Let $F$ be a finite subset of $\mathbb{Z}^d$. We say that $F$ is a translational tile of $\mathbb{Z}^d$ if it is possible to cover $\mathbb{Z}^d$ by translates of $F$ without any overlaps. The periodic tiling conjecture, which is perhaps the most well-known conjecture in the area, suggests that any translational tile admits at least one periodic tiling. In the talk, we will motivate and discuss the study of this conjecture. We will also present some new results, joint with Terence Tao, on the structure of translational tilings in lattices and introduce some applications.

We present a perspective on neural networks stemming from quiver representation theory. This point of view emphasizes the symmetries inherent in neural networks, interacts nicely with gradient descent, and has the potential to improve training algorithms. As an application, we prove an analogue of the QR decomposition for radial neural networks, which leads to a dimensional reduction result. We assume a basic machine learning background, while explaining all necessary representation theory concepts from first principles.
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The talk is based on joint work-in-progress with Robin Walters.

After discussing the motivation behind the talk and some necessary preliminaries, we will consider the ``stabilization" of a countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group. Using a new local characterization of the Schmidt property, we show that this always gives rise to a so-called stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. We will also discuss some new structural results for product equivalence relations, which we will obtain using von Neumann algebraic techniques.

The Generalized Nash Equilibrium Problem (GNEP) is a kind of games to find strategies for a group of players such that each player’s objective function is optimized, given other players’ strategies. If all the objective and constraining functions involved are polynomials, we call the problem a Generalized Nash Equilibrium Problem of Polynomials (GNEPP). When the constraining functions of each player are independent of other player’s strategies, the GNEP is called a (standard) Nash Equilibrium Problem (NEP). The GNEP is said to be convex if each player’s optimization is a convex optimization problem, given other players’ strategies.
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For nonconvex Nash equilibrium problems that are given by polynomial functions, we formulate efficient polynomial optimization problems for computing Nash equilibria. We show that under generic assumptions, the method can find one or even all Nash equilibria if they exist, or detect nonexistence of Nash equilibria. For convex GNEPPs, we introduce rational and parametric expressions for Lagrange multipliers to formulate polynomial optimization for computing Generalized Nash Equilibria (GNEs). We prove that under some specific assumptions, the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the methods. The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization.
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Moreover, we study the Gauss-Seidel method for solving the nonconvex GNEPPs. We give a certificate for a class of GNEPPs such that the Gauss-Seidel method is guaranteed to converge, and the numerical experiments show that the Gauss-Seidel method can solve many GNEPPs efficiently.

We discuss how knot Floer homology can be used to study equivariant knots. We introduce some large-surgery correction terms that obstruct equivariant sliceness (and more generally, bound equivariant genus, following work of Juhasz and Zemke). We describe some crossing-change inequalities for these invariants. We also describe an amusing application to distinguishing (up to isotopy rel boundary) pairs of slice disks related by symmetries of a knot.
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This is work in progress with Abhishek Mallick and Matthew Stoffregen.

Fermat's Last Theorem that $x^n + y^n = z^n$ has no positive integer solutions for n$>$2 was first written down by Fermat himself in the early 17th century and resisted proof until Andrew Wiles' monumental 1994 paper. During the 300 years in between, many others tried their hand and along the way developed a lot of interesting number theory. We will discuss more classical topics in algebraic number theory - cyclotomic fields, higher reciprocity laws, class field theory, etc. - in the context of historical attempts to prove the theorem. We will be able to verify (the first case of) Fermat's Last Theorem for pretty high prime exponents (p $<$= 156,442,236,847,241,729).

The well-known Simons's cone suggests that minimal hypersurfaces could be possibly singular in a Riemannian manifold with dimension greater than 7, unlike the lower dimensional case. Nevertheless, it was conjectured that one could perturb away these singularities generically. In this talk, I will discuss how to perturb them away to obtain a smooth minimal hypersurface in an 8-dimension closed manifold, by induction on the ``capacity" of singular sets. This result generalizes the previous works by N. Smale and by Chodosh-Liokumovich-Spolaor to any 8-dimensional closed manifold.
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This talk is based on joint work with Zhihan Wang.

MemComputing [1, 2] is a novel physics-based approach to computation that employs time non-locality (memory) to both process and store information on the same physical location. Its digital version [3, 4] is designed to solve combinatorial optimization problems. A practical realization of digital memcomputing machines (DMMs) can be accomplished via circuits of non-linear dynamical systems with memory engineered so that periodic orbits and chaos can be avoided. A given logic problem is first mapped into this type of dynamical system whose point attractors represent the solutions of the original problem. A DMM then finds the solution via a succession of elementary instantons whose role is to eliminate solitonic configurations of logical inconsistency (``logical defects") from the circuit [5, 6]. I will discuss the physics behind memcomputing and show many examples of its applicability to various combinatorial optimization and Machine Learning problems demonstrating its advantages over traditional approaches [7, 8]. Work supported by DARPA, DOE, NSF, CMRR, and MemComputing, Inc.
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{[1]} M. Di Ventra and Y.V. Pershin, Computing: the Parallel Approach, Nature Physics 9, 200 (2013).
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{[2]} F. L. Traversa and M. Di Ventra, Universal Memcomputing Machines, IEEE Transactions on Neural Networks and Learning Systems 26, 2702 (2015).
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{[3]} M. Di Ventra and F.L. Traversa, Memcomputing: leveraging memory and physics to compute efficiently, J. Appl. Phys. 123, 180901 (2018).
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{[4]} F. L. Traversa and M. Di Ventra, Polynomial-time solution of prime factorization and NP-complete problems with digital memcomputing machines, Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 023107 (2017).
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{[5]} M. Di Ventra, F. L. Traversa and I.V. Ovchinnikov, Topological field theory and computing with instantons, Annalen der Physik 529,1700123 (2017).
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{[6]} M. Di Ventra and I.V. Ovchinnikov, Digital memcomputing: from logic to dynamics to topology, Annals of Physics 409, 167935 (2019).
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{[7]} F. L. Traversa, P. Cicotti, F. Sheldon, and M. Di Ventra, Evidence of an exponential speed-up in the solution of hard optimization problems, Complexity 2018, 7982851 (2018).
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{[8]} F. Sheldon, F.L. Traversa, and M. Di Ventra, Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks, Phys. Rev. E 100, 053311 (2019).

Nori proved in 2002 that given a complex algebraic variety
$X$, the bounded derived category of the abelian category of constructible sheaves on $X$ is
equivalent to the usual triangulated category $D(X)$ of bounded constructible complexes on $X$.
He moreover showed that given any constructible sheaf $\mathcal F$ on
$\mathbb{A}^n$, there is an injection $\mathcal F\hookrightarrow\mathcal G$ with
$\mathcal G$ constructible and $H^i(\mathbb{A}^n,\mathcal G)=0$ for $i>0$.
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In this talk, I'll discuss how to extend Nori's theorem to the case of a
variety over an algebraically closed field of positive characteristic, with
Betti constructible sheaves replaced by $\ell$-adic sheaves.
This is the case $p=0$ of the general problem which asks whether the bounded
derived category of $p$-perverse sheaves is equivalent to $D(X)$, resolved
affirmatively for the middle perversity by Beilinson.

In this talk I will discuss various topological and geometric consequences of the existence of zeros of global holomorphic 1-forms on smooth projective varieties. Such consequences have been indicated by a plethora of results. I will present some old and new results in this direction. One highlight of the topic is an interesting connection between two sets of such 1-forms, one that arises out of the generic vanishing theory and the other that falls out of Hodge theory of algebraic maps.
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This is joint work with Feng Hao and Yongqiang Liu.

In this talk, after discussing the necessary background material, I will discuss the main results of my recent work \emph{Variational Structures in Cochain Projection Based Discretizations of Lagrangian PDEs} and \emph{Multisymplectic Hamiltonian Variational Integrators}; this is joint work with Prof. Melvin Leok.
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Building on these ideas, I will conclude by discussing future research directions.