Seminars and Colloquia Schedule

Shimura operators and interpolation polynomials

Series
Algebra Seminar
Time
Monday, September 30, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Songhao ZhuGeorgia Tech

This talk starts at 1pm rather than the usual time.

The late Goro Shimura proposed a question regarding certain invariant differential operators on a Hermitian symmetric space. This was answered by Sahi and Zhang by showing that the Harish-Chandra images of these namesake operators are specializations of Okounkov's BC-symmetric interpolation polynomials. We prove, in the super setting, that the Harish-Chandra images of super Shimura operators are specializations of certain BC-supersymmetric interpolation polynomials due to Sergeev and Veselov. Similar questions include the Capelli eigenvalue problems which are generalized to the quantum and/or super settings. This talk is based a joint work with Siddhartha Sahi.

Existence of optimal flat ribbons

Series
Geometry Topology Seminar
Time
Monday, September 30, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matteo RaffaelliGeorgia Tech

We revisit the classical problem of constructing a developable surface along a given Frenet curve $\gamma$ in space. First, we generalize a well-known formula, introduced in the literature by Sadowsky in 1930, for the Willmore energy of the rectifying developable of $\gamma$ to any (infinitely narrow) flat ribbon along the same curve. Then we apply the direct method of the calculus of variations to show the existence of a flat ribbon along $\gamma$ having minimal bending energy. Joint work with Simon Blatt.

Exploring Conditional Computation in Transformer models

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 30, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and ONLINE
Speaker
Xin WangGoogle Research

Transformer (Vaswani et al. 2017) architecture is a popular deep learning architecture that today comprises the foundation for most tasks in natural language processing and forms the backbone of all the current state-of-the-art language models. Central to its success is the attention mechanism, which allows the model to weigh the importance of different input tokens. However, Transformers can become computationally expensive, especially for large-scale tasks. To address this, researchers have explored techniques for conditional computation, which selectively activate parts of the model based on the input. In this talk, we present two case studies of conditional computation in Transformer models. In the first case, we examine the routing mechanism in the Mixture-of-Expert (MoE) Transformer models, and show theoretical and empirical evidence that the router’s ability to route intelligently confers a significant advantage to MoE models. In the second case, we introduce Alternating Updates (AltUp), a method to take advantage of increased residual stream width in the Transformer models without increasing the computation cost.

 

Speaker's brief introduction: Xin Wang is a research engineer in the Algorithms team at Google Research. Xin finished his PhD in Mathematics at Georgia Institute of Technology before coming to Google. Xin's research interests include efficient computing, memory mechanism for machine learning, and optimization.

The talk will be presented online at

 https://gatech.zoom.us/j/93087689904

Proof of the Goldberg-Seymour conjecture (Guantao Chen)

Series
Graph Theory Seminar
Time
Tuesday, October 1, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guantao ChenGeorgia State University

The Goldberg-Seymour Conjecture asserts that if the chromatic index $\chi'(G)$ of a loopless multigraph $G$ exceeds its maximum degree $\Delta(G) +1$, then it must be equal to another well known lower bound $\Gamma(G)$, defined as

$\Gamma(G) = \max\left\{\biggl\lceil  \frac{ 2|E(H)|}{(|V (H)|-1)}\biggr\rceil \ : \  H \subseteq G \mbox{ and } |V(H)| \mbox{ odd }\right\}.$

 

In this talk, we will outline a short proof,  obtained recently with  Hao, Yu, and Zang. 

Bifurcation for hollow vortex desingularization

Series
PDE Seminar
Time
Tuesday, October 1, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ming ChenUniversity of Pittsburgh

A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; we can think of it as a spinning bubble of air in water. In this talk, we present a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. The machinery simultaneously treats the translating, rotating, and stationary regimes. Through global bifurcation theory, we further obtain maximal curves of solutions that continue until the onset of a singularity. As specific examples, we obtain the first existence theory for co-rotating hollow vortex pairs and stationary hollow vortex tripoles, as well as a new construction of Pocklington’s classical co-translating hollow vortex pairs. All of these families extend into the non-perturbative regime, and we obtain a rather complete characterization of the limiting behavior along the global bifurcation curve. This is a joint work with Samuel Walsh (Missouri) and Miles Wheeler (Bath).

Smooth Discrepancy and Littlewood's Conjecture

Series
Analysis Seminar
Time
Wednesday, October 2, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Niclas TechnauUniversity of Bonn

Given x in $[0,1]^d$, this talk is about the fine-scale distribution of the Kronecker sequence $(n x mod 1)_{n\geq 1}$.
After a general introduction, I will report on forthcoming work with Sam Chow.
Using Fourier analysis, we establish a novel deterministic analogue of Beck’s local-to-global principle (Ann. of Math. 1994),
which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation.
This opens up a new avenue of attack for Littlewood’s conjecture.

Introduction to Bergman geometry

Series
Geometry Topology Seminar
Time
Wednesday, October 2, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jihun YumGyeongsang National University in South Korea

The Poincaré metric on the unit disc $\mathbb{D} \subset \mathbb{C}$, known for its invariance under all biholomorphisms (bijective holomorphic maps) of $\mathbb{D}$, is one of the most fundamental Riemannian metrics in differential geometry.

In this presentation, we will first introduce the Bergman metric on a bounded domain in $\mathbb{C}^n$, which can be viewed as a generalization of the Poincaré metric. We will then explore some key theorems that illustrate how the curvature of the Bergman metric characterizes bounded domains in $\mathbb{C}^n$ and more generally, complex manifolds. Finally, I will discuss my recent work related to these concepts. 

Harmonic Functions and Beyond

Series
School of Mathematics Colloquium
Time
Thursday, October 3, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yanyan LiRutgers University

A harmonic function of two variables is the real or imaginary part of an analytic function. A harmonic function of $n$ variables is a function $u$ satisfying

$$

\frac{\partial^2 u}{\partial x_1^2}+\ldots+\frac{\partial^2u}{\partial x_n^2}=0.

$$

We will first recall some basic results on harmonic functions: the mean value property, the maximum principle, the Liouville theorem, the Harnack inequality, the Bocher theorem, the capacity and removable singularities. We will then present a number of more recent results on some conformally invariant elliptic and degenerate elliptic equations arising from conformal geometry. These include results on Liouville theorems, Harnack inequalities, and Bocher theorems.

Minimum Norm Interpolation Meets The Local Theory of Banach Spaces

Series
Stochastics Seminar
Time
Thursday, October 3, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gil KurETH
Minimum-norm interpolators have recently gained attention primarily as an analyzable model to shed light on the double descent phenomenon observed for neural networks.  The majority of the work has focused on analyzing interpolators in Hilbert spaces, where typically an effectively low-rank structure of the feature covariance prevents a large bias. More recently, tight vanishing bounds have also been shown for isotropic high-dimensional data  for $\ell_p$-spaces with $p\in[1,2)$, leveraging the sparse structure of the ground truth.
This work takes a first step towards establishing a general framework that connects generalization properties of the interpolators to well-known concepts from high-dimensional geometry, specifically, from the local theory of Banach spaces.

Shock formation in weakly viscous conservation laws

Series
PDE Seminar
Time
Friday, October 4, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cole GrahamUniversity of Wisconsin–Madison

The compressible Euler equations readily form shocks, but in 1D the inclusion of viscosity prevents such singularities. In this talk, we will quantitatively examine the interaction between generic shock formation and viscous effects as the viscosity tends to zero. We thereby obtain sharp rates for the vanishing-viscosity limit in Hölder norms, and uncover universal viscous structure near shock formation. The results hold for large classes of viscous hyperbolic conservation laws, including compressible Navier–Stokes with physical rather than artificial viscosity. This is joint work with John Anderson and Sanchit Chaturvedi.

CANCELLED - Tight minimum colored degree condition for rainbow connectivity

Series
Graph Theory Seminar
Time
Friday, October 4, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiaofan YuanArizona State University

Let $G = (V,E)$ be a graph on $n$ vertices, and let $c : E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$.  In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $\delta^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$.
In this paper, we show that the same bound for $\delta^c(G)$ implies that any two vertices are connected by a rainbow path. This is joint work with Andrzej Czygrinow.


This is to note that the graph theory seminar for Friday the 4th has been CANCELLED. This is due to the cancellation of the AMS sectional meeting due to Hurricane Helene. I apologize for any inconvenience. We intend to reschedule the talk for next semester.