This Week's Seminars and Colloquia

Inference, Computation, and Games

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 20, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://bluejeans.com/457724603/4379
Speaker
Florian SchaeferGT CSE

Note the hybrid mode. The speaker will be in person in Skiles 005.

In this talk, we develop algorithms for numerical computation, based on ideas from competitive games and statistical inference. 

 

In the first part, we propose competitive gradient descent (CGD) as a natural generalization of gradient descent to saddle point problems and general sum games. Whereas gradient descent minimizes a local linear approximation at each step, CGD uses the Nash equilibrium of a local bilinear approximation. Explicitly accounting for agent-interaction significantly improves the convergence properties, as demonstrated in applications to GANs, reinforcement learning, and computer graphics.

 

In the second part, we show that the conditional near-independence properties of smooth Gaussian processes imply the near-sparsity of Cholesky factors of their dense covariance matrices. We use this insight to derive simple, fast solvers with state-of-the-art complexity vs. accuracy guarantees for general elliptic differential- and integral equations. Our methods come with rigorous error estimates, are easy to parallelize, and show good performance in practice.

Mitsumatsu's Liouville domains are stably Weinstein

Series
Geometry Topology Seminar
Time
Monday, September 20, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Austin ChristianGeorgia Tech

In 1995, Mitsumatsu constructed a large family of Liouville domains whose topology obstructs the existence of a Weinstein structure.  Stabilizing these domains yields Liouville domains for which the topological obstruction is no longer in effect, and in 2019 Huang asked whether Mitsumatsu's Liouville domains were stably homotopic to Weinstein domains.  We answer this question in the affirmative.  This is joint work-in-progress with J. Breen.

Geometric equations for matroid varieties

Series
Algebra Seminar
Time
Tuesday, September 21, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ashley K. WheelerGeorgia Tech

Each point x in Gr(r,n) corresponds to an r×n matrix A_x which gives rise to a matroid M_x on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets {y∈Gr(r,n)|M_y=M_x} form a stratification of Gr(r,n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals I_x of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of I_x geometrically when the combinatorics of the matroid is sufficiently rich. 

The feasible region of induced graphs

Series
Graph Theory Seminar
Time
Tuesday, September 21, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xizhi LiuUniversity of Illinois at Chicago

Fix a graph $F$. A classical problem in extremal graph theory asks about how many induced copies of $F$ can a graph with edge density $\rho$ have? The only case in which we know the asymptotic solution is when $F$ is a complete graph, and it was solved completely only recently by Reiher using the flag algebra machinery. We will consider the other cases and show some results when $F$ is a complete bipartite graph or a complete graph minus one edge. Many interesting related open problems will also be introduced. Joint work with Dhruv Mubayi and Christian Reiher.

Geometric equations for matroid varieities

Series
Research Horizons Seminar
Time
Wednesday, September 22, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ashley WheelerGeorgia Institute of Technology

Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.

Mapping Class Group of 4-Manifolds

Series
Geometry Topology Student Seminar
Time
Wednesday, September 22, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anubhav MukherjeeGeorgia Tech

One interesting question in low-dimensional topology is to understand the structure of mapping class group of a given manifold. In dimension 2, this topic is very well studied. The structure of this group is known for various 3-manifolds as well (ref- Hatcher's famous work on Smale's conjecture). But virtually nothing is known in dimension 4. In this talk I will try to motivate why this problem in dimension 4 is interesting and how it is different from dimension 2 and 3. I will demonstrate some "exotic" phenomena and if time permits, I will talk a few words on my upcoming work with Jianfeng Lin. 

Tropical intersection theory I

Series
Algebra Student Seminar
Time
Friday, September 24, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Trevor GunnGeorgia Tech

This is the first part of a two part introduction to tropical intersection theory. The first part will review some of the classical theory. We will mostly focus on the parts of the classical theory that have counterparts in the tropical theory but we may also cover some elements of the classical theory which do not have tropical analogues.

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An atomic matrix norm regularizer for sparse phase retrieval and PCA

Series
ACO Student Seminar
Time
Friday, September 24, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Andrew McraeGeorgia Tech ECE

Stream online at https://bluejeans.com/520769740/

We present a mixed atomic matrix norm that, when used as regularization in optimization problems, promotes low-rank matrices with sparse factors. We show that in convex lifted formulations of sparse phase retrieval and sparse principal component analysis (PCA), this norm provides near-optimal sample complexity and error rate guarantees. Since statistically optimal sparse PCA is widely believed to be NP-hard, this leaves open questions about how practical it is to compute and optimize this atomic norm. Motivated by convex duality analysis, we present a heuristic algorithm in the case of sparse phase retrieval and show that it empirically matches existing state-of-the-art algorithms.

Whitney Towers, Higher Order Intersections, and Tree-Valued Invariants, Part 2

Series
Geometry Topology Working Seminar
Time
Friday, September 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Miriam KuzbaryGeorgia Tech

In this pair of talks I will survey some of the machinery developed by Conant, Schneiderman, and Teichner to study Whitney towers, and their applications to the study of knot and link concordance. Whitney towers can be thought of as measuring the failure of the Whitney trick in dimension 4 and can be used, in a sense, to approximate slice disks. The talks will be based on various papers of Schneiderman, Conant-Schneiderman-Teichner, Cochran-Orr-Teichner and lecture notes by those authors. 

Mathematical approaches to Imaging and data

Series
SIAM Student Seminar
Time
Friday, September 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Sung-Ha KangSchool of Math, Georgia Tech,

 

I will talk about introduction to mathematical image processing, and cover how numerical PDE can be used in data understanding.  This talk will present some of variational/PDE-based methods for image processing, such as denoising, inpainting, colorization.  If time permits, I will introduce identification of differential equation from given noisy data.