Date/Time : 04 Apr 2023 16:00
Venue : via Zoom
Speaker : Johannes Flake, MPIM Bonn

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https://nus-sg.zoom.us/j/86446315086?pwd=amtmalVXUnA0bkRnT0haRmZjeGsrQT09

Meeting ID: 864 4631 5086
Passcode: 280937

I will explain a construction of categories with a tensor product using topological operations involving manifolds, or more precisely, cobordisms. Then I will show how to analyze categories like these using algebraic and representation-theoretic ideas. The categories in question include Deligne’s interpolation categories and Khovanov-Sazdanovic’s cobordism categories. We will discuss their Grothendieck rings and, if time permits, their monoidal centers.

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Date/Time : 03 Apr 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Xuhua He, Chinese University of Hong Kong

An invertible matrix is called totally positive if all its minors are positive. In 1994, Lusztig developed the theory of total positivity for arbitrary split real reductive groups and their flag manifolds. He further generalized the theory to arbitrary Kac-Moody groups in 2019. The theory of total positivity has found important applications in different areas: cluster algebras, higher Teichmuller theory, the theory of amplituhedron in physics, etc.

In this talk, we will discuss some remarkable combinatorial, geometric and representation-theoretic aspects of total positivity. This talk is based on some recent works with Huanchen Bao.

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Date/Time : 30 Mar 2023 10:00
Venue : via Zoom
Speaker : Xun Huan, University of Michigan

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https://nus-sg.zoom.us/j/9775606531?pwd=cVZ6eTRYS3ZzVUlFNXRuSlhMVjJSUT09

Meeting ID: 977 560 6531
Passcode: 123321

Experiments are indispensable for learning and developing models in engineering and science. A careful design of these often-expensive data-acquisition opportunities can be immensely beneficial. Simulation-based optimal experimental design, while leveraging a predictive model, offers a framework to systematically quantify and maximize the value of experiments. We focus on designing a finite sequence of experiments, seeking optimal design policies that can (a) adapt to newly collected data during the sequence (i.e. feedback) and (b) anticipate future changes (i.e. lookahead). We cast this sequential decision-making problem in a Bayesian setting with information-based utilities, and solve it numerically via policy gradient methods from reinforcement learning. In particular, we directly parameterize the policies and value functions by neural networks—thus adopting an actor-critic approach—and improve them using gradient estimates produced from simulated design and observation sequences. We demonstrate the overall method on a problem of optimal sensor movement for contaminant source inversion.

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Date/Time : 30 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Simon Eberle, Basque Center for Applied Mathematics, Spain

We will investigate an old problem posed by Isaac Newton on shapes that do not admit gravity in the cavity. We will see that it can be equivalently formulated as a question on the rigidity of global solutions of the obstacle problem, a question that recently has been answered in full generality.

This is joint work with A. Figalli, H. Shahgholian and G. S. Weiss.

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Date/Time : 30 Mar 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Song Jinfeng, National University of Singapore

Let G be a reductive group over an algebraically closed field with positive characteristic, and N be the unipotent radical of a Borel subgroup. The quantum Frobenius splitting introduced by Lusztig gives an algebraic construction for a Frobenius splitting of the unipotent group N. Another important construction on N is the cluster algebra structure established by Geiss-Leclerc-Schroer. In this talk, I will explain that the algebraic Frobenius splitting on N is compatible with its cluster structure. This is based on an ongoing work.

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Date/Time : 29 Mar 2023 17:00
Venue : S17 #05-12 (SR4)
Speaker : Ruofei Xie,

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

In the article “Highness Properties Close to PA-Completeness”, Greenberg, Miller, and Nies left the following questions:

–Is there a finite collection of functionals {Gamma_i:i<N} such that for every martingale M majorizing the optimal c.e. supermartingale m, Gamma_i(M) is a DNC_k function for some i<N?

–Is there a uniform way to compute a DNC_k function from a martingale M majorizing m whose initial capital is bounded by 1?

In the talk, I will try to answer these questions.

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Date/Time : 29 Mar 2023 11:00
Venue : S17 #04-06 (SR1)
Speaker : George Marinescu, Universität zu Köln

We introduce a probability measure on the space of square integrable holomorphic sections of a prequantum line bundle with the help of a Toeplitz operator and we study the behaviour of random sections for high tensor powers
of the bundle. We can recover the zero set of the symbol of the Toeplitz operator from this asymptotic.

The talk is based on joint work with Alexander Drewitz and Bingxiao Liu (arXiv:2302.08426).

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Date/Time : 29 Mar 2023 16:00
Venue : via Zoom
Speaker : Xizhi Liu, University of Warick

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https://nus-sg.zoom.us/j/81046462775?pwd=aDhaYnhQOTgybzdYS21yajlQdnBmZz09

Meeting ID: 810 4646 2775
Passcode: 23571113

We present a general approach for determining the maximum number of edges in an n-vertex r-uniform hypergraph that does not contain t+1 vertex-disjoint copies of F, when n is large, t = o(n), and F is nondegenerate. This extends old results of Simonovits and Moon on complete graphs.

Based on joint work with Jianfeng Hou, Heng Li, Long-Tu Yuan, and Yixiao Zhang.

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Date/Time : 23 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Tan Nguyen, University of California, Los Angeles

Self-attention is key to the remarkable success of transformers in sequence modeling tasks including many applications in natural language processing and computer vision. Like neural network layers, these attention mechanisms are often developed by heuristics and experience. In this talk, I present two novel mathematically principled frameworks to facilitate the process of designing efficient and robust transformer models with better expressivity.

In my first framework, I interpret self-attention in transformers as the Nadaraya-Watson estimator in a nonparametric regression setting. In the context of transformers, this Nadaraya-Watson estimator is constructed from the kernel density estimators (KDE) of the keys and queries in self-attention. However, such KDEs are not robust to contaminated samples. By viewing KDE as the solution to the kernel regression problem in a reproducing kernel Hilbert space, I adopt the state-of-the-art robust KDE methods based on, e.g., robust kernel regression and median-of-mean estimator, to design new robust self-attention mechanisms. The resulting family of robust self-attentions can be tailored to various transformer architectures.

In my second framework, I show that the self-attention corresponds to the support vector expansion derived from a support vector regression problem, whose primal formulation has the form of a neural network layer. Thus, I prove the primal-dual relationship between a self-attention layer and a neural network layer. Using my framework, I derive popular attention layers used in practice and propose two new attention mechanisms: 1) the Batch Normalized Attention (Attention-BN) derived from the batch normalization layer and 2) the Attention with Scaled Head (Attention-SH) derived from using less training data to fit the model. I empirically demonstrate the advantages of the Attention-BN and Attention-SH in reducing head redundancy, increasing the model’s accuracy, and improving the model’s efficiency in a variety of practical applications.

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