The problem of estimating an unknown vector from linear measurements has a long history in statistics, machine learning, and signal processing. Classical studies focus on the "n >> p" regime (#measurements >> #parameters), and more recent studies handle the "n << p" regime by exploiting low-dimensional structure such as sparsity or low-rankness. Such variants are commonly known as compressive sensing.
In this talk, I will overview recent methods that move beyond these simple notions of structure, and instead assume that the underlying vector is well-modeled by a generative model (e.g., produced by deep learning methods such as GANs). I will highlight algorithmic works that demonstrated up to 5-10x savings in the number of measurements over sparsity-based methods, and then move on to our theoretical work characterizing the order-optimal sample complexity in terms of quantities such as (i) the Lipschitz constant of the model, or (ii) the depth/width in a neural network model. I will also briefly highlight some recent results on non-linear observation models.

Link Floer homology of links in S^3 can be computed as the homology of a grid chain complex defined using grid diagrams. I will describe a construction of a CW spectrum whose cells correspond to the generators of the grid chain complex, and whose cellular chain complex is the grid chain complex (and therefore, the homology is link Floer homology). This is joint with Ciprian Manolescu.

The Robinson-Schensted (RS) correspondence is a famous bijection between permutations and pairs (P,Q) of standard tableaux of the same shape, called the RS partition. The RS partition and its conjugate record certain permutation statistics called Greene’s theorem statistics.
A box-ball system is a discrete dynamical system which can be thought of as a collection of time states. A permutation on n objects gives a box-ball system state by assigning its one-line notation to n consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau (not necessarily standard) called the soliton decomposition. The shape of the soliton decomposition is called the BBS partition. An exciting discovery (made in 2019 by Lewis, Lyu, Pylyavskyy, and Sen) is that the BBS partition and its conjugate record a localized version of Greene’s theorem statistics.
We will discuss a few new results:
(1) The Q tableau of a permutation completely determines the dynamics of the corresponding box-ball system.
(2) The permutations whose BBS partitions are L-shaped have steady-state time at most 1. This large class of permutations include column reading words and noncrossing involutions.
(3) If the soliton decomposition of a permutation is a standard tableau or if its BBS partition coincides with its RS partition, then its soliton decomposition and its P tableau are equal.
(4) Finally, we study the permutations whose P tableaux and soliton decompositions coincide and refer to them as “good". These “good” permutations are closed under consecutive pattern containment. Furthermore, we conjecture that the “good” Q tableaux are counted by the Motzkin numbers.
This talk is based on REU projects with Ben Drucker, Eli Garcia, Aubrey Rumbolt, Rose Silver (UConn Math REU 2020) and Marisa Cofie, Olivia Fugikawa, Madelyn Stewart, David Zeng (SUMRY 2021).

The Kaczmarz algorithm is an iterative method for solving linear systems of equations of the form Ax=y. Owing to its low memory footprint, the Kaczmarz algorithm has gained popularity for its practicality in applications to large-scale data, acting only on single rows of A at a time. In this talk, we discuss selecting rows of A randomly (Randomized Kaczmarz), selecting rows in a greedy fashion (Motzkin's Method), and selecting rows in a partially greedy fashion (Sampling Kaczmarz-Motzkin algorithm). Despite their variable computational costs, these algorithms have been proven to have the same theoretical upper bound on the convergence rate. Here we present an improvement upon previous known convergence bounds of the Sampling Kaczmarz-Motzkin algorithm, capturing the benefit of partially greedy selection schemes. Time permitting, we also will discuss an extension of the Kaczmarz algorithm to the setting where data takes on the form of a tensor and make connections between the new Tensor Kaczmarz algorithm and previously established algorithms. This presentation contains joint work with Jamie Haddock and Denali Molitor.

We'll discuss the background material for the main result of the following paper: https://arxiv.org/abs/2107.07351. We'll start by talking about quasi-split special unitary groups and the associated Steinberg groups. If we have time, we'll talk about algebraic rings. This will be the first in a sequence of two talks on this paper.

In this series of talks we show a necessary and sufficient condition for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Specifically, the n-dimensional continuous Hochschild cohomology vanishes if and only if every norm continuous n-linear map from the uniform Roe algebra to itself is equivalent to a weakly continuous n-linear map.
In our first talk we will begin defining and discussing derivations as they are an important building block of Hochschild cohomology. Motivated by the needs of mathematical physics and the study of one-parameter automorphism groups, it is interesting to study whether all derivations are inner (i.e. given by the commutator bracket) for a particular C*-algebra. In the 1970s, a complete solution to this problem was obtained in the separable case via the work of several authors. For non-separable C*-algebras the picture is murkier. Our main goal in this talk is to give a new class of examples that only have inner derivations: uniform Roe algebras, which are separable only in the trivial finite dimensional case. Uniform Roe algebras were originally introduced for index-theoretic purposes but are now studied for their own sake as a bridge between C*-algebra theory and coarse geometry, as well as having interesting applications to single operator theory. Lastly, we will briefly explain how the uniform Roe algebra only having inner derivations is equivalent to the first Hochschild cohomology vanishing.

A fundamental result in 3-manifold topology due to Lickorish and Wallace says that every closed oriented connected 3-manifold can be realized as surgery on a link in the 3-sphere. One may therefore ask: which 3-manifolds can be obtained by surgery on a link with a single component, i.e. a knot, in the 3-sphere? More specifically, one can ask: which 3-manifolds are obtained by zero surgery on a knot in the 3-sphere? In this talk, we give a brief outline of some known results to this question in the context of small Seifert fibered spaces. We then sketch a new method, using involutive Heegaard Floer homology, to show that certain 3-manifolds cannot be obtained by zero surgery on a knot in the three sphere. In particular, we produce a new infinite family of weight 1 irreducible small Seifert fibered spaces with first homology Z which cannot be obtained by zero surgery on a knot in the 3-sphere, extending a result of Hedden, Kim, Mark and Park.

Jenna Rajchgot observed that the Castelnuovo-Mumford regularity of matrix Schubert varieties is computed by the degrees of the corresponding Grothendieck polynomials. We give a formula for these degrees. Indeed, we compute the leading terms of the top degree pieces of Grothendieck polynomials and give a complete description of when two Grothendieck polynomials have the same top degree piece (up to scalars). Our formulas rely on some new facts about major index of permutations. (Joint work with David Speyer and Anna Weigandt.)

A compact hyper-Kahler manifold is a higher dimensional generalization of a K3 surface. An elliptic fibration of a K3 surface correspondingly generalizes into the so-called Lagrangian fibration of a compact hyper-Kahler manifold. It is known that an elliptic fibration of a K3 surface is always "self-dual" in a certain sense. This turns out to be not the case for higher-dimensional Lagrangian fibrations. In this talk, we will explicitly construct the dual of Lagrangian fibrations of all currently known examples of compact hyper-Kahler manifolds.
Passcode: MSUALG

In this talk I will discuss energy decay of solutions of the Damped wave equation. After giving an overview of classical results I'll focus on the torus with damping that does not satisfy the geometric control condition. In this setup properties of the damping at the boundary of its support determine the decay rate, however a general sharp rate is not known.
I will discuss damping which is 0 on a strip and vanishes either like a polynomial x^b or an oscillating exponential e^{-1/x} sin^2(1/x). Polynomial damping produces decay of the semigroup at exactly t^{-(b+2)/(b+3)}, while oscillating damping produces decay at least as fast as t^{-4/5+\delta} for any \delta>0. I will explain how these model cases are proved and how they direct further study of the general sharp rate.

The success of operator splitting techniques for convex optimization has led to an explosion of methods for solving large-scale and nonconvex optimization problems via convex relaxation. This success is at the cost of overlooking direct approaches to operator splitting that embrace some of the more inconvenient aspects of many model problems, namely nonconvexity, nonsmoothness and infeasibility. I will introduce some of the tools we have developed for handling these issues, and present sketches of the basic results we can obtain. The formalism is in general metric spaces, but most applications have their basis in Euclidean spaces. Along the way I will try to point out connections to other areas of intense interest, such as optimal mass transport.

Title: An introduction to Instanton Floer homology

Date: 09/16/2021

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

Contact: Danika Keala Van Niel ()

Instanton Floer homology is a three-manifold invariant connected to Donaldson theory introduced by Floer himself which turns out to be a powerful tool to study 3-manifolds. In today’s talk, I will first briefly review the (ordinary) Morse background and to introduce basic notions about connections. Then we will see the definition and basic properties of the Chern-Simons invariant, and a sketch of the construction of the instanton homology of a homology 3-sphere. Lastly, we will see some applications of the Instanton Floer homology and some other versions of Instanton Floer homology if time permits.

In this series of talks we show a necessary and sufficient condition for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Specifically, the n-dimensional continuous Hochschild cohomology vanishes if and only if every norm continuous n-linear map from the uniform Roe algebra to itself is equivalent to a weakly continuous n-linear map.
In our second talk we will continue discussing derivations as they are an important building block of Hochschild cohomology. Motivated by the needs of mathematical physics and the study of one-parameter automorphism groups, it is interesting to study whether all derivations are inner (i.e. given by the commutator bracket) for a particular C*-algebra. In the 1970s, a complete solution to this problem was obtained in the separable case via the work of several authors. For non-separable C*-algebras the picture is murkier. Our main goal in this talk is to give a new class of examples that only have inner derivations: uniform Roe algebras, which are separable only in the trivial finite dimensional case. Uniform Roe algebras were originally introduced for index-theoretic purposes but are now studied for their own sake as a bridge between C*-algebra theory and coarse geometry, as well as having interesting applications to single operator theory. We will then briefly explain how the uniform Roe algebra only having inner derivations is equivalent to the first Hochschild cohomology vanishing. Lastly, we will discuss the Hochschild cohomology in higher dimensions.

I will exhibit an action of the d-strand extended affine braid group on the top-dimensional positroid variety in the Grassmannian Gr(k,n), where d = gcd(k,n). Roger Casals and Honghao Gao showed that this action can be used to exhibit infinitely many distinct Lagrangian fillings of (most) Legendrian torus knots. The braid group action is compatible with the cluster variety structure on Gr(k,n). Finally, I will discuss joint work in progress with Bernhard Keller which interprets this action as a composition of spherical twists in bounded derived categories which additively categorify the cluster algebra.

Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, as a continued fraction, for a 14-parameter family of integer sequences and interpret these in terms of statistics on permutations and other combinatorial objects. Special cases include several classical and noncommutative probability laws, and a substantial subset of the orthogonalizing measures in the q-Askey scheme of orthogonal polynomials.
This continued fraction captures a variety of combinatorial sequences. In particular, it characterizes the moment sequences associated to the numbers of permutations avoiding (classical and vincular) patterns of length three. This connection between pattern avoidance and classical and noncommutative probability is among several consequences that generalize and unify previous results in the literature.
The fourteen combinatorial statistics further generalize to colored permutations, and, as an infinite family of statistics, to the k-arrangements: permutations with k-colored fixed points, introduced here. This is joint work with Natasha Blitvić, Lancaster University.

Take a square and consider the damped waves with boundary damping $a>0$ on the top side only. We will discuss my recent result implying that the energy of those waves must uniformly decay no faster than $t^{-1/2}$, and no slower than it. We will also discuss this result in the context of product manifolds where the transverse geometric control is sufficient but not necessary for such energy decay.
Zoom passcode: A*****-P**

Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This talk describes a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment problems. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over 10^14 entries.
This talk will highlight the ideas behind the algorithm in a streamlined setting. The insights include a careful problem formulation, design of a bespoke optimization method, and use of randomized matrix computations.
Joint work with Alp Yurtsever, Olivier Fercoq, Madeleine Udell, and Volkan Cevher. Based on arXiv 1912.02949 (Scalable SDP, SIMODS 2021) and other papers (SketchyCGM in AISTATS 2017, Nyström sketch in NeurIPS 2017).

Hypergraphs are generalizations of both graphs and simplicial complexes. They are often used to represent data for which graphs or simplices do not tell the whole story. As with many data structures, the new hotness is to do TDA (Topological Data Analysis) on hypergraphs. In this talk, I will introduce hypergraphs, why they are useful, and talk about their homology.
https://msu.zoom.us/j/91485321701
Meeting ID: 914 8532 1701
Passcode: SGTS

Tomita–Takesaki theory is a powerful (and often necessary) tool for studying von Neumann algebras that lack tracial states. It shows that any state on a von Neumann algebra $M$ automatically induces an action $\mathbb{R}\curvearrowright M$ (i.e. a non-commutative dynamical system) that then allows one to construct a crossed product von Neumann algebra $M\rtimes \mathbb{R}$. As the notation suggests, this crossed product is an analogue of semidirect products for groups, and it both contains $M$ and encodes the action via unitary elements. It turns out this crossed product always admits a trace (albeit an infinite one) and a dual action $\mathbb{R}\curvearrowright (M\rtimes \mathbb{R})$, and this structure was used by Alain Connes in 1973 to give a classification of the so-called type III von Neumann algebras. In this expository talk, I will provide an introduction to these ideas that does not require any previous experience with von Neumann algebras aside from some functional analysis (e.g. Hilbert spaces, bounded operators, and dual spaces).

The A_n type cluster variety, denoted by X_n, is the affine scheme associated to the upper cluster algebra of the A_n quiver (of n unfrozen vertices and one frozen vertex). The dual of X_n is denoted as Y_n, and is isomorphic but not canonically to X_n. For example, the variety X_1 is given by the equation {x y = 1 + q} where x, y are in \C and q is in \C^*. We prove that the homological mirror symmetry (HMS) conjecture for X_n, Y_n: the coherent sheaf category of Coh(X_n) is equivalent to the wrapped Fukaya category WFuk(Y_n), generalizing the known case for n=1 and 2. This is based on a recent work of Gammage-Le on HMS for truncated cluster variety, by putting back the 'truncated' part. This is work in progress, and is joint with Linhui Shen and Zhe Sun.

The prevalence of graph-based data has spurred the rapid development of graph neural networks (GNNs) and related machine learning algorithms. These methods extend convolutions to graphs either in the spatial domain as a localized averaging operator or in the spectral domain via the eigendecomposition of a suitable Laplacian. However, most popular GNNs have two limitations. i) The filters used in these networks are essentially low-pass filters (i.e. averaging operators). This leads to the so called ``oversmoothing problem'' and the loss of high-frequency information. ii) If the graph is directed, as is the case in many applications including citation, website, and traffic networks, these networks are unable to effectively encode directional information. In this talk, discuss how we can overcome these limitations via i) the graph scattering transform, which uses band-pass filters rather than low-pass, and ii) MagNet, a network designed for directed graphs based on a complex Hermitian matrix known as the magnetic Laplacian.