and symmetric powers over the Quot scheme of zero dimensional quotients over the projective line. We prove several results regarding the vanishing of their higher cohomology, and we describe the spaces of global sections via tautological constructions. This is based on joint work with Alina Marian, Shubham Sinha and Steven Sam.
Recently several streaming algorithms for computing low-rank Tucker approximations of large tensors have been introduced by several researchers including Tropp, Udell, et al., De Lathauwer et al., and others. In this talk we will review the basic approach used by these works highlighting their similarities, and then present generalized theoretical guarantees proving their accuracy on full-rank and noisy data with high probability from a wide range of measurements. Some of our newly proposed structured measurements have the advantage of reducing the overall memory needed to sketch and recover large, potentially streamed, tensors and can also be chosen to economize sketching and recovery time. In particular, our recovery algorithms (after measurements have been collected) are sublinear-time in the overall tensor size. In addition, we improve upon prior works by providing stronger recovery error bounds which apply to a much more general class of measurements than previously analyzed. Our numerical experiments further highlight the value of these new measurement choices in various problem scenarios.
In this talk I will present spectral features of the Dirac operator with infi nite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. A non-linear variational formulation to characterize its principal eigenvalue will be presented. This characterization allows for a simple proof of a Szegö type inequality as well as a new formulation of a Faber-Krahn type inequality for this operator. Moreover, strong numerical evidence supporting the existence of a Faber-Krahn type inequality, will be given.
This talk is based on joint work with Pedro Antunes, Vladimir Lotoreichik, and Thomas Ourmieres-Bonafos.
Exact WKB analysis consists in the Borel-resummation of the divergent series appearing in the context of singular perturbation theory of differential equations. However, beyond the obvious perks for those interested in genuine solutions, e.g. of some equivariant differential systems, it also provides a way to study asymptotic properties of certain enumerations through the generating function method. Applications of these procedures include for instance constructing physical solutions to Schrödinger equations in finite-dimensional Hilbert spaces, as well as obtaining large-genus asymptotics of generalised Catalan numbers.
We will discuss variants of matrix and tensor CUR decompositions and algorithms for Robust PCA and matrix completion that allow one to observe only submatrices or subtensors of the data. By subsampling modes, we can obtain algorithms with state-of-the-art runtime for these tasks. Sample applications to image and video processing will be discussed.
Indigenous cultures have been using mathematics in their lives for generations. In this talk, we present methods of using place-based examples in a trigonometry classroom using Native Hawaiian culture. Through these examples, we hope to increase student participation in the classroom from both Indigenous and non-Indigenous students as well as spread awareness of various cultures.
The characteristic cycle of a constructible sheaf on a projective smooth algebraic variety is an algebraic cycle on the cotangent bundle that computes the Euler characteristic of the sheaf. In this talk, we consider a rank 1 sheaf on the variety. For a computation of the characteristic cycle of a rank 1 sheaf, we introduce a general theory called partially logarithmic ramification theory, and construct an algebraic cycle using several invariants measuring the ramification of the sheaf, which is compared with the characteristic cycle.
I will discuss joint work with Kenny Ascher, Dori Bejleri, Harold Blum, Giovanni Inchiostro, Yuchen Liu, and Xiaowei Wang on construction of moduli stacks and moduli spaces of log Calabi Yau pairs that can be realized as slc log Fano pairs with complements. Unlike moduli of canonically polarized varieties (respectively, Fano varieties) in which the moduli stack of KSB stable (respectively, K semistable) objects is bounded for fixed volume, dimension, the objects here form unbounded families. Despite this unbounded behavior, in the case of plane curve pairs (P2, C), we construct a projective good moduli space parameterizing S-equivalence classes of these slc Fanos with complements.
In this talk, we will explore the celebrated Almgren’s monotonicity formula. This beautiful result with far-reaching consequences states that if u is harmonic in the unit ball, then a certain frequency function $N(r)$ is non-decreasing. Moreover, $N(r)=k$ for all $r<1$ if, and only if, $u$ is homogeneous of degree $k$. We will then discuss some of the many applications of this formula, and recent developments connected to it.
We study compressed sensing when the signal structure is the range of a (pre-trained) generative neural network (GNN) and give the first signal recovery guarantees when the measurements are subsampled from an orthonormal basis. This includes the subsampled Fourier transform as a special case, thus allowing a measurement model in line with applications such as MRI. We define a coherence parameter which depends upon the alignment of the orthonormal basis and the range of the GNN and show that small coherence implies robust signal recovery from relatively few measurements. Numerical experiments verify that regularizing to keep the coherence parameter small while training the GNN can improve performance in the signal recovery stage.
In this talk, motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, I will introduce the noncommutative Bloch transform of Fuchsian groups, that we call the hyperbolic Bloch transform. I will prove that the hyperbolic Bloch transform is injective and “asymptotically unitary” and I will introduce a modified, geometric, Bloch transform, that transforms wave functions to sections of irreducible, flat, Hermitian vector bundles over the orbit space and transforms the hyperbolic Laplacian into the covariant Laplacian. If time permits, I will talk about potential applications to hyperbolic band theory. This is a joint work with Steve Rayan.
We will introduce a mathematical signal transform based on Optimal Transport Theory. It builds upon the existing Cumulative Distribution Transform by Park, Kolouri, Kundu, and Rohde (2018). We will present both forward (analysis) and inverse (synthesis) formulas for this nonlinear transform, and describe several of its properties including translation, scaling, convexity, and linear separability. Indeed, since this tool is a new signal representation based on Optimal Transport theory, it has suitable properties for decoding information related to certain signal displacements. We will describe a Wasserstein-type metric in transform space, and show applications in classifying (detecting) signals under random displacements, and parameter estimation problems for certain types of generative models.
The construction of Kummer K3 surfaces from abelian surfaces can be generalized to yield higher dimensional varieties known as hyperk"ahler varieties of Kummer type. Hassett and Tschinkel showed that a portion of the middle cohomology of generalized Kummer 4-folds may be understood as fixed loci of symplectic involutions corresponding to the three-torsion points of the abelian surface. In recent work with Sarah Frei, we have extended this result, allowing us to characterize the Galois action on the cohomology when working over non-closed fields.
Geometric Deep Learning is an emerging field of research that aims to extend the success of convolutional neural networks (CNNs) to data with non-Euclidean geometric structure such as graphs and manifolds. Despite being in its relative infancy, this field has already found great success and is utilized by e.g., Google Maps and Amazon’s recommender systems.
In order to improve our understanding of the networks used in this new field, several works have proposed novel versions of the scattering transform, a wavelet-based model of neural networks for graphs, manifolds, and more general measure spaces. In a similar spirit to the original Euclidean scattering transform, these geometric scattering transforms provide a mathematically rigorous framework for understanding the stability and invariance of the networks used in geometric deep learning. Additionally, they also have many interesting applications such as discovering new drug-like molecules, solving combinatorial optimization problems, and using single-cell data to predict whether or not a cancer patient will respond to treatment.
Grothendieck’s Existence Theorem asserts that a coherent sheaf on a scheme proper over a complete local noetherian ring is the same as a compatible system of coherent sheaves on the thickenings of its central fiber. This is a fundamental result with important applications to moduli theory. We will discuss generalizations of this result to algebraic stacks beginning with a review of the characteristic 0 situation where a satisfactory answer is known: any quotient stack $[{\rm Spec} A/G]$ whose invariant ring $A^G$ is a complete local k-algebra is coherently complete along its unique closed point. We will report on partial progress in joint work with Hall and Lim on extending this result to positive characteristic.
Geometric quantisation has proven an effective approach to many problems in mathematical physics. Many examples have been shown of theories whose classical solutions form geometric spaces with rich and interesting structures, which may then be used for quantisation. Sometimes, however, there is just too much structure, and it can become difficult to pick on to use. This is the case, for example, for hyper-Kähler manifolds, which come with infinite families of symplectic forms.
In a recent work with J.E. Andersen and G. Rembado, we proposed a new paradigm for quantisation of hyper-Kähler spaces assuming sufficient symmetry, which opens the way to exploration in many different directions. There are many examples of spaces to which this quantisation can be applied, including several from mathematical physics, and there are many famous results in “ordinary” quantisation that should be tested for this new version, notably the famous statement that quantisation commutes with reduction. In this talk I will give a panoramic of known results and possible research directions, including ongoing work with M. Mayrand.
We consider the nonlinear inverse problem of learning a transition operator ${A}$ from partial observations across different time scales, or in other words, from {sparse observations of its powers ${A},\cdots,{A}^{T-1}$. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the non-linearity issue by embedding the problem into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if ${A}$ is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We establish quadratic local convergence analysis of a suitable non-convex iterative reweighted least squares (IRLS) algorithm, and show that in optimal scenarios, no more than $\mathcal{O}(rn \log(nT))$ space-time samples are sufficient to ensure accurate recovery of a rank-$r$ operator ${A}$ of size $n \times n$. This establishes that {spatial samples} can be substituted by a comparable number of {space-time samples}. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order $O(r n T)$ and whose per-iteration time complexity is {linear} in $n$. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability of the proposed algorithm.
Inspired by Perelman’s work on the entropy formula for Ricci flow, we introduce the $W$-entropy and prove its monotonicity and rigidity theorem for the geodesics flow on the Wasserstein space over Riemannian manifold. This improves an earlier result due to Lott and Villani on the displacement of the Boltzmann type entropy on Riemannian manifolds with non-negative Ricci curvature. We then introduce the Langevin deformation on the Wasserstein space, which interpolates the Wasserstein geodesic flow and the gradient flow of the Boltzmann entropy (i.e., the heat equation on the underlying manifold). Moreover, we present a $W$-entropy type formula for the Langevin deformation. Joint work with Songzi Li.
Optimal transport studies the most economical movement of resources. In other words, one considers a pile of raw material and wants to transport it to a final configuration in a cost-efficient way. Under quite general assumptions, the solution to this problem will be induced by a transport map where the mass at each point in the initial distribution is sent to a unique point in the target distribution. In this talk, we will discuss the regularity of this transport map (i.e., whether nearby points in the first pile are sent to nearby points in the second pile). $$ $$ It turns out there are both local and global obstructions to establishing smoothness for the transport. When the cost is induced by a convex potential, we show that the local obstruction corresponds to the curvature of an associated Kähler manifold and discuss the geometry of this curvature tensor. In particular, we show (somewhat surprisingly) that its negativity is preserved along Kähler-Ricci flow.
We discuss the gradient flow structure of hydrodynamic limit equations obtained from microscopic interacting particle systems. We first show that for a wide class of reversible non-degenerate interacting particle systems, their hydrodynamic limit equations are formally given as a gradient flow with respect to the Wasserstein metric with mobility. Then we discuss when such formulation can be justified rigorously, and how it can be applied for the study of hydrodynamic limits. This talk is based on an ongoing work with Kohei Hayashi.
Multi-marginal optimal transport, a natural extension of the well-known classical optimal transport problem, is the problem of correlating given probability measures as efficiently as possible relative to a given cost function. Although a variety of applications have arisen over the past twelve years, the structure of solutions for the multi-marginal case has been difficult to address, mainly due to the strong dependence on the cost function. In this talk, I will briefly outline the known theory for uniqueness of this problem. Next, I will present a recent joint work with Brendan Pass based on a general condition on the cost function that provides uniqueness.
Consider the problem of matching two independent sets of $N$ i.i.d. observations from two densities $\rho_0$ and $\rho_1$ in $\mathbb{R}^d$. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that minimizes the total cost. We consider instead the problem where each matching is endowed with a Gibbs probability weight proportional to the exponential of the negative total cost of that matching. Viewing each matching as a joint distribution with $N$ atoms, we then take a convex combination with respect to the above Gibbs probability measure. We show that this resulting random joint distribution converges, as $N\rightarrow \infty$, to the solution of a variational problem, introduced by Föllmer, called the Schrödinger problem. Finally, we prove limiting Gaussian fluctuations for this convergence in the form of Central Limit Theorems for integrated test functions. This is enabled by a novel chaos decomposition for permutation symmetric statistics, generalizing the Hoeffding decomposition for U-statistics. Our results establish a novel passage for the transition from discrete to continuum in Schrödinger’s lazy gas experiment.
In this talk, we will introduce some interesting applications of optimal transportation in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric inequality in geometry; and an application in image recognition relating to a transport between hypercubes. This talk is based on a series of joint work with Shibing Chen, Xu-Jia Wang, and with Gregoire Loeper.