Date/Time : 25 Jan 2023 17:00
Venue : via Zoom
Speaker : Leonardo Mainardi,

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

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

When the Canonical Ramsey’s Theorem by Erdös and Rado is applied to regressive functions, one obtains the Regressive Ramsey’s Theorem by Kanamori and McAloon. Taylor proved a “canonical” version of Hindman’s Theorem, analogous to the Canonical Ramsey’s Theorem. We introduce the restriction of Taylor’s Canonical Hindman’s Theorem to a subclass of the regressive functions, the λ-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman’s Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that this same principle reduces the well-ordering-preservation principle for base-ω exponentiation by a uniform reduction.

Add to calendar :

Date/Time : 25 Jan 2023 15:30
Venue : S17 #04-05 (SR2)
Speaker : Wang Tianqi, National University of Singapore

Minsky shows that there is an open set of PSL(2,C) characters of a free group F_n, namely the primitive-stable character, which is strictly larger than the convex-cocompact character and on which the action of Out(F_n) is properly discontinuous. In this talk, we introduce the notion of simple Anosov representations of closed surface groups and give an analogue of Minsky’s theorem, that is, the set of simple Anosov representations of the fundamental group of a closed hyperbolic surface into a higher rank Lie group is open in the character variety, which is strictly larger than the set of Anosov representations, with the mapping class group action properly discontinuous. This is joint work with Nicholas Tholozan.

Add to calendar :

Date/Time : 20 Jan 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Sergiu Klainerman, Princeton University

The question whether black holes are real can be approached mathematically by addressing basic issues concerning their rigidity, stability and how they form in the first place. I will review these and focus on recent results concerning the nonlinear stability of slowly rotating Kerr black holes.

Add to calendar :

Date/Time : 19 Jan 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Chenyan Wu, University of Melbourne

Let $\pi$ be a cuspidal automorphic representation of a classical group or a metaplectic group. We show an exact relation between two invariants associated to $\pi$, one called the lowest occurrence index of $\pi$ which is an invariant arising from theta lifts and the other the location of the maximal positive pole of an Eisenstein series attached to $\pi$ and a character. As an application, We show that certain global Arthur packets cannot contain cuspidal automorphic representations.

Add to calendar :

Date/Time : 18 Jan 2023 17:00
Venue : S17 #04-05 (SR2) via Zoom
Speaker : Bakhadyr Khoussainov, University of Auckland

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

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

We aim to describe the isomorphism types of algorithmically presented structures in the language of the first order logic.

This is one of the key research topics related to the expressive power of the first order logic. The focus is on two classes of structures. The first class is the class of structures for which the positive atomic diagrams are computably enumerable.

We call these structures positive structures. The second class is the class of structures for which the negative atomic diagrams are computably enumerable. We call these structures negative structures. We investigate definability of positive and negative structures by sets of sentences quantified with Existential, Universal, Existential-Universal and Universal-Existential quantifiers using expansions of languages.

Add to calendar :

Date/Time : 18 Jan 2023 15:30
Venue : S17 #04-05 (SR2)
Speaker : Yap Jit Wu, Harvard University

Let f,g be two rational maps on P^1. Then Baker-Demarco, and independently Yuan-Zhang, showed that either they have exactly the same set of preperiodic points, or only finitely many in common. More recently, Demarco-Krieger-Ye showed the existence of an uniform bound on number of common preperiodic points for the quadratic family z^2 + c and also the Lattes family. In this talk, we will discuss the ideas behind these two results along with some work in progress by the author on related questions.

Add to calendar :

Date/Time : 17 Jan 2023 10:00
Venue : via Zoom
Speaker : Tan Nguyen, University of California, Los Angeles

Join Zoom Meeting
https://nus-sg.zoom.us/j/82637054452?pwd=ZkpEVllwN3g1YUNYRHIwMXF3TmUzdz09

Meeting ID: 826 3705 4452
Passcode: 332211

Designing deep learning models for practical applications, including those in computer vision, natural language processing, and mathematical modeling, is an art that often involves an expensive search over candidate architectures. In this talk, I present novel frameworks to facilitate the process of designing efficient and robust deep learning models with better representation capacity via three principled approaches: optimization, differential equation, and statistical modeling.

From an optimization viewpoint, I develop a gradient descent analogy of the recurrent cell in the recurrent neural network (RNN). I then integrate the momentum used for accelerating gradient dynamics into the recurrent cell, which results in the MomentumRNN. MomentumRNN can alleviate the vanishing gradient problem and accelerate training by choosing the appropriate momentum coefficients (Efficiency).
From a differential equation approach, I present a random walk interpretation of graph neural networks (GNNs), revealing a potentially inevitable over-smoothing phenomenon. Based on this random walk viewpoint of GNNs, I then propose the graph neural diffusion with a source term (GRAND++) that overcomes the over-smoothing issue and achieves better accuracy in low-labeling rate regimes (Robustness).

Using statistical modeling as a tool, I show that the attention in transformer models can be derived from solving a nonparametric kernel regression problem. I then propose the FourierFormer, a new class of transformers in which the softmax kernels are replaced by the novel generalized Fourier integral kernels. The generalized Fourier integral kernels can automatically capture the dependency of the features of data and remove the need to tune the covariance matrix (Representability).

Add to calendar :

Date/Time : 12 Jan 2023 10:00
Venue : S17 #04-05 (SR2)
Speaker : Stefan Dawydiak, Max Planck Institute for Mathematics in Bonn (MPIM)

In the 1980s, Kazhdan-Lusztig and independently Ginzburg showed that the affine Hecke algebra $H$ can be realized as the Grothendieck group of the category of coherent sheaves on the Steinberg variety. The correct coherent Hecke category was however found only later, by Bezrukavnikov. A crucial ingredient in this coherent categorification of $H$ is viewing the Steinberg variety as a derived scheme.

On the other hand, Lusztig’s asymptotic Hecke algebra $J$ is a certain based ring receiving a map from $H$, via which its representation theory is closely related to that of $H$. In the 1990s, Xi showed that the best-understood direct summand $J_0$ of $J$ can be realized as the Grothendieck group of coherent sheaves on the square of the flag variety, and Ginzburg gave a K-theoretic definition of the map from $H$ to $J_0$. In this talk we will explain how to upgrade this realization to a monoidal category receiving a natural monoidal functor from the coherent Hecke category. A crucial ingredient is again to consider a derived enhancement, this time of the flag variety.

Add to calendar :

Date/Time : 12 Jan 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Van Tien Nguyen, National Taiwan University

In this talk I will present constructive examples and numerical illustrations of blowup solutions to the Keller-Segel system in R^d.

• L^1-critical (d = 2): There exist finite time blowup solutions that are of Type II with finite mass. Blowup rates are quantized according to the spectrum of a linearized operator in the self-similar setting. There is also the case of multiple collapsing blowup solutions formed by a collision of single-solutions.

• L^1$-supercritical (d >= 3): We exhibit finite time blowup solutions that are completely unrelated to the self-similar scale, in particular, they are of Type II with finite mass. Interestingly, the radial blowup profile is linked to the traveling wave of the 1D viscous Burgers equation. There also exist solutions that blow up in finite time with infinite mass. The solution is asymptotically self-similar with a logarithmic correction to its profile for d = 3,4. We found such an asymptotic profile can be either radial or completely non-radial.

The talk is based on results obtained in collaboration with C. Collot (Paris Cergy), T-E. Ghoul (NYU Abu Dhabi), N. Masmousdi (NYU), N. Nouaili (Paris Dauphine), H. Zaag (Paris Nord).

Add to calendar :