Date/Time : 16 Oct 2023 16:00
Venue : S16 #04-35
Speaker : Lin Yi-Hsuan, National Yang Ming Chiao Tung University

We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce the nonlocal parabolic inverse problems to the corresponding local inverse problems by using the extension problem for nonlocal parabolic equations. This talk is based on joint works with Ching-Lung Lin and Gunther Uhlmann.

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Date/Time : 13 Oct 2023 15:00
Venue : S17 #05-11 (SR5)
Speaker : Yi Shan, Ecole Normake Superieur Paris

Arthur’s conjectures on endoscopic classification enable us to determine whether a representation with a given Arthur parameter occurs in the automorphic spectrum. Chenevier and Renard use Arthur’s results to classify level one automorphic representations for definite orthogonal groups over Q with small ranks and (conjecturally) for anisotropic G_2. In this talk, I will give an overview of their method and then talk about my recent work for the anisotropic semisimple Q-group of type F_4. If time permits, we will discuss the relation of my results with exceptional theta correspondences.

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Date/Time : 13 Oct 2023 16:30
Venue : S17 #05-11 (SR5)
Speaker : Ed Karasiewicz, National University of Singapore

The uniqueness of Whittaker models plays an important role in the representation theory of linear reductive groups due to its relation to L-functions. However such uniqueness fails in general for nonlinear covering groups. We investigate this failure of uniqueness through the Gelfand-Graev representation, which is the dual of the Whittaker space.

Using the pro-p Iwahori-Hecke algebra, we describe the Iwahori-fixed vectors in the Gelfand-Graev representation of covering groups as a module over the Iwahori-Hecke algebra, generalizing work of Barbasch-Moy and Chan-Savin for linear groups. As an application we 1) compute the dimension of the space of Whittaker models for constituents of unramified principal series for regular and unitary characters; 2) describe a connection to quantum affine Schur Weyl duality in the case of covers of GL(r).

This is joint work with Fan Gao and Nadya Gurevich.

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Date/Time : 12 Oct 2023 14:00
Venue : S16 #03-04
Speaker : Wang Peng, University of Michigan

Over the past decade, deep learning has proven to be a highly effective method for extracting meaningful features from high-dimensional data. This work attempts to unveil the mystery of feature learning in deep networks. Specifically, for a multi-class classification problem, we explore how the features of training data evolve across the intermediate layers of a trained neural network. We investigate this problem based on simple deep linear networks trained on linearly separable data, and we analyze how the output features in each layer concentrate around the means of their respective classes. Remarkably, when the deep linear network is trained using gradient descent from a small orthogonal initialization, we theoretically prove that the features exhibit a linear decay in the measure of within-class feature variability as we move from shallow to deep layers. Moreover, our extensive experiments not only validate our theoretical findings numerically but also reveal a similar pattern in deep nonlinear networks which well aligns with recent empirical studies.

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Date/Time : 12 Oct 2023 15:30
Venue : S17 #06-11 (SR6)
Speaker : Luo Yujie, National University of Singapore

A curve on a surface is called a negative curve if its normal bundle has negative degree. We study negative curves on P^2 blowing up some general points, and we deduce some bounds and inequalities on the numerical invariants of those curves. This is an ongoing project with Runcao Lyu.

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Date/Time : 11 Oct 2023 14:30
Venue : S17 #05-11 (SR5)
Speaker : Chen Shibing, University of Science and Technology of China

Optimal partial transportation is an important generalization of the classical optimal transport problem, and it arises in many applications, such as image retrieval. Mathematically, the problem was first studied by Caffarelli and McCann. They established the existence, uniqueness, and Holder regularity of the free normal and the map near the free boundary while leaving open the question of their higher regularity. In this talk, we will introduce our recent work in this direction.

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Date/Time : 10 Oct 2023 10:30
Venue : S17 #04-06 (SR1)
Speaker : Charles Cheung, NVIDIA AI Technology Center

In this talk, I will talk about a few directions and use cases of AI for Science. PINNs and neural operator are used for solving many engineering problems that involves differential equations. We will walk through the basic concept of PINNs and neural operator and introduce NVIDIA modulus, an SDK for training PINNs and neural operator. We will also share a recent project about transfer learning.

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Date/Time : 06 Oct 2023 15:00
Venue : S17 #05-11 (SR5)
Speaker : Gordan Savin, University of Utah

I will review basic definitions of Fell topology and results of Milicic and Tadic, then discuss several results and problems involving the Fell topology on unitary generic representations:
Does a sequence of unitary generic representations contains a generic representation as a cluster point? Is a generic discrete series isolated in the unitary dual?

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Date/Time : 06 Oct 2023 16:30
Venue : S17 #05-11 (SR5)
Speaker : Bryan Wang Peng Jun, National University of Singapore

Recent work of Ben-Zvi, Sakellaridis and Venkatesh (BZSV) predicts the existence of a duality in the relative Langlands program, indexed by a class of Hamiltonian spaces with reductive group action. I will first give a rough introduction to the BZSV-duality, focusing on motivations and a series of motivating examples. Then, I will explain a local incarnation of the BZSV-duality in the setting of branching problems, and how the generalised Whittaker (or Gelfand-Graev) models serve as the prototypical example. Finally, I will present a new family of examples of BZSV-duality, which can so far be verified in this local setting. Time permitting, I will explain the proof, which proceeds via results of Gomez and Zhu, and an interesting interpretation in the framework of Hamiltonian spaces. This is based on my thesis work supervised by Wee Teck Gan.

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