Date/Time : 18 Jan 2023 17:00
Venue : S17 #04-05 (SR2) via Zoom
Speaker : Bakhadyr Khoussainov, University of Auckland

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We aim to describe the isomorphism types of algorithmically presented structures in the language of the first order logic.

This is one of the key research topics related to the expressive power of the first order logic. The focus is on two classes of structures. The first class is the class of structures for which the positive atomic diagrams are computably enumerable.

We call these structures positive structures. The second class is the class of structures for which the negative atomic diagrams are computably enumerable. We call these structures negative structures. We investigate definability of positive and negative structures by sets of sentences quantified with Existential, Universal, Existential-Universal and Universal-Existential quantifiers using expansions of languages.

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Date/Time : 18 Jan 2023 15:30
Venue : S17 #04-05 (SR2)
Speaker : Yap Jit Wu, Harvard University

Let f,g be two rational maps on P^1. Then Baker-Demarco, and independently Yuan-Zhang, showed that either they have exactly the same set of preperiodic points, or only finitely many in common. More recently, Demarco-Krieger-Ye showed the existence of an uniform bound on number of common preperiodic points for the quadratic family z^2 + c and also the Lattes family. In this talk, we will discuss the ideas behind these two results along with some work in progress by the author on related questions.

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Date/Time : 17 Jan 2023 10:00
Venue : via Zoom
Speaker : Tan Nguyen, University of California, Los Angeles

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Designing deep learning models for practical applications, including those in computer vision, natural language processing, and mathematical modeling, is an art that often involves an expensive search over candidate architectures. In this talk, I present novel frameworks to facilitate the process of designing efficient and robust deep learning models with better representation capacity via three principled approaches: optimization, differential equation, and statistical modeling.

From an optimization viewpoint, I develop a gradient descent analogy of the recurrent cell in the recurrent neural network (RNN). I then integrate the momentum used for accelerating gradient dynamics into the recurrent cell, which results in the MomentumRNN. MomentumRNN can alleviate the vanishing gradient problem and accelerate training by choosing the appropriate momentum coefficients (Efficiency).
From a differential equation approach, I present a random walk interpretation of graph neural networks (GNNs), revealing a potentially inevitable over-smoothing phenomenon. Based on this random walk viewpoint of GNNs, I then propose the graph neural diffusion with a source term (GRAND++) that overcomes the over-smoothing issue and achieves better accuracy in low-labeling rate regimes (Robustness).

Using statistical modeling as a tool, I show that the attention in transformer models can be derived from solving a nonparametric kernel regression problem. I then propose the FourierFormer, a new class of transformers in which the softmax kernels are replaced by the novel generalized Fourier integral kernels. The generalized Fourier integral kernels can automatically capture the dependency of the features of data and remove the need to tune the covariance matrix (Representability).

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Date/Time : 12 Jan 2023 10:00
Venue : S17 #04-05 (SR2)
Speaker : Stefan Dawydiak, Max Planck Institute for Mathematics in Bonn (MPIM)

In the 1980s, Kazhdan-Lusztig and independently Ginzburg showed that the affine Hecke algebra $H$ can be realized as the Grothendieck group of the category of coherent sheaves on the Steinberg variety. The correct coherent Hecke category was however found only later, by Bezrukavnikov. A crucial ingredient in this coherent categorification of $H$ is viewing the Steinberg variety as a derived scheme.

On the other hand, Lusztig’s asymptotic Hecke algebra $J$ is a certain based ring receiving a map from $H$, via which its representation theory is closely related to that of $H$. In the 1990s, Xi showed that the best-understood direct summand $J_0$ of $J$ can be realized as the Grothendieck group of coherent sheaves on the square of the flag variety, and Ginzburg gave a K-theoretic definition of the map from $H$ to $J_0$. In this talk we will explain how to upgrade this realization to a monoidal category receiving a natural monoidal functor from the coherent Hecke category. A crucial ingredient is again to consider a derived enhancement, this time of the flag variety.

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Date/Time : 12 Jan 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Van Tien Nguyen, National Taiwan University

In this talk I will present constructive examples and numerical illustrations of blowup solutions to the Keller-Segel system in R^d.

• L^1-critical (d = 2): There exist finite time blowup solutions that are of Type II with finite mass. Blowup rates are quantized according to the spectrum of a linearized operator in the self-similar setting. There is also the case of multiple collapsing blowup solutions formed by a collision of single-solutions.

• L^1$-supercritical (d >= 3): We exhibit finite time blowup solutions that are completely unrelated to the self-similar scale, in particular, they are of Type II with finite mass. Interestingly, the radial blowup profile is linked to the traveling wave of the 1D viscous Burgers equation. There also exist solutions that blow up in finite time with infinite mass. The solution is asymptotically self-similar with a logarithmic correction to its profile for d = 3,4. We found such an asymptotic profile can be either radial or completely non-radial.

The talk is based on results obtained in collaboration with C. Collot (Paris Cergy), T-E. Ghoul (NYU Abu Dhabi), N. Masmousdi (NYU), N. Nouaili (Paris Dauphine), H. Zaag (Paris Nord).

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Date/Time : 11 Jan 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Omar Kidwai, University of Birmingham

Starting from the data of a quadratic (or higher) differential, Gaiotto-Moore-Neitzke’s famous study of four-dimensional N=2 supersymmetric QFTs outputs data known as a BPS structure (the same structure describes the output of Donaldson-Thomas theory). To solve a totally different problem 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 (through residue calculus and combinatorics) an infinite tower of geometric invariants, often with an enumerative interpretation.

We will describe recent work connecting these two different theories. In the simplest cases, when the spectral curve is of degree two and genus zero, we describe a simple explicit formula for the “free energies” of the topological recursion purely in terms of a corresponding BPS structure constructed from the same initial data. Time permitting, we will sketch the relation to the WKB analysis of quantum curves, and suggest a wide range of generalizations. Based on joint work with K. Iwaki and K. Osuga.

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Date/Time : 09 Jan 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Jonathan Eckstein, Rutgers University

This talk describes the solution of convex optimization problems that include uncertainty modeled by a finite but potentially very large multi-stage scenario tree.

In 1991, Rockafellar and Wets proposed the progressive hedging (PH) algorithm to solve such problems. This method has some advantages over other standard methods such as Benders decomposition, especially for problems with large numbers of decision stages. The talk will open by showing that PH is an application of the Alternating Direction Method of Multipliers (ADMM), and thus of the “Douglas-Rachford” operator splitting method. The equivalence of PH to the ADMM has long been known but not explicitly published.

Next, the talk will apply a different, more recent operator splitting method called “projective splitting” to the same problem. The resulting method is called “asynchronous projective hedging” (APH). Unlike most decomposition methods, it does not need to solve every subproblem at every iteration; instead, each iteration may solve just a single subproblem or a small subset of the available subproblems.

Finally, the talk will describe work integrating the APH algorithm into mpi-sppy, a Python package for modeling and distributed parallel solution of stochastic programming problems. Mpi-sppy uses the Pyomo Python-based optimization modeling sytem. Our experience includes using up to 2,400 processor cores to solve 2-stage and 4-stage test problem instances with as many as 1,000,000 scenarios.

Portions of the work described in this talk are joint with Patrick Combettes (North Carolina State University), Jean-Paul Watson (Lawrence Livermore National Laboratory, USA), and David Woodruff (University of California, Davis).

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Date/Time : 06 Jan 2023 14:00
Venue : LT34 (S17 Level 3)
Speaker : George Lusztig, MIT

According to Chevalley, semisimple groups can be defined over any field. More recently they were defined over some structures called semifields (which include the set of strictly positive real numbers under the usual addition and multiplication). It turns out that the theory over semifields can be used to get a better understanding of the theory over fields.

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Date/Time : 06 Jan 2023 14:00
Venue : S17 #05-12 (SR4)
Speaker : Tanya Veeravalli, University of Illinios, Urbana-Champaign

There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include “infinitely wide” neural nets, where the underlying nonlinearity is given by the activation function of an individual neuron. In this talk, we will bring this perspective to function representation by neural stochastic differential equations (SDEs). A neural SDE is an Itô diffusion process whose drift and diffusion matrix are elements of some parametric families. We will show that the ability of a neural SDE to realize nonlinear functions of its initial condition can be related to the problem of optimally steering a certain deterministic dynamical system between two given points in finite time. This auxiliary system is obtained by formally replacing the Brownian motion in the SDE by a deterministic control input. We derive upper and lower bounds on the minimum control effort needed to accomplish this steering; these bounds may be of independent interest in the context of motion planning and deterministic optimal control.

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