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 : 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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Date/Time : 22 Mar 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : Kihara Takayuki,

In this talk, we examine the topos-theoretic aspect of the degrees of (computable) unsolvability. One of the main interpretations of constructive mathematics is Kleene’s realizability interpretation, which, as is well known, can be relativized by an oracle. In this sense, an oracle can be a factor that causes a change in a model of constructive mathematics.

Let us review this observation from another point of view: there is a topos, called the effective topos, based on Kleene’s realizability interpretation. And relativizing the realizability interpretation to an oracle yields a subtopos of the effective topos. Thus, the structure of oracles, i.e., the structure of the degrees of unsolvability, is expected to be closely related to the structure of the subtoposes of the effective topos.

In this talk, we give a complete correspondence, in a strict sense, between the structure of the degrees of unsolvability and the structure of subtoposes of the effective topos (or its relatives).

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Date/Time : 22 Mar 2023 15:00
Venue : via Zoom
Speaker : Huy Tuan Pham, Stanford University

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Meeting ID: 810 4646 2775
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A graph is Ramsey if its largest clique or independent set is of size logarithmic in the number of vertices. While almost all graphs are Ramsey, there is still no known explicit construction of Ramsey graphs. We discuss recent progress on finding these “dark matter” of graphs. Along the way, we study some fundamental problems in additive combinatorics, and discover that group structure is superfluous for these problems. Joint work with David Conlon, Jacob Fox, and Liana Yepremyan.

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Date/Time : 21 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Jie Shen, Purdue University

Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and/or positivity/maximum principle preserving. It is desirable, sometimes necessary, to preserve these structures in a numerical scheme.

I will present some recent advances on using the scalar auxiliary variable (SAV) approach and Lagrange multiplier approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, only require solving decoupled linear equations with constant coefficients at each time step, and can achieve higher-order accuracy.

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Date/Time : 16 Mar 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Elden Elmanto, Harvard University

Motivic cohomology is an invariant of smooth schemes which simultaneously “glues” together its etale cohomology and Chow groups. I will give a gentle introduction to the theory, its motivation, its genesis, as well as some its highlights and achievements. All this leads up to recent joint work with Matthew Morrow on its extension to singular schemes, which uses prismatic cohomology in a crucial way.

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Date/Time : 15 Mar 2023 10:00
Venue : via Zoom
Speaker : Cosmin Pohoata, Institute for Advanced Study

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Meeting ID: 810 4646 2775
Passcode: 23571113

Suppose we are given matchings M_1,….,M_N of size t in some r-uniform hypergraph, and let us think of each matching having a different color. How large does N need to be (in terms of t and r) such that we can always find a rainbow matching of size t? This problem was first introduced by Aharoni and Berger, and has since been studied by several different authors. While interesting for its own sake in the context of finding transversals in Latin Squares (e.g. the Ryser–Brualdi–Stein Conjecture), it is perhaps not too surprising that this question is also connected with different sides of additive combinatorics. In this talk, we will survey some of these connections, discuss some recent results that more or less completely answer the original question when one of the parameters t or r is fixed, and then end with some (fairly arbitrary) speculations/open problems. Joint work with Lisa Sauermann and Dmitrii Zakharov.

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Date/Time : 15 Mar 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : Liu Shixiao ,

In the first half of the talk I shall be proving the properness of Mathias forcing with lower density requirement. In the second half of the talk I shall demonstrate the difficulty in applying this method (and some other commonly used methods) to Silver forcing with lower density requirement.

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Date/Time : 10 Mar 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jie Ma, University of Science and Technology of China

A fundamental theorem of Rademacher from 1941 led to the study of supersaturation problems of graphs, which aim to count the minimum number of copies of a given graph F among all graphs with n vertices and m edges. This is closely related to a central concept in Extremal Graph Theory — the Turán number of F, which denotes the maximum number of edges in an n-vertex graph which does not contain F as a subgraph. Famous results of Erdős, and Lovász and Simonovits determine the minimum number of cliques K_r in graphs whose number of edges exceed the Turán number of K_r. Subsequent works of Mubayi as well as Pikhurko and Yilma extend these classical results from cliques to color-critical graphs, a rich family playing important roles in extremal problems. In this talk, we will discuss supersaturation problems beyond color-critical graphs and investigate natural enumerative parameters.

Our results go beyond the previous results and show that supersaturation problems for general graphs can be rather complicated. Among others, we disprove a conjecture of Mubayi. Joint work with Long-Tu Yuan.

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