Title: Fundamental limits of learning in deep neural networks; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 08/20/2020

Time: 2:30 PM - 3:30 PM

Place:

(Part of One World MINDS seminar:
https://sites.google.com/view/minds-seminar/home)
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\]
We develop a theory that allows to characterize the fundamental limits of learning in deep neural networks. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the epsilon-entropy of the hypothesis class to be learned and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the quantized weights and biases. The theory we develop educes remarkable universality properties of deep networks. Specifically, deep networks can Kolmogorov-optimally learn essentially any hypothesis class. In addition, we find that deep networks provide exponential approximation accuracy—i.e., the approximation error decays exponentially in the number of non-zero weights in the network—of widely different functions including the multiplication operation, polynomials, sinusoidal functions, general smooth functions, and even one-dimensional oscillatory textures and fractal functions such as the Weierstrass function, both of which do not have any known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks. We conclude with an outlook on the further role our theory could play.

Title: Unsupervised deep learning of forward and inverse solutions for PDE-based imaging; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 08/27/2020

Time: 2:30 PM - 3:30 PM

Place:

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home)
\[
\]
Many imaging modalities are based on inverse problems of physical processes that are given as PDEs. Traditional methods for solving these PDE-based forward and inverse problems are based on discretizations of the domain. Deep learning methods are based on an excessive amount of input-output pairs. Both approaches encounter problems either by numerical instabilities and by being limited to low dimensions or by the lack of sufficient data. We suggest an alternative method of unsupervised deep learning method were the network parametrizes the solution and the loss function minimizes the deviation from the PDE. The input set are points sampled randomly in the domain and the output is the deviation from the PDE, namely zero. One key issue in the loss function is the introduction of the L_infty term that guaranty the uniform convergence of the network to the solution. We demonstrate our method on the Electrical Impedance Tomography (EIT).

Title: Bipartite dimer model and minimal surfaces in the Minkowski space

Date: 09/01/2020

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

We discuss a new approach to the convergence of height fluctuations in the bipartite dimer model considered on big planar graphs. This viewpoint is based upon special embeddings of weighted planar graphs into the complex plane known under the name Coulomb gauges or, equivalently, t-embeddings. The long-term motivation comes from trying to understand fluctuations on irregular graphs, notably on random planar maps equipped with the dimer (or, similarly, the critical Ising) model.
When the dimer model is considered on subgraphs of refining lattices, a classical conjecture due to Kenyon--Okounkov predicts the convergence of fluctuations to the Gaussian Free Field in a certain conformal structure. However, the latter is defined via a lattice-dependent entropy functional, which makes the analysis of irregular graphs highly problematic. To overcome this difficulty, we introduce a notion of 'perfect t-embeddings' of abstract weighted bipartite graphs and develop new discrete complex analysis techniques to handle correlation functions of the dimer model on t-embeddings. Though in full generality the existence of perfect embeddings remains an open question, we prove that - at least in some concrete cases - they reveal the relevant conformal structure in a lattice-independent way: as that of a related Lorentz-minimal surface in the Minkowski space.
Based upon joint works with Benoît Laslier, Sanjay Ramassamy and Marianna Russkikh.

Title: High dimensional approximation with trigonometric polynomials; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 09/03/2020

Time: 2:30 PM - 3:30 PM

Place:

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home)
\[
\]
In this talk, we present fast Fourier based methods for the approximation of multivariate functions. Our aim is to learn the support of the Fourier coefficients in the frequency domain of high-dimensional functions. We are interested in two different approximation scenarios. The first case is black-box approximation where the user is allowed to sample the unknown function at any point and in the second case we are working with fixed scattered data. For black-box approximation we employ quasi Monte-Carlo methods on rank-1 lattice points. The fast algorithms are then based on one-dimensional fast Fourier transforms (FFT). In the second case, which is much more difficult, we will couple truncated ANOVA (analysis of variance) decompositions with the fast Fourier transform on nonequispaced data (NFFT). In both cases, we present error estimates and numerical results. The presented methods can be understood as sparse high dimensional FFT’s.
This talk based on joint work with Lutz Kämmerer, Michael Schmischke, Manfred Tasche, and Toni Volkmer.

Let K be a null-homologous knot in a closed 3-manifold Y, and F be a Seifert surface. One can cap off the boundary of F with a disk in the zero surgery on K to get a closed surface F_0. If we know that F is Thurston norm minimizing, we can ask whether F_0 is also Thurston norm minimizing. A classical theorem of Gabai says that the answer is Yes when Y is the 3-sphere. Gabai's theorem can be generalized to many other 3-manifolds using Heegaard Floer homology. In this talk, we will discuss a sufficient condition for F_0 to be Thurston norm minimizing which relates this property to the 4-genus of the knot.
Zoom: https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

A sheaf quantization is a sheaf associated to a Lagrangian brane. This sheaf conjecturally has information as much as Floer theory of the Lagrangian. On the other hand, exact WKB analysis is an analysis of differential equations containing the Planck constant hbar.
In this talk, I will explain how to construct a sheaf quantization over the Novikov ring of the spectral curve of an hbar-differential equation, by using the ideas of exact WKB analysis and spectral network. In the construction, one can see how (conjecturally) the convergence in WKB analysis are related to the convergence of Fukaya category. In degree 2, the sheaf quantization associates a cluster coordinate which is the same as Fock—Goncharov coordinate. I will also mention about some relationships to Riemann—Hilbert correspondence of D’Agnolo—Kashiwara and Kontsevich—Soibelman.
https://msu.zoom.us/j/95159415920?pwd=bUlETkdpazdiWGNjZnNkUWNIaXRFQT09

Title: On the homology of subword order. Zoom https://msu.zoom.us/j/5476724571

Date: 09/09/2020

Time: 4:10 PM - 5:00 PM

Place:

In this talk we examine the homology representation of the symmetric group $S_n$ on rank-selected subposets of subword order. We show that the action on the rank-selected chains is a nonnegative integer combination of tensor powers of the reflection representation $S_{(n-1,1)}$ indexed by the partition $(n-1,1)$, and that its Frobenius characteristic is $h$-positive and supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$
We give an explicit formula for the homology module for words of bounded length, as a sum of tensor powers of $S_{(n-1,1)}$. This recovers, as a special case, a theorem of Bj\"orner and Stanley for words of length at most $k.$ We exhibit a curious duality in homology in the case when one rank is deleted. We also show that in many cases, the rank-selected homology modules, modulo one copy of the reflection representation, are $h$-positive and supported on the set $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
Our analysis of the homology also uncovers curious enumerative formulas that may be interesting to investigate combinatorially.

We'll introduce and motivate techniques of Galois descent for classifying twisted forms of algebras, using non-abelian group cohomology. We'll also describe a connection to relative Brauer groups.
Zoom link: https://msu.zoom.us/j/96111069403

Title: The phase factor: recovery from magnitudes of signal representations; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 09/10/2020

Time: 2:30 PM - 3:30 PM

Place:

We study the problem of phase retrieval from a deterministic viewpoint, in which the magnitudes of a time-frequency or time-scale representation of a signal are known. From an inverse problems perspective, the questions of uniqueness and stability are crucial to theoretically guarantee meaningful reconstruction. In this talk, we present results on these two questions for Gabor frames and wavelet frames and conclude by discussing some open problems.

Title: Relative Reshetikhin-Turaev invariants and hyperbolic cone metrics on 3-manifolds

Date: 09/15/2020

Time: 3:00 PM - 4:00 PM

Place: Online (virtual meeting)

We propose the Volume Conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern-Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring, and prove the conjecture for a number of families of examples. This provides a possible approach of solving the Volume Conjecture for the Reshetikhin-Turaev invariants of closed oriented hyperbolic 3-manifolds. A large part of this work is joint with Ka Ho Wong. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! What's worse, this doesn't just happen for some exotic theories, but in the standard theories describing some of the most fundamental aspects of nature. Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will dip our toes into some of the conceptual and mathematical aspects of these techniques and we will see how they have recently been used in probability theory to study equations whose meaning was not even clear until now.

Speaker: Samantha Dahlberg, Arizona State University

Title: Diameters of Graphs of Reduced Words of Permutations, Zoom https://msu.zoom.us/j/5476724571

Date: 09/16/2020

Time: 3:00 PM - 3:50 PM

Place: Online (virtual meeting)

It is a classical result that any two reduced words of a permutation in the symmetric group can be transformed into one another by a sequence of long braid moves and commutation moves. In this talk we will discuss the diameters of these connected graphs formed from the reduced words connected by single moves. Recently, the diameter has been calculated for the longest permutation $n\ldots 21$ by Reiner and Roichman as well as Assaf. In this talk we present our results on diameters for certain classes or permutation . We also make progress on conjectured bounds of the diameter by Reiner and Roichman, which are based on the underlying hyperplane arrangement.

Classical (contravariant) Dieudonne theory establishes an antiequivalence of categories between finite commutative group schemes over a perfect field $k$ and Dieudonne modules. In this talk we will talk about this antiequivalence and some simple applications of it.

Title: Learning Interaction laws in particle- and agent-based systems; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 09/17/2020

Time: 2:30 PM - 3:30 PM

Place:

Interacting agent-based systems are ubiquitous in science, from modeling of particles in Physics to prey-predator and colony models in Biology, to opinion dynamics in economics and social sciences. Oftentimes the laws of interactions between the agents are quite simple, for example they depend only on pairwise interactions, and only on pairwise distance in each interaction. We consider the following inference problem for a system of interacting particles or agents: given only observed trajectories of the agents in the system, can we learn what the laws of interactions are? We would like to do this without assuming any particular form for the interaction laws, i.e. they might be “any” function of pairwise distances. We consider this problem both the mean-field limit (i.e. the number of particles going to infinity) and in the case of a finite number of agents, with an increasing number of observations, albeit in this talk we will mostly focus on the latter case. We cast this as an inverse problem, and study it in the case where the interaction is governed by an (unknown) function of pairwise distances. We discuss when this problem is well-posed, and we construct estimators for the interaction kernels with provably good statistically and computational properties. We measure their performance on various examples, that include extensions to agent systems with different types of agents, second-order systems, and families of systems with parametric interaction kernels. We also conduct numerical experiments to test the large time behavior of these systems, especially in the cases where they exhibit emergent behavior. This is joint work with F. Lu, J.Miller, S. Tang and M. Zhong.

Title: Operator Spaces and Operator Systems: An Exposition

Date: 09/21/2020

Time: 2:00 PM - 2:50 PM

Place: Online (virtual meeting)

During this lecture I will give an overview of the history and theory of operator spaces and operator systems. These "matrix normed spaces'' and "matrix ordered $*$-vector spaces'' arose in a somewhat natural fashion and the study of these objects is motivated by problems that arise when studying the "classical'' theory such as C*-algebras. After going over necessary background for both objects I will discuss how operator space and operator system theory were applied to approaching and solving problems in operator algebras.
Join via Zoom: https://msu.zoom.us/j/95716797501

Title: An Abstract Characterization for Projections in Operator Systems

Date: 09/21/2020

Time: 3:00 PM - 3:50 PM

Place: Online (virtual meeting)

Given an abstract operator system V it is not clear how one would go about defining the notion of a projection. During this talk I will present an answer and some recent results on this question. This is done by first considering abstract compression operator systems associated with a positive contraction in V and then determining when we have a realization of V in such an abstract compression operator system. It then follows that there is a one-to-one correspondence between abstract and concrete projections, and in particular, that every abstract projection is a concrete projection in the C*-envelope of V. I will then conclude with some applications to quantum information theory. In particular, the study of certain correlation sets. This is joint work with Travis Russell (West Point).
Join via Zoom: https://msu.zoom.us/j/95716797501

Title: A geometric approach to fractional powers of the Laplacian and sharp Sobolev trace inequalities

Date: 09/22/2020

Time: 3:00 PM - 4:00 PM

Place: Online (virtual meeting)

A seminal paper of Caffarelli and Silvestre identifies fractional powers of the Laplacian on Euclidean space as Dirichlet-to-Neumann operators. In this talk, I will use conformal geometry to generalize their approach to Riemannian manifolds. More specifically, I will present multiple (equivalent) definitions of (conformally covariant) operators with principal symbol that of a fractional power of the Laplacian. I will also discuss how these operators lead to a simple derivation of a broad family of sharp Sobolev trace inequalities.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Title: Polynomial interpolation is harder than it sounds

Date: 09/22/2020

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

Suppose that $(x_1,y_1),\ldots,(x_r,y_r)$ is a set of points in the plane. Given a degree $d$ and multiplicities $m_i$, does there a nonzero polynomial in two variables of degree at most $d$ which vanishes to order at least $m_i$ at $(x_i,y_i)$? What is the dimension of the space of such polynomials, and how does it vary with the parameters? I will explain some of the basic results and conjectures and show how this problem is connected to some questions of current interest in algebraic geometry.

Speaker: Samin Aref, Max Planck Institute for Demographic Research

Title: Structural analysis of signed graphs: a talk on methods and applications, Zoom https://msu.zoom.us/j/5476724571

Date: 09/23/2020

Time: 3:00 PM - 3:50 PM

Place: Online (virtual meeting)

This talk focuses on positive and negative ties in networks (signed graphs) resulting in a common structural configuration. We analyze signed networks from the perspective of balance theory which predicts structural balance as a stable configuration. A signed network is balanced iff its set of vertices can be partitioned into two groups such that positive edges are within the groups and negative edges are between the groups.
The scarcity of balanced configurations in networks inferred from empirical data (real networks) requires us to define the notion of partial balance in order to quantify the extent to which a network is balanced. After evaluating several numerical measures of partial balance, we recommend using the frustration index, which equals the minimum number of edges whose removal results in a balanced network [arxiv.org/abs/1509.04037].
We use the definition of balance to optimally partition nodes of signed networks into two internally solidary but mutually hostile groups. An optimal partitioning leads to an exact value for the frustration index. We tackle the intensive computations of finding an optimal partition by developing efficient mathematical models and algorithms [arxiv.org/abs/1710.09876] [arxiv.org/abs/1611.09030]. We then extend the concepts of balance and frustration in signed networks to applications beyond the classic friend-enemy interpretation of balance theory in the social context. Using a high-performance computer, we analyze large networks to investigate a range of applications from biology, chemistry and physics to finance, international relations, and political science [arxiv.org/abs/1712.04628].
In another project manly focused on a political science application, we focus on the challenge of quantifying political polarization in the US Congress, and analyzing its relationship to the fraction of introduced bills that are passed into law (bill passage rate). We use signed graph models of political collaboration among legislators to show that changes in bill passage rates are better explained by the partisanship of a chamber's largest coalition, which we identify by partitioning signed networks of legislators according to balance theory [arxiv.org/abs/1906.01696].
In another project, we expand the evaluation of balance to incorporate directionality of the edges and consider three levels of analysis: triads, subgroups, and the whole network. Through extensive computational analysis, we explore common structural patterns across a range of social settings from college students and Wikipedia editors to philosophers and Bitcoin traders. We then apply our multilevel framework of analysis to examine balance in temporal and multilayer networks which leads to new observations on balance with respect to time and layer dimensions [arxiv.org/abs/2005.09925].

Title: Algebraic groups with good reduction. Zoom https://msu.zoom.us/j/97573873209

Date: 09/23/2020

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

Techniques involving reduction are very common in number theory and arithmetic geometry. In particular, elliptic curves and general abelian varieties having good reduction have been the subject of very intensive investigations over the years. The purpose of this talk is to report on recent work that focuses on good reduction in the context of reductive linear algebraic groups over finitely generated fields. In addition, we will highlight some applications to the study of local-global principles and the analysis of algebraic groups having the same maximal tori. (Parts of this work are joint with V. Chernousov and A. Rapinchuk.)

A path connected topological space is simply connected if the space of based paths is path connected. Equivalently, the fundamental group is zero or any connected covering space is trivial. However, these notions do not capture the correct notion in the world of algebraic geometry. For example, if $X$ is a Riemann surface then the Zariski topology (the usual topology in algebraic geometry) on $X$ is equivalent to the cofinite topology, so $X$ is simply connected.
In this talk, we will introduce a few definitions of simply connectedness in algebraic geometry - each corresponding to one of the equivalent definitions above. We will then compare these definitions and discuss how their consequences differ from their topological counterparts.

Title: Regularization in Infinite-Width ReLU Networks; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 09/24/2020

Time: 2:30 PM - 3:30 PM

Place: Online (virtual meeting)

A growing body of research illustrates that neural network generalization performance is less dependent on the network size (i.e. number of weights or parameters) and more dependent on the magnitude of the weights. That is, generalization is not achieved by limiting the size of the network, but rather by explicitly or implicitly controlling the magnitude of the weights. To better understand this phenomenon, we will explore how neural networks represent functions as the number of weights in the network approaches infinity. Specifically, we characterize the norm required to realize a function f as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate functions in Savarese et al. (2019), where it was shown that the required norm is determined by the L1-norm of the second derivative of the function. We extend the characterization to multivariate functions (i.e., networks with d input units), relating the required norm to the L1-norm of the Radon transform of a (d+1)/2-power Laplacian of the function. This characterization allows us to show that all functions in certain Sobolev spaces can be represented with bounded norm and to obtain a depth separation result. These results have important implications for understanding generalization performance and the distinction between neural networks and more traditional kernel learning.

Title: Fundamental groups of certain property (T) factors

Date: 09/28/2020

Time: 2:00 PM - 2:50 PM

Place: Online (virtual meeting)

Calculation of fundamental groups of type $\rm II_1$ factor is, in general, an extremely hard and central problem in the field of von Neumann algebras. In this direction, a conjecture due to A. Connes states that the fundamental group of the group von Neumann algebra $L(\Gamma)$ associated to any icc property (T) group $\Gamma$ is trivial. Up to now, there was no single example of property (T) group factor satisfying the conjecture. In this talk, I shall provide the first examples of property (T) group factors with trivial fundamental group. This talk is based on a joint work with Ionut Chifan, Sayan Das and Cyril Houdayer.
Join via Zoom: https://msu.zoom.us/j/98441498789

Title: Skein theory in the geometric Langlands TFT

Date: 09/29/2020

Time: 3:00 PM - 4:00 PM

Place: Online (virtual meeting)

I will overview several appearances (some recently established, some conjectural) of skein theory in the so-called quantum geometric Langlands fully extended TFT. The talk will be mostly elementary, and I'll highlight an application to a conjecture of Witten concerning the finite-dimensionality of skein modules of 3-manifolds at generic values of the quantum parameter. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Informally, Kakeya type problems ask whether tubes with different positions and directions can overlap a lot. One usually expects the answer to be no in an appropriate sense. Thanks to the uncertainty principle, such a quantified non-overlapping theorem would often see powerful applications in analysis problems that have Fourier aspects. Perhaps the most well-known Kakeya type problem is the Kakeya conjecture. It remains widely open in $\Bbb{R}^n (n>2)$ as of today. Nevertheless, in the recent few decades people have been able to prove new Kakeya type theorems that led to improvements or complete solutions to analysis problems that appeared out of reach before. I will give an introduction to Kakeya type problems/theorems and analysis problems that see their applications. Potentially reporting some recent progress joint with Du, Guo, Guth, Hickman, Iosevich, Ou, Rogers, Wang and Wilson.

Title: Whitney Duals of Graded Posets, Zoom https://msu.zoom.us/j/5476724571

Date: 09/30/2020

Time: 3:00 PM - 3:50 PM

Place: Online (virtual meeting)

To each graded poset one can associate two sequences of numbers; the Whitney numbers of the first kind and the Whitney numbers of the second kind. One sequence keeps track of the Möbius function at each rank level and the other keeps track of the number of elements at each rank level. The Whitney numbers appear in many contexts in combinatorics. For example, they appear as the coefficients of the chromatic polynomial of a graph and can be used to compute the number of regions in a real hyperplane arrangement.
We say that posets P and Q are Whitney duals if the Whitney numbers of the first kind of P are the Whitney numbers of the second kind of Q and vice-versa. In this talk, we will discuss a method to construct Whitney duals using edge labelings and quotient posets. We will also discuss some applications of Whitney duals.
This is joint work with Rafael S. González D'León.

Schramm-Loewner evolutions (SLE) are probabilistic models of simple planar curves. They first arise as interfaces in scaling limits of 2D statistical mechanics lattice models which exhibit conformal invariance. In this talk, I will explain how asymptotic behaviors of SLE give rise to an interesting quantity (multichordal Loewner potential), which connects to rational function, zeta-regularized determinants of Laplacian, and Belavin-Polyakov-Zamolodchikov equations in conformal field theory. This is a joint work with Eveliina Peltola (Bonn).

Title: Reconstruction of a variety from the derived category and groups of autoequivalences

Date: 10/01/2020

Time: 1:00 PM - 2:00 PM

Place: Online (virtual meeting)

This talk is mainly based on Bondal and Orlov’s works. We consider smooth algebraic varieties with ample either canonical or anticanonical sheaf. We prove that such a variety is uniquely determined by its derived category of coherent sheaves.We also calculate the group of exact autoequivalences for these categories

Title: On the Regularization Properties of Structured Dropout; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 10/01/2020

Time: 2:30 PM - 3:30 PM

Place: Online (virtual meeting)

Dropout and its extensions (e.g. DropBlock and DropConnect) are popular heuristics for training neural networks, which have been shown to improve generalization performance in practice. However, a theoretical understanding of their optimization and regularization properties remains elusive. This talk will present a theoretical analysis of several dropout-like regularization strategies, all of which can be understood as stochastic gradient descent methods for minimizing a certain regularized loss. In the case of single hidden-layer linear networks, we will show that Dropout and DropBlock induce nuclear norm and spectral k-support norm regularization, respectively, which promote solutions that are low-rank and balanced (i.e. have factors with equal norm). We will also show that the global minimizer for Dropout and DropBlock can be computed in closed form, and that DropConnect is equivalent to Dropout. We will then show that some of these results can be extended to a general class of Dropout-strategies, and, with some assumptions, to deep non-linear networks when Dropout is applied to the last layer.

Every toric variety is a GIT quotient of an affine space by an algebraic torus. In this talk, I will discuss a way to understand and compute the symplectic mirrors of toric varieties from this universal perspective using the concept of window subcategories. The talk is based on results from a work of myself and a joint work in progress with Peng Zhou.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Speaker: Colin McLarty, Case Western Reserve University

Title: Grothendieck's personal idea of a topos as a space

Date: 10/06/2020

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

In 33 hours of tape recordings in 1973 Grothendieck described his view of topos beyond what is in the collective volume Théorie des topos et cohomologie étale (SGA 4). In particular, this shows how Grothendieck got his idea of a "generalized topological space" simultaneously with what became etale cohomology during a 1958 talk by Jean-Pierre Serre.

Using anti-diagonal Gröbner geometry, Knutson and Miller explained how Grothendieck polynomials arise as K-polynomials of matrix Schubert varieties. We will discuss how Lascoux's transition equations for Grothendieck polynomials can be realized geometrically through "almost anti-diagonal" Gröbner degeneration of matrix Schubert varieties. In particular, under this strange term order, Fulton's generators form a Gröbner basis. Our proof involves studying the lattice of alternating sign matrices.

26894

Wednesday 10/7 4:00 PM

Bruce E. Sagan, MSU https://msu.zoom.us/j/93361845408

Let $\alpha=(a,b,\ldots)$ be a composition. Consider the associated poset $F(\alpha)$, called a fence, whose covering relations are
$$
x_1\lhd x_2 \lhd \ldots\lhd x_{a+1}\rhd x_{a+2}\rhd \ldots\rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots\ .
$$
We study the associated distributive lattice $L(\alpha)$ consisting of all lower order ideals of $F(\alpha)$.
These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define $q$-analogues of rational numbers.
In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that $L(\alpha)$ is rank unimodal. We show that if one of the parts of $\alpha$ is greater than the sum of the others, then the conjecture is true. We conjecture that $L(\alpha)$ enjoys the stronger properties of having a nested chain decomposition and having a rank sequence which is either top or bottom interlacing, the latter being a recently defined property of sequences. We verify that these properties hold for compositions with at most three parts and for what we call $d$-divided posets, generalizing work of Claussen and simplifying a construction of Gansner.

In this talk I will discuss the optimal transport problem with ''storage fees.'' This is a variant of the semi-discrete optimal transport (a.k.a. Monge-Kantorovich) problem, where instead of transporting an absolutely continuous measure to a fixed discrete measure and minimizing the transport cost, one must choose the weights of the target measure, and minimize the sum of the transport cost and some given ''storage fee function'' of the target weights. This problem arises in queue penalization and semi-supervised data clustering. I will discuss some basic theoretical questions, such as existence, uniqueness, a dual problem, and characterization of solutions. Then, I will present a numerical algorithm which has global linear and local superlinear convergence for a subcase of storage fee functions. All work in this talk is joint with M. Bansil (UCLA).

This will be an expository talk, where we'll begin by introducing Fourier series for periodic functions. After an effusive discussion of their nice properties, we'll move to computer-land and talk about how to approximate coefficients of Fourier series "digitally". This will bring us around to what Gilbert Strang described as "the most important numerical algorithm of our lifetime": the Fast Fourier Transform. We'll talk about how it works and just how fast it really is.
https://msu.zoom.us/j/99892700137

Given a nontrivial band sum of two knots, we may add full twists to the band to obtain a family of knots indexed by the integers. In this talk, I'll show that the knots in this family have the same Heegaard and instanton knot Floer homology but distinct Khovanov homology, generalizing a result of M. Hedden and L. Watson. A key component of the argument is a proof that each of the three knot homologies detects the trivial band. The main application is a verification of the generalized cosmetic crossing conjecture for split links.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

One real-life situation application of graph theory is the study of electrical grids: they have to be constructed carefully since unstable grids can lead to brownouts, blackouts, damaged equipment, or other possible problems. If we know the connections in the grid that we want, how can the voltages at each node be coordinated in a way that makes sure the network stays stable? This is a difficult question, but even knowing the number of ways to keep a network stable can help.
In this talk, we will see how to count the number of "stable solutions" using geometric and algebraic methods. These methods will help us obtain recurrences for networks satisfying mild conditions. Consequently, we obtain explicit, non-recursive formulas for the number of stable solutions for a large class of outerplanar graphs, and conjecture that the formula holds for all outerplanar graphs. One of the keys to our results: studying dragons and the havoc they wreak on fictional medieval villages.

Gelfand’s trick shows that the spherical Hecke algebra of a
p-adic split reductive group such as $\mathrm{GL}_n(\mathbf{Q}_p)$ is commutative.
In this talk I will present a generalization of this result to the
setting of derived Hecke algebras as introduced by Venkatesh, i.e. I
show that the spherical derived Hecke algebra is graded-commutative
under mild assumptions on the coefficient ring.

It is an introduction for K3 surfaces. I will give the definition of algebraic K3 surfaces and compute the hodge diamond over different base fields. After showing several examples, I will show some important properties of K3 surfaces from my interest.

Convolutional Neural Networks are highly successful tools for image recovery and restoration. A major contributing factor to this success is that convolutional networks impose strong prior assumptions about natural images—so strong that they enable image recovery without any training data. A surprising observation that highlights those prior assumptions is that one can remove noise from a corrupted natural image by simply fitting (via gradient descent) a randomly initialized, over-parameterized convolutional generator to the noisy image.
In this talk, we discuss a simple un-trained convolutional network, called the deep decoder, that provably enables image denoising and regularization of inverse problems such as compressive sensing with excellent performance. We formally characterize the dynamics of fitting this convolutional network to a noisy signal and to an under-sampled signal, and show that in both cases early-stopped gradient descent provably recovers the clean signal.
Finally, we discuss our own numerical results and numerical results from another group demonstrating that un-trained convolutional networks enable magnetic resonance imaging from highly under-sampled measurements, achieving results surprisingly close to trained networks, and outperforming classical untrained methods.

This talk is motivated by applications to music theory, especially the construction of musical scales. We'll describe how to think of "good" scales as a subgroups of S^1, then dip our toes into the theory of continued fractions to understand why some "good" scales are "better" than others. In particular, we can get a satisfying mathematical reason for the ubiquity of 12-tone equal temperament scale system.

In dimension 3, the theory of codimension 2 contact submanifolds is better known as the Transverse Knot Theory of a contact manifold, a theory which has a complete description in terms of braids in S^3 and braids in open books more generally. In higher dimensions, little is known about the structure of codimension 2 contact submanifolds, but the list of results is growing. I will explain a method developed with A. Kaloti to use open books and lefschetz fibrations to study codimension 2 contact embeddings in all dimensions. I will give a lot of background, present some initial applications and will highlight the similarities and differences from the relatively complete picture in dimension 3.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure.
In the joint paper with Paata Ivanisvili and Ramon Van Handel we prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the discrete cube, which, in its turn, is based on a certain formula that we used before in improving the constants in the scalar Poincaré inequality on the Hamming cube. I will also show several extensions of Pisier's inequality with ultimate assumptions on a Banach space structure.
Some of our results use approach via quantum random variables.

26908

Wednesday 10/21 3:00 PM

Clifford Smyth, University of North Carolina, Greensboro

Given a set R of natural numbers let S(n,k,R) be the restricted Stirling number of the second kind: the number of ways of partitioning a set of size n into k non-empty subsets with the sizes of these subsets restricted to lie in R. Let S(R) be the matrix with S(n,k,R) in its (n,k) entry. If R contains 1, S(R) has an inverse T(R) with integer entries. We find that, for many R, the entries T(n,k,R) of T(R) are expressible (up to sign) as the cardinalities of explicitly defined sets of trees and forests. For example, this is the case when R has no exposed odds, i.e. R contains 1 and 2 and R never contains an odd number n greater than 1 without also containing n+1 and n-1. We have similar results for restricted Stirling numbers of the first kind (partitions into cycles) and Lah numbers (partitions into ordered lists). Our proofs depend in part on a combinatorial formula for the coefficients of the compositional inverse of a power series.
This is joint work with David Galvin of the University of Notre Dame and John Engebers of Marquette University.

This is an introduction talk for Khovanov homology. Since Khovanov homology categorifies Jones polynomial, I will start with a brief discussion about Jones polynomial and the enhanced states for the bracket state sum. After that I will show how to construct Khonvanov homology from the enhanced states and its invariance under Reidemeister moves, which makes it a well defined link invariant.

Rapoport-Zink formal schemes are moduli spaces of p-divisible groups equipped with additional structures coming from an algebraic group. In order to define these formal schemes in general, one is led to search for the "right" way to endow a p-divisible group with the structure of an arbitrary algebraic group. We discuss a Tannakian approach to this problem using Zink's theory of displays, and we explain how it recovers another approach using crystalline Tate tensors in the Hodge-type case.

The Multiple Access Channel (MAC) is one of the most well studied and understood network information theoretic models, describing a scenario where K users wish to deliver their information message to one receiver, sharing the same transmission channel. Beyond the multitude of practical and somwho heuristic MAC protocols (e.g., TDMA, FDMA, CDMA, CSMA, Aloha, and variations thereof), the information theoretic capacity region is well understood under many situations of interest, and in particular in the Gaussian case, modeling the uplink of a wireless system with one access point or base station (receiver) and several users, sharing the same frequency band.
More recently, a variant of this model has been proposed for a situation where a very large (virtually unlimited) number of users wish to communicate only very sporadically, such that at any point in time only a finite and relatively small number of users are ``active''. This scenario is appropriate for machine-type communications and Internet of Things, where a multitude of sensors and objects have only rather sporadic data to send, but they need to send them when they are created, at random times.
The identification of the active user set, or the active message set (the list of messages transmitted, irrespectively of who is transmitting them) has some points in common with a compressed sensing problem, where the activity vector (entry 1 if a user/message is active and 0 otherwise) is the key object to be estimated at the receiver side.
In this talk we review the basic MAC model and results, a variant for massive random access called ``unsourced random access'' where all users use the same codebook, and related recent results and algorithms. including some capacity scaling for this model, which remains quite open as far as a full information theoretic characterization is concerned.

Bipartite graphs can be used to capture social networks through event participation. By letting one set of vertices be participants and the other set be events, each edge represents an individual participating in an event. These bipartite graphs can be projected into a weighted graph by multiplying the bipartite adjacency matrix by its transpose. In the projection, an edge between two individuals represents the number of times they participated in the same event. We can now ask, how many times do two people have to participate in events together before we can assume they have some sort of relationship? We will discuss ways to decide if an edge weight is strong enough and deduce friendship ties.

An interesting problem in contact topology is to understand the Lagrangian surfaces that fill a given Legendrian link in a contact 3-manifold. A key breakthrough this year is that we now know some families of Legendrian links that have infinitely many different fillings. This is due to various work by Honghao Gao, Linhui Shen, Daping Weng, and some people who aren't currently at Michigan State, and the common approach is through microlocal sheaf theory and clusters. I'll describe a different, Floer-theoretic approach to the same sort of result, using augmentations of Legendrian contact homology. The Floer approach recovers some of the results from the sheaf approach and also produces some new examples of links with infinitely many fillings. This is joint work in progress with Roger Casals.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Emmy Noether is famous as the “mother of modern algebra,” but her influence extended far beyond algebra alone. This talk, based on my recent book with the title above, will focus on Noether’s broader influence as an international figure in the 1920s. Beyond her immediate circle of students, Noether’s courses drew talented mathematicians from all over the world. Four of the most important were B.L. van der Waerden, Pavel Alexandrov, Helmut Hasse, and Olga Taussky. Noether’s classic papers on ideal theory inspired van der Waerden to recast his research in algebraic geometry. Her lectures on group theory motivated Alexandrov to develop links between point set topology and combinatorial methods. Noether’s vision for a new approach to algebraic number theory gave Hasse the impetus to pursue a line of research that led to the Brauer-Hasse-Noether Theorem, whereas her abstract style clashed with Taussky’s approach to classical class field theory during a difficult time when both were trying to find their footing in a foreign country. Hermann Weyl, her colleague before both fled to the United States in 1933, fully recognized that Noether’s dynamic school was the very heart and soul of the famous Göttingen community.
Two recent books on Emmy Noether:
Emmy Noether – Mathematician Extraordinaire
https://www.springer.com/gp/book/9783030638092
Proving It Her Way: Emmy Noether, a Life in Mathematics
https://www.springer.com/gp/book/9783030628109

Legendrian graphs naturally appear in the study of Weinstein manifolds with a singular Lagrangian skeleton, and a tangle decomposition of Legendrian submanifolds. I will introduce various invariants of Legendrian graphs including DGA type, polynomial type, sheaf theoretic one, and their relationship. This is joint work with Byunghee An, and partially with Tamas Kalman and Tao Su.
https://msu.zoom.us/j/95159415920?pwd=bUlETkdpazdiWGNjZnNkUWNIaXRFQT09

Arithmetical structures on finite connected graphs are generalization of the Laplacian of a graph. Dino Lorenzini originally defined them in order to answer some questions in algebraic geometry, but more recently, they have been studied on their own, particularly with a combinatorics lens. In this talk, we will discuss how to count the number of arithmetical structures on different types of graphs and discuss why it is a hard but interesting question for other families. If time permits, we will talk about their corresponding critical groups.

(Palais, 1979) Many problems are set up as variational problems. That is, on a (possibly infinite-dimensional) manifold M with a group of symmetries G, we look for critical points of a G-invariant functional f. In order to do this, we might first restrict ourselves to looking at the set S of G-symmetric points of M (points p such that gp=p for all g). The principle asserts that if p is a critical point of f restricted to S, then p is in fact a critical point in M. In other words: “Critical symmetric points are symmetric critical points”.
For example, harmonic functions on a space X are critical points of an energy functional on a space of functions M = {X to R}. To find a harmonic map on X, we might start by considering only maps which are rotationally symmetric. The principle states that it suffices to consider only rotationally symmetric variations as well. This reduces the problem from a PDE to an ODE, how nice!
Anyway, the Principle does not always hold, but in some very general situations it does. Let’s find out about them.

We will show how to construct hyperkähler manifolds of O'Grady 10 type out of smooth cubic fourfolds. Applications to classical algebraic geometry questions on intermediate jacobian fibrations of families of cubic threefolds, and quintic elliptic curves on cubic fourfolds will be explained. This is based on joint work with Chunyi Li and Laura Pertusi.
Meeting passcode: MSU ALG

26844

Thursday 10/29 4:30 AM

Yang Wang, Hong Kong University of Science and Technology

(Note the unusual time: 4:30pm Shanghai, 10:30am Paris.)
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Generative Adversarial Nets (GAN) have been one of the most exciting developments in machine learning and AI. But training of GAN is highly nontrivial. In this talk I will give an introduction to GAN, and propose a framework to learn deep generative models via Variational Gradient Flow (Vgrow) on probability measure spaces. Connections of our proposed VGrow method with other popular methods, such as VAE, GAN and flow-based methods, have been established in this framework, gaining new insights of deep generative learning.

I will discuss random projections as a method for dimension reduction, and review one of the main theorems in this topic which is the Johnson-Lindenstrauss lemma. I will explain the lemma and some of the ideas in its proof.

I will give an overview of some relations between finite group theory, G-equivariant topological quantum field theory, and the computational complexity of invariants of 3-manifolds, both classical and quantum. We will start with one of the simplest kinds of invariants in knot theory: the coloring invariants, introduced by Fox when giving a talk to undergraduates in the 1950s. We will then build up to the idea of G-equivariant TQFT (aka homotopy QFT with target K(G,1)), which mathematically describes the topological order determined by a symmetry-enriched topological phase of matter. Physicists have studied these in part motivated by the search for new universal topological quantum computing architectures.
Our goal will be to convey two complexity-theoretic lessons. First, when G is sufficiently complicated (nonabelian simple), the simple-to-define coloring invariants associated to G are, in fact, very difficult to compute, even on a quantum computer. Second, no matter what finite group G one uses, a 3-dimensional G-equivariant TQFT can not be used for universal topological quantum computation if the underlying non-equivariant theory is not already universal. This talk is based on joint works with Greg Kuperberg and Colleen Delaney.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

The first part of the talk will review background material on the differential geometry of 7-dimensional manifolds with the exceptional holonomy group $G_{2}$. There are now many thousands of examples of deformation classes of such manifolds and there are good reasons for thinking that many of these have fibrations with general fibre diffeomorphic to a K3 surface and some singular fibres: higher dimensional analogues of Lefschetz fibrations in algebraic geometry. In the second part of the talk we will discuss some questions which arise in the analysis of these fibrations and their "adiabatic limits". The key difficulties involve the singular fibres. This brings up a PDE problem, analogous to a free boundary problem, and similar problems have arisen in a number of areas of differential geometry over the past few years, such as in Taubes' work on gauge theory. We will outline some techniques for handling these questions.

A grid associahedron is a simple polytope whose faces correspond to nonkissing collections of routes inside a grid graph. The usual associahedron arises as a special case when the graph is a 2 by n rectangle. Other examples of grid associahedra have been considered in connection with combinatorial properties of Grassmannians and with the representation theory of gentle algebras. The 1-skeleton of the grid associahedron has a natural orientation that induces the grid-Tamari order, a poset with many remarkable properties. I will present a new construction of the grid associahedron as a Minkowski sum of order polytopes of fence posets. Using this construction, I will show that the grid-Tamari order has the non-revisiting chain property. This is based on joint work with Alexander Garver.

First, I will introduce Hilbert spaces, reproducing kernel Hilbert spaces, and von Neumann algebras. Then, I will show how one may construct a sequence of RKHS's associated to a finite-dimensional von Neumann algebra, which serves as a complete algebraic invariant. Finally, I will describe an RKHS structure which recovers whether or not an arbitrary von Neumann algebra has property Gamma.

What's the difference between a continuous function and a Hermitian matrix? From the perspective of operator algebras, not much! Operator algebras is a branch of mathematics that is equal parts analysis and linear algebra, and operator algebraists spend a lot of time thinking about mathematical objects called C*-algebras. If you've taken a course in calculus, then you are already familiar with one example of a C*-algebra: the continuous functions on a closed interval [a,b]. If you've taken linear algebra, then you're familiar with another example: the nxn square matrices. In this talk I will introduce the definition of a C*-algebra by way of these examples, and show how each example can provide insights into the other.

The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on $L^{2}$-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud.

The alternating genus of a knot is the minimum genus of a surface onto which the knot has an alternating diagram satisfying certain conditions. Very little is currently known about this knot invariant. We study spanning surfaces for knots, and define an alternating distance from the extremal spanning surfaces. This gives a lower bound on the alternating genus and can be calculated exactly for torus knots. We prove that the alternating genus can be arbitrarily large, find the first examples of knots where the alternating genus is equal to n for each n>2, and classify all toroidally alternating torus knots.
Zoom: https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Starting from a short review of Riemann's sketch of a geometry on manifolds (1854ff.) this talk discusses how Hermann Weyl proposed to take up Riemann's ideas and how to modify them in the light of his views of the new developments in the foundations of analysis and in physics (the general theory of relativity, GTR). Although in his early work (1913) Weyl contributed to the later axiomatisation of the concept of manifolds, his own preferences lay clearly in a constructive approach (for a time even an intuitionistic one). Moreover, he saw a basic problem for accepting Riemannian geometry as an adequate geometric framework for a field theoretic foundation of physics. As an alternative he proposed a ``purely infinitesimal'' approach to the metric -- later called Weyl geometry. It contained the first instance of an (explicit) gauge structure which --in transformed form -- became the grandfather of the later gauge theories in physics.

We consider posets of (projective) sign vectors ordered by reverse inclusion of zero entries with restrictions on sign variation. We show many of these posets are partitionable and give an interpretation of the h-vector in terms of descents in signed permutations. This is joint work with Nantel Bergeron and Aram Dermenjian. Time permitting we will discuss conjectural (strong) Spernicity.

I’ll discuss some explicit examples of Calabi-Yau threefolds (CY3s) with torsion in various cohomology groups and consequences of such torsion; in particular, I’ll present computations using p-adic Hodge theory which answer some outstanding questions about CY3s over finite fields. I will not assume familiarity with Calabi-Yau manifolds or p-adic Hodge theory.

I will discuss recent joint work with Lewis Bowen and Frank Lin. In it, we consider a natural metric satisfying the CAT(0) condition (a certain natural negative curvature condition) on a space of operators affiliated to a tracial von Neumann algebra (a version of this space appeared in previous work of Andruchow-Larotonda). We also investigate how the geometric properties of this CAT(0) space reflect algebraic/analytic properties of the underlying von Neumann algebra.

A central problem of machine learning is the following. Given data of the form (y_i, f(y_i) + ϵ_i)_{i = 1}^M, where y_i’s are drawn randomly from an unknown (marginal) distribution μ* and ϵ_i are random noise variables from another unknown distribution, find an approximation to the unknown function f, and estimate the error in terms of M. The approximation is accomplished typically by neural/rbf/kernel networks, where the number of nonlinear units is determined on the basis of an estimate on the degree of approximation, but the actual approximation is computed using an optimization algorithm. Although this paradigm is obviously extremely successful, we point out a number of perceived theoretical shortcomings of this paradigm, the perception reinforced by some recent observations about deep learning. We describe our efforts to overcome these shortcomings and develop a more direct and elegant approach based on the principles of approximation theory and harmonic analysis.

The talk starts an introduction to weights and some history in the one and two weight theory and Calderon-Zygmund operators. There is also an introduction to T1 and Tb Theorems and I present a recent result of mine together with Grigoriadis, Sawyer, Shen and Uriarte-Tuero about the two-weight local Tb theorem for fractional singular integral operators.

26922

Friday 11/13 11:00 AM

Dr. Anna DeJarnette , University of Cincinnati ; Dr. Casey Hord, University of Cincinnati

In this talk we will share how our work as researchers in math education and, respectively, special education has led to an ongoing collaboration to support pre-service teachers to facilitate meaningful mathematical conversations and conceptual learning for all students. Anna will describe how she has applied techniques of discourse analysis to document different aspects of students’ talk, including how students negotiate authority and co-construct mathematical ideas. Casey will summarize his research on mathematics interventions for students with mild disabilities, including the use of gestures and visual representations. We will discuss how our research trajectories have led to our collaborative work, as well as the affordances and challenges of bridging different perspectives. Finally, we will share an example from our current research to illustrate how we integrate our analysis discourse and gesturing between students and pre-service teachers.
Register in advance for the colloquium:
https://msu.zoom.us/meeting/register/tJEsce2pqjwoGtRHK0KTvrsyYiH6GvjuNHSU
After registering, you will receive a confirmation email containing information about joining the meeting.

An operator algebra is an algebra of bounded linear operators acting on a Hilbert space that is closed in a certain norm topology. When that algebra is closed with respect to the adjoint operation (an abstract conjugate transpose), we call it a C*-algebra. The prototypical examples of C*-algebras include the ring of n x n matrices over the complex numbers and the ring of complex-valued continuous functions on a compact Hausdorff space. The latter example gives an algebraic perspective for studying topological dynamics. In particular, one can build an operator algebra called a crossed product that encodes the dynamical information of a group of homeomorphisms acting on a topological space.
In the 1960s, W. Arveson determined that the action of a homeomorphism on a topological space is better encoded in a crossed product via the action of a semigroup on that space, rather than a group, which led to many important results in operator algebra theory.
I will discuss how and why operator algebraists have been returning in recent years to crossed products in the context of groups acting on non-adjoint closed operator algebras, and I will discuss a recent partial solution to when dynamics are encoded fully in this crossed product context.

Not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we show that all augmentations come from possibly immersed exact Lagrangian fillings. We first associate an immersed cobordism with a DGA map from the top Legendrian knot to the DGA of the cobordism. This gives a functor from a Legendrian category whose morphisms involve immersed Lagrangian cobordisms to a DGA category. With this functorality, an immersed filling L together with an augmentation of L induce an augmentation of the top Legendrian knot. This is a joint work with Dan Rutherford.
Zoom: https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

26927

Tuesday 11/17 4:00 PM

Catherine Goldstein, CNRS, Institut de mathématiques de Jussieu Paris Gauche

Charles Hermite’s name has been attributed to several objects and results in mathematics, from Hermitian matrices to Hermite polynomials to Hermite’s identity or Hermite-Minkowski theorem. Despite his achievements and central role in the mathematical life of the 19th century, he often appears as an anti-hero, opposed to anything modern, be it ideals, non-Euclidean geometry or set theory. I will try to explain his point of view which is linked to a vision of mathematics as a natural, observational science, and show how this perspective shaped his mathematical work and his requirements on what good mathematics should be.

A Schwarz function on an open domain $\Omega$ is a holomorphic function satisfying $S(\zeta)=\overline{\zeta}$ on the boundary $\Gamma$ of $\Omega$. Sakai in 1991 managed to give a complete characterisation of the boundary of a domain admitting a Schwarz function. In fact, if $\Omega$ is simply connected $\Gamma$ has to be regular real analytic. Here we try to describe $\Gamma$ when the boundary condition is slightly relaxed. In particular, we are interested in three different conditions over a simply connected domain $\Omega$: When $f_1(\zeta)=\overline{\zeta}f_2(\zeta)$ with $f_1,f_2$ holomorphic, when $\mathcal{U}/\mathcal{V}$ equals some real analytic function on $\Gamma$ with $\mathcal{U},\mathcal{V}$ harmonic and when $S(\zeta)=\Phi(\zeta,\overline{\zeta})$ with $\Phi$ a holomorphic function of two variables. It turns out the boundary can be from analytic to just $C^1$, regular except finitely many points, or regular except for a measure zero set, respectively.

Work in progress joint with Samuele Anni and Alexandru Ghitza. For N
prime to p, we count the number of classical modular forms of level Np
and weight k with fixed residual Galois representation and
Atkin-Lehner-at-p sign, generalizing both recent results of Martin (no
residual representation constraint) and rhobar-dimension-counting
formulas of Jochnowitz and Bergdall-Pollack. One challenge is the
tension between working modulo p and the need to invert p when working
with the Atkin-Lehner involution. To address this, we use the trace
formula to establish up-to-semisimplifcation isomorphisms between
certain mod-p Hecke modules (namely, refinements of weight-graded
pieces of spaces of mod-p forms) by exhibiting ever-deeper congruences
between traces of prime-power Hecke operators acting on
characteristic-zero Hecke modules. This last technique is new and
combinatorial in nature; it relies on a theorem discovered by the
authors and beautifully proved by Gessel, and may be of independent
interest.

The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal and image processing. However, in many modern applications, the signal bandwidths have increased tremendously, while the acquisition capabilities have not scaled sufficiently fast. Consequently, conversion to digital has become a serious bottleneck. Furthermore, the resulting digital data requires storage, communication and processing at very high rates which is computationally expensive and requires large amounts of power. In the context of medical imaging sampling at high rates often translates to high radiation dosages, increased scanning times, bulky medical devices, and limited resolution.
In this talk, we present a framework for sampling and processing a large class of wideband analog signals at rates far below Nyquist in space, time and frequency, which allows to dramatically reduce the number of antennas, sampling rates and band occupancy. Our framework relies on exploiting signal structure and the processing task. We consider applications of these concepts to a variety of problems in communications, radar and ultrasound imaging and show several demos of real-time sub-Nyquist prototypes including a wireless ultrasound probe, sub-Nyquist MIMO radar, super-resolution in microscopy and ultrasound, cognitive radio, and joint radar and communication systems. We then discuss how the ideas of exploiting the task, structure and model can be used to develop interpretable model-based deep learning methods that can adapt to existing structure and are trained from small amounts of data. These networks achieve a more favorable trade-off between increase in parameters and data and improvement in performance, while remaining interpretable.

I have often been vexed by textbooks or professors offering one word proofs to quickly justify concepts that do not come so lazily to me. (That's a pangram!) The purpose of this talk will be to shed some light on some of these one word proofs, by giving an introduction to categories. This talk will be accessible to anyone who has ever been similarly frustrated (so all of us), as my examples include things such as the integers, partially ordered sets, and matrices. On the other hand, as a warning, if you can define a natural transformation without looking it up, then this talk will probably be boring for you.

This is the first talk in a series on "ergodic quantum processes." A quantum process is a sequence of quantum channels, which in turn are completely positive, trace preserving maps. In this talk, I will discuss the definition of a quantum channel and describe a general ergodic theorem for a quantum process formed from an ergodic sequence of stochastic channels. The proof of the result and applications to Matrix Product States will be discussed if time permits (but likely in a future talk).
(Zoom password: CPTP)

The stable Gauss map of a Lagrangian $L$ in a cotangent $T^*M$ is a map $g
\colon L \to U/O$ obtained by stabilization of the usual Gauss map from $L$ to
the Lagrangian Grassmannian of $T^*M$. Arnold's conjecture on nearby
Lagrangians implies in particular that $g$ is homotopic to a constant map. We
will see the weaker result that the map induced by $g$ on the homotopy groups is
trivial.
By a theorem of Giroux and Latour $g$ is homotopic to a constant map if and only
if $L$ admits a generating function. We introduce ``twisted'' generating
functions as a tool to the study of $L$ and make the link with difficult results
of pseudo-isotopy theory.
This is a joint work with Mohammed Abouzaid, Sylvain Courte and Thomas Kragh.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

This talk is about the coherent-constructible correspondence (CCC). CCC is a version of homological mirror symmetry for toric varieties. It equates the derived category of coherent sheaves on a toric variety and the category of constructible sheaves on a torus that satisfy some condition on singular support. Recently, Harder-Katzarkov conjectured that there should be a version of CCC for toric fiber bundles and they proved their conjecture for $\mathbb{P}_1$-bundles. I will explain how we can prove (half of) their conjecture for
$\mathbb{P}_n$-bundles. If time permits, I will give a more precise version of the conjecture for arbitrary toric fiber bundles.

Emmy Noether is famous as the “mother of modern algebra,” but her influence extended far beyond algebra alone. This talk, based on my recent book with the title above, will focus on Noether’s broader influence as an international figure in the 1920s. Beyond her immediate circle of students, Noether’s courses drew talented mathematicians from all over the world. Four of the most important were B.L. van der Waerden, Pavel Alexandrov, Helmut Hasse, and Olga Taussky. Noether’s classic papers on ideal theory inspired van der Waerden to recast his research in algebraic geometry. Her lectures on group theory motivated Alexandrov to develop links between point set topology and combinatorial methods. Noether’s vision for a new approach to algebraic number theory gave Hasse the impetus to pursue a line of research that led to the Brauer-Hasse-Noether Theorem, whereas her abstract style clashed with Taussky’s approach to classical class field theory during a difficult time when both were trying to find their footing in a foreign country. Hermann Weyl, her colleague before both fled to the United States in 1933, fully recognized that Noether’s dynamic school was the very heart and soul of the famous Göttingen community.
Two recent books on Emmy Noether:
Emmy Noether – Mathematician Extraordinaire
https://www.springer.com/gp/book/9783030638092
Proving It Her Way: Emmy Noether, a Life in Mathematics
https://www.springer.com/gp/book/9783030628109

In complex algebraic geometry, positivity of direct images of relative canonical bundles are important for the study of geometry of algebraic morphisms. In this talk, I would like to discuss a notion of metric positivity for coherent sheaves and prove that a large class of sheaves from Hodge theory, including the direct images of relative canonical bundles, always satisfy the metric positivity. This result unifies and strengthens several results of positivity on the algebraic side (i.e. weak positivity). Based on joint work with Christian Schnell.

26845

Thursday 11/26 3:30 AM

Man-Cho Anthony So, Chinese University of Hong Kong

(Note the unusual time: 4:30pm Shanghai, 10:30am Paris.)
The group synchronization problem calls for the estimation of a ground-truth vector from the noisy relative transforms of its elements, where the elements come from a group and the relative transforms are computed using the binary operation of the group. Such a problem provides an abstraction of a wide range of inverse problems that arise in practice. However, in many instances, one needs to tackle a non-convex optimization formulation. It turns out that for synchronization problems over certain subgroups of the orthogonal group, a simple projected gradient-type algorithm, often referred to as the generalized power method (GPM), is quite effective in finding the ground-truth when applied to their non-convex formulations. In this talk, we survey the related recent results in the literature and focus in particular on the techniques for analyzing the statistical and optimization performance of the GPM.
This talk covers joint works with Huikang Liu, Peng Wang, Man-Chung Yue, and Zirui Zhou.

26924

Monday 11/30 2:00 PM

Lara Ismert, Embry–Riddle Aeronautical University, Prescott

Permutations can be described in many ways, including as words and as graphs. The graphical perspective lets us think about "heights" and "depths" of a permutation. Peak sets of permutations have a long history in the literature. Inspired by work of Billey, Burdzy, and Sagan on those sets, we introduced the pinnacle sets of permutations. Despite natural symmetries, pinnacle sets and peak sets have notably different properties. We will explore some of those differences in this talk, giving characterization and enumerative results about pinnacle sets. Very recently, several papers have added to the literature on pinnacle sets, and we will describe those updates here.
This talk includes joint work with Robert Davis, Sarah Nelson, Kyle Petersen, and Irena Rusu.

The colored Jones polynomial is a generalization of the Jones polynomial from the finite-dimensional representations of $U_q(sl_2)$. One motivating question in quantum topol- ogy is to understand how the polynomial relates to other knot invariants. An interesting example is the strong slope conjecture, which relates the asymptotics of the degree of the polynomial to the slopes of essential surfaces of a knot. As motivated by the recent progress on the conjecture, we discuss a connection from the colored Jones polynomial of a knot to the normal surface theory of its complement. We give a map relating generators of a state-sum expansion of the polynomial to normal subsets of an ideal triangulation of the knot complement. Besides direct applications to the slope conjecture, we will also discuss potential applications to colored Khovanov homology.

(Note the unusual time: 4:30pm Shanghai, 10:30am Paris.)
Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, natural language processing, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoretical foundation for understanding the modelling, approximation or generalization ability of deep learning models with network architectures. Here we are interested in deep convolutional neural networks (CNNs) with convolutional structures. The convolutional architecture gives essential differences between deep CNNs and fully-connected neural networks, and the classical approximation theory for fully-connected networks developed around 30 years ago does not apply. This talk describes an approximation theory of deep CNNs associated with the rectified linear unit (ReLU) activation function. In particular, we prove the universality of deep CNNs, meaning that a deep CNN can be used to approximate any continuous function to an arbitrary accuracy when the depth of the neural network is large enough. We also show that deep CNNs perform at least as well as fully-connected neural networks for approximating general functions, and much better for approximating radial functions in high dimensions.

For historians, the thirties are arguably the most difficult decade of the twentieth century to capture. Crisis and consolidation coexisted. The talk will illustrate this general impression with respect to the international mathematical community and the development of mathematics.
A broad variety of events will be discussed, such as the beginning of the IAS in Princeton, two brilliant ICMs (1932 in Zurich and 1936 in Oslo), a memorable conference in Moscow; the axiomatization of probability, the beginning of Bourbaki, the rewriting of Algebraic Geometry; politically motivated migrations, the fate of international mathematical review journals, etc. etc.
Many of these events have left their mark on mathematics as we know it today.