In this talk, we offer an entirely “white box’’ interpretation of deep (convolution) networks from the perspective of data compression (and group invariance). In particular, we show how modern deep layered architectures, linear (convolution) operators and nonlinear activations, and even all parameters can be derived from the principle of maximizing rate reduction (with group invariance). All layers, operators, and parameters of the network are explicitly constructed via forward propagation, instead of learned via back propagation. All components of so-obtained network, called ReduNet, have precise optimization, geometric, and statistical interpretation. There are also several nice surprises from this principled approach: it reveals a fundamental tradeoff between invariance and sparsity for class separability; it reveals a fundamental connection between deep networks and Fourier transform for group invariance – the computational advantage in the spectral domain (why spiking neurons?); this approach also clarifies the mathematical role of forward propagation (optimization) and backward propagation (variation). In particular, the so-obtained ReduNet is amenable to fine-tuning via both forward and backward (stochastic) propagation, both for optimizing the same objective.
This is joint work with students Yaodong Yu, Ryan Chan, Haozhi Qi of Berkeley, Dr. Chong You now at Google Research, and Professor John Wright of Columbia University.

This is joint work with Joanna Kania-Bartoszynska and Thang Le $\\$
I will discuss the representation theory of the Kauffman bracket skein algebra of a finite type surface at a root of unity whose order is not divisible by 4. $\\$
Specifically, the Kauffman bracket skein algebra is an algebra with trace in the sense of De Concini, Procesi, Reshetikhin and Rosso, so it has a well defined character variety of trace preserving representations, which can be identified with a branched cover of the SL(2,C)-character variety of the fundamental group of the underlying surface. $\\$
In the case of a closed surface the branched cover is trivial so its just the character variety of the fundamental group of the surface. $\\$
The skein algebra is also a Poisson order, so the character variety representations of the Kauffman bracket skein algebra of a closed surface decomposes into representations corresponding to irreducible, abelian and central representations of the fundamental group of the underlying surface. The irreducible representations of the fundamental group of the surface correspond to irreducible representations of the skein algebra. $\\$
We then use this as basic data to define an invariant of framed links in a three-manifold equipped with an irreducible representation of its fundamental group. The invariant satisfies the Kauffman bracket skein relations. $\\$
Such a representation could be the lift of the holonomy of a hyperbolic structure on the three-manifold, hence the title : A Geometric Kauffman Bracket. $\\$

Confusingly for the uninitiated, experts in weak infinite-dimensional category theory make use of different definitions of an ∞-category, and theorems in the ∞-categorical literature are often proven "analytically", in reference to the combinatorial specifications of a particular model. In this talk, we present a new point of view on the foundations of ∞-category theory, which allows us to develop the basic theory of ∞-categories --- adjunctions, limits and colimits, co/cartesian fibrations, and pointwise Kan extensions --- "synthetically" starting from axioms that describe an ∞-cosmos, the infinite-dimensional category in which ∞-categories live as objects. We demonstrate that the theorems proven in this manner are "model-independent", i.e., invariant under change of model. Moreover, there is a formal language with the feature that any statement about ∞-categories that is expressible in that language is also invariant under change of model, regardless of whether it is proven through synthetic or analytic techniques. This is joint work with Dominic Verity.

I will present a rational function valuation on strict partitions as shapes
that is defined via shifted standard tableaux. The goal is to determine
explicit expressions for this valuation by solving a linear system that
reflects the lattice structure of strict partitions.
Somewhat surprisingly, this leads into the area of symmetric functions.
The problem as such may appear as an isolated curiosity, yet is motivated
by the extension of the model for an asymmetric annihilation process originally proposed by A. Ayyer and K. Mallick (2010).

Schramm-Loewner evolution (SLE for short) is a family of random fractal curves, which describe the scaling limits of some lattice models. When one studies SLE with additional marked points, and requires that those SLE satisfy certain nice properties, some special functions come into play. They arise as the solution of some second-order partial differential equations. In this talk, I will describe how the one-variable and multi-variable hypergeometric functions are used to study the time-reversal of SLE$_\kappa(\rho_1,\dots,\rho_m)$ curves.
We use the same zoom link ant passcode as before.

In this talk, I will report on ongoing work with Ryan Takahashi in which we study Brauer-Manin obstructions to the existence of rational points on moduli spaces of sheaves on K3 surfaces. There are Brauer classes naturally arising out of geometric constructions, and we aim to find conditions under which these Brauer classes obstruct the existence of certain kinds of sheaves on a K3 surface over a number field.

Despite breakthrough performance, modern learning models are known to be highly vulnerable to small adversarial perturbations in their inputs. While a wide variety of recent adversarial training methods have been effective at improving robustness to perturbed inputs (robust accuracy), often this benefit is accompanied by a decrease in accuracy on benign inputs (standard accuracy), leading to a tradeoff between often competing objectives. Complicating matters further, recent empirical evidence suggests that a variety of other factors (size and quality of training data, model size, etc.) affect this tradeoff in somewhat surprising ways. In this talk we will provide a precise and comprehensive understanding of the role of adversarial training in the context of linear regression with Gaussian features and binary classification in a mixture model. We precisely characterize the standard/robust accuracy and the corresponding tradeoff achieved by a contemporary mini-max adversarial training approach in a high-dimensional regime where the number of data points and the parameters of the model grow in proportion to each other. Our theory for adversarial training algorithms also facilitates the rigorous study of how a variety of factors (size and quality of training data, model overparametrization etc.) affect the tradeoff between these two competing accuracies.

While we have undoubtedly mastered quantitative problem solving, in the coming months many of us will deal with some very common and personal quantitative decisions that we haven't encountered before: retirement benefits. The sooner you have a basic grasp of what benefits are available to you and their impact, the sooner you can make informed decisions and potential compare employment opportunities. As potential new employees, learn the basics of common retirement offerings for US employment and associated tax implications from a student with first-hand experience.

In this talk, I will introduce correspondence modules (c-modules), which generalize persistence and zigzag modules. I also will introduce the persistence sheaf of sections of a c-module, which is used to analyze its structure and prove an interval decomposition theorem. Several applications in which c-modules arise naturally will be discussed.

We will present a full spectral analysis for a model of graphene in magnetic fields with constant flux through every hexagonal comb. In particular, we provide a rigorous foundation for self-similarity by showing that for any irrational flux, the spectrum of graphene is a zero measure Cantor set. I will also discuss the spectral decomposition, Hausdorff dimension of the spectrum and existence of Dirac cones. This talk is based on joint works with S. Becker and S. Jitomirskaya.

One of the central problems in the interface of deep learning and mathematics is that of building learning systems that can automatically uncover underlying mathematical laws from observed data. In this work, we make one step towards building a bridge between algebraic structures and deep learning, and introduce\textbf {AIDN},\textit {Algebraically-Informed Deep Networks}.\textbf {AIDN} is a deep learning algorithm to represent any finitely-presented algebraic object with a set of deep neural networks. The deep networks obtained via\textbf {AIDN} are\textit {algebraically-informed} in the sense that they satisfy the algebraic relations of the presentation of the algebraic structure that serves as the input to the algorithm. Our proposed network can robustly compute linear and non-linear representations of most finitely-presented algebraic structures such as groups, associative algebras, and Lie algebras. We evaluate our proposed approach and demonstrate its applicability to algebraic and geometric objects that are significant in low-dimensional topology. In particular, we study solutions for the Yang-Baxter equations and their applications on braid groups. Further, we study the representations of the Temperley-Lieb algebra. Finally, we show, using the Reshetikhin-Turaev construction, how our proposed deep learning approach can be utilized to construct new link invariants. We believe the proposed approach would tread a path toward a promising future research in deep learning applied to algebraic and geometric structures.

Title: Classification and rigidity for group von Neumann algebras

Date: 04/13/2021

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

Contact: Aaron D Levin ()

Any countable group G gives rise to a von Neumann algebra L(G). The classification of these group von Neumann algebras is a central theme in operator algebras. I will survey recent rigidity results which provide instances when various algebraic properties of groups, such as the presence or absence of a direct product decomposition, are remembered by their von Neumann algebras. I will also explain the strongest such rigidity results, where L(G) completely remembers G, and discuss some of the open problems in the area.

We will present the notion of a conveyor belt configuration on disjoint disks in the plane, which means a tight simple closed curve that touches the boundary of each disk. An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, possibly touched multiple times. We will present three main results. 1) For unit disks whose centers are both x-monotone and y-monotone, or whose centers have x-coordinates that differ by at least two units, a conveyor belt always exists and can be found efficiently. 2) It is NP-complete to determine whether disks of arbitrary radii have a conveyor belt, and it remains NP-complete when we constrain the belt to touch disks exactly once. 3) Any disjoint set of n disks of arbitrary radii can be augmented by O(n) “guide” disks so that the augmented system has a conveyor belt touching each disk exactly once, answering a conjecture of Demaine, Demaine, and Palop. Many open problems remain on this topic and we will share some of our favorites. This talk is based on joint work with Molly Baird, Erik D. Demaine, Martin L. Demaine, David Eppstein, Sándor Fekete, Graham Gordon, Sean Griffin, Joseph S. B. Mitchell, and Joshua P. Swanson.

I will give an overview of spectral theory of second order elliptic operators on $\mathbb R^2$ on the example of Schrodinger operators and explain the main steps of the proof that spectral band edged are attained at finitely many values of the quasimomenta. If time permits, I will discuss related result and work in progress. The talk is based on joint work with N. Filonov.
Note: The meeting time is 3:00 pm instead of 4:00 pm.
We use the same zoom link and passcode as before.

Affine Deligne-Lusztig varieties (ADLV) naturally arise from the study of Shimura varieties. We prove a formula for the number of their irreducible components, which was a conjecture of Miaofen Chen and Xinwen Zhu. Our method is to count the number of F_q points, and to relate it to certain twisted orbital integrals. We then study the growth rate of these integrals using the Base Change Fundamental Lemma of Clozel and Labesse. In an ongoing work we also give the number of irreducible components in the basic Newton stratum of a Shimura variety. This is joint work with Rong Zhou and Xuhua He. Password: MSUALG

Toeplitz operators are fundamental and ubiquitous in signal processing and information theory as models for convolutional (filtering) systems. Due to the fact that any practical system can access only signals of finite duration, however, time-limited restrictions of Toeplitz operators are also of interest. In the discrete-time case, time-limited Toeplitz operators are simply Toeplitz matrices. In this talk we survey existing and present new bounds on the eigenvalues (spectra) of time-limited Toeplitz operators, and we discuss applications of these results in various signal processing contexts. As a special case, we discuss time-frequency limiting operators, which alternatingly limit a signal in the time and frequency domains. Slepian functions arise as eigenfunctions of these operators, and we describe applications of Slepian functions in spectral analysis of multiband signals, super-resolution SAR imaging, and blind beamforming in antenna arrays. This talk draws from joint work with numerous collaborators including Zhihui Zhu from the University of Denver.

Given a group $G$ acting on measure space $(X,\mu)$ Murray and von Neumann’s group-measure space construction describes a von Neumann algebra $L^\infty(X,\mu)\rtimes G $ which encodes both the group, the space and the action. The special case where the space is a singleton and the action is trivial produces the group von Neumann algebra $L(G) $.
In this talk, we will aim to describe properties of $L^\infty(X,\mu)\rtimes G $ in terms of the group, the space and the action; compute $L^\infty(X,\mu)\rtimes G $ in special cases; and describe how the group-measure space varies or the group von Neumann algebra varies with $G$. All this serves to illustrate the fundamental problem in this area: von Neumann algebras tend to have poor memory of their generating data.
This talk assumes a working knowledge of group theory and linear algebra, and while knowledge of measure theory may be helpful, it is not required.

I will introduce some of the basic ideas behind network embedding, and show how a complicated or high-dimensional graph can be represented as a collection of points in a lower-dimensional vector space. This will be an expository talk focusing on random walk-based methods, and how these methods are related to certain matrix-based methods and diffusion maps. Throughout the talk, I'll show how these techniques are useful for machine learning tasks like social network analysis and image recognition.

In this presentation, I will invite participants to follow me as I narrate in words and images my transition from colonial, imperialist research practices to creating knowledge with knowers. As part of this transition—which I see as continuous and liberating—I will discuss multiple powerful forces that have contributed to reinvigorate my eyes, ears, and all my senses, leading me to find my desire to do research in mathematics education desde adentro; that is, from inside the practices and wisdoms that I am trying to make sense. Two research projects, one recently completed and one that starts this Fall, will serve as important referents for gauging the intensities of my desire to do research. Participants are strongly encouraged to bring to this informal conversation insights, experiences, questions, wonderings, and contributions related to their desire to conduct research in education. There is no need to register! Just join the Zoom room at:
https://msu.zoom.us/j/91681702869
Passcode: 731530

A series of conjectures of Braverman-Kazhdan, Sakellaridis and Sakellaridis-Venkatesh propose that affine spherical varieties should provide a source for new integral representations of automorphic L-functions. This global problem is conjecturally (and sometimes provably) related to a certain local problem in harmonic analysis. In particular, it is conjectured that unramified local L-factors are related to intersection complexes of formal arc spaces of spherical varietes. I will explain how we establish this connection for a large class of spherical varieties over a local function field, using techniques from geometric representation theory. This is joint work with Yiannis Sakellaridis.