Georgia Institute of Technology College of Sciences

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. 

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).

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

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.