Date/Time : 18 Oct 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ruobing Zhang, Princeton University

This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics.

My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress and propose several open questions in the field.

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Date/Time : 18 Oct 2023 13:00
Venue : S17 #05-11 (SR5)
Speaker : Chandrima Kayal, Indian Statistical Insitute, Kolkata

Suppose we want to cover all the vertices of the n-dimensional Boolean cube using a minimum number of hyperplanes. Observe that this can be done using only two hyperplanes. Moreover, no single hyperplane can cover all the vertices. Now what if we want to cover only a subset of the Boolean cube? For example, suppose we want to cover all the vertices except one, viz. the origin. One can observe that n hyperplanes are sufficient. But can we do better? The celebrated covering result by Alon and Füredi shows that at least n hyperplanes will be required to cover all the vertices of the n-dimensional cube leaving out exactly one vertex.

We shall discuss different versions of this covering problem, and give a generalization of Alon and Füredi’s covering result for any symmetric subset of the Boolean cube. Also, we shall show a strict separation between the size of a polynomial cover and a hyperplane cover.

This work was jointly done with Arijit Ghosh, Soumi Nandi and S. Venkitesh.

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Date/Time : 17 Oct 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Frank Stephan, National University of Singapore

Floyd and Knuth studied in their 1990 paper addition machines which are machines which can add, subtract and compare integers (<,=,>) in unit time; also the reading or writing of an integer is in unit time. They showed that multiplying and dividing can be done in linear time with six registers and asked in their Open Problem (2) whether this bound can be broken; furthermore, they asked in Open Problem (5) of their paper whether there is a register machine which can output in subquadratic time the powers of two occurring in the binary representation of an integer.

The talk answers both questions affirmatively and presents the key ideas and programs to witness this.

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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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