Date/Time : 03 Oct 2024 17:00
Venue : S17 #05-12 (SR4)
Speaker : Meng Fai Lim, Central China Normal University

In this talk, we begin reviewing the non-commutative Iwasawa theory as developed by Coates, Fukaya, Kato, Sujatha and Venjakob which is concerned with attaching suitable characteristic elements to modules of arithmetic interest (for instance, Selmer groups). By their construction, these characteristic elements are realized as element in an appropriate localized K_1 group. We then introduce a notion of “taking modulo p” of these characteristic elements and discuss some of its properties. This is an ongoing work with Chao Qin.

Add to calendar :

Date/Time : 03 Oct 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Dihua Jiang, University of Minnesota

I will discuss relations between non-abelian Fourier transforms and the Langlands conjecture on automorphic L-functions and associated local factors in the framework of Braverman-Kazhdan-Ngo.

Add to calendar :

Date/Time : 02 Oct 2024 14:00
Venue : via Zoom
Speaker : Jing Yu, Fudan University

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

Meeting ID: 810 4646 2775
Passcode: 23571113

In 1995, Levin and Linial, London, and Rabinovich conjectured that every connected graph $G$ of polynomial growth admits an injective homomorphism to the $n$-dimensional grid graph for some $n$. Moreover, they conjected that if every ball of radius $r$ in $G$ contains at most $O(r^\rho)$ vertices, then one can take $n = O(\rho)$. Krauthgamer and Lee confirmed the first part of this conjecture and refuted the second in 2007. By constructing some finite expander graphs, they showed best possible upper bound on $n$ is $O(\rho \log \rho)$.

Prompted by these results, Papasoglu asked whether a graph $G$ of polynomial growth admits a coarse embedding into a grid graph. We give an affirmative answer to this question. Moreover, it turns out that the dimension of the grid graph only needs to be linear in the asymptotic growth rate of $G$, which confirms the original Levin–Linial–London–Rabinovich conjecture “on the large scale.”

Besides, we find an alternative proof of the result of Papasoglu that graphs of polynomial growth rate $\rho < \infty$ have asymptotic dimension at most $\rho$.

Furthermore, our proof works in the Borel setting and shows that Borel graphs of polynomial growth rate $\rho < \infty$ have Borel asymptotic dimension at most $\rho$.

(This is joint work with Anton Bernshteyn.)

Add to calendar :

Date/Time : 02 Oct 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Hideo Kojima, Niigata University

A cuspidal curve means a projective curve whose singular points are locally irreducible. The study of rational cuspidal plane curves has had a long and interesting history. In the first half of the talk, I will introduce some of recent progress on the study. Then, I will give some results on rational cuspidal curves on Q-homology projective planes. Here, a Q-homology plane is a normal projective complex surface with quotient singularities having the same rational homology as the projective plane. Some results on rational cuspidal plane curves will be generalized.

Add to calendar :

Date/Time : 25 Sep 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Hsuan-Tien Lin, National Taiwan University

Complementary-Label Learning (CLL) is a weakly supervised learning paradigm that is claimed to be useful in situations where collecting true labels is expensive. This talk begins by walking the audience through two advances in CLL. The first work discovered that the CLL loss design from Unbiased Risk Estimator (URE) suffers from high variance in gradient estimation. To address this, a novel surrogate complementary loss (SCL) framework is proposed, which reduces variance and improves gradient alignment, mitigating the overfitting issue. The second work introduces a new approach to CLL by reducing the problem to estimating the probability of complementary classes. This framework sidesteps the limitations of traditional methods and improves robustness in noisy environments, offering a broader perspective for both deep and non-deep models. Empirical results demonstrate the efficacy of both approaches in improving CLL performance. Finally, the speaker will share some of the ongoing attempts in making CLL more realistic, including some first-hand experience in collecting real-world datasets and releasing an open-source library.

Add to calendar :

Date/Time : 23 Sep 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : David Schwein, University of Bonn

Supercuspidal representations are the elementary particles in the representation theory of reductive p-adic groups. Constructing such representations explicitly, via (compact) induction, is a longstanding open problem, solved when p is large. When p is small, the remaining supercuspidals are expected to have an arithmetic source: wildly ramified field extensions. In this talk I’ll discuss ongoing work joint with Jessica Fintzen that identifies a second, Lie-theoretic, source of new (tame!) supercuspidals: special features of root systems at very small primes.

Add to calendar :

Date/Time : 18 Sep 2024 12:00
Venue : via Zoom
Speaker : Peter van Hintum, Institute for Advanced Study

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

Meeting ID: 810 4646 2775
Passcode: 23571113

The Brunn-Minkowski inequality is a foundational result in convex geometry asserting that if sets A and B in R^n have the same volume and t is some number in (0,1), then |tA+(1-t)B|>|A|. We’ll prove the following folklore conjecture on the stability of the Brunn-Minkowski inequality that if in fact |tA+(1-t)B|<(1+d) |A|, then there is a common convex hull K containing A and B so that |K|<(1+ O(sqrt(d)))|A| and |co(A)|<(1+O(d))|A|. We’ll additionally describe how this result neatly fits in the additive combinatorics study of sets with small sumset. (based on joint work with Alessio Figalli and Marius Tiba)

Add to calendar :

Date/Time : 18 Sep 2024 16:45
Venue : S17 #04-04 (SR3)
Speaker : Le Quy Thuong,

Since 1995, motivic integration has been a powerful tool in algebraic geometry and other branches of mathematics. In particular, it has many important applications to singularity theory. For instance, Denef-Loeser around 2000 gave a breakthrough point of view in the study of singularities, by introducing the so-called motivic Milnor fiber, with the philosophy that this is a motivic incarnation of the classical Milnor fiber. One shows that many singularity invariants can be easily recovered from motivic zeta function and motivic Milnor fiber employing an appropriate Hodge realization. Furthermore, there are important problems concerning singularity theory such as monodromy conjecture, the integral identity conjecture, and the Thom-Sebastiani theorem that are waiting for new methods in motivic integration to have a solution.

In this talk, we will describe some surprising interactions between motivic integration, model theory and singularity theory that lead to our proofs for the integral identity conjecture, and the motivic Thom-Sebastiani theorem, as well as other applications to singularities. The talk will avoid technical aspects and emphasize key ideas in motivic integration and singularity theory, which may be friendly to a general audience.

Note that this week the seminar starts 15 minutes earlier than usual.

Add to calendar :

Date/Time : 12 Sep 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Kyo Nishiyama, Aoyama Gakuin University

Let $ G = Sp_{2n}(\R) $ (or $ U(n, n) $) and $ P $ a maximal parabolic subgroup stabilizing a Lagrangian subspace (Siegel parabolic subgroup). Let $ H \simeq GL_n(\R) $ be a Levi subgroup of $ P $, which is also a symmetric subgroup of $ G $. We choose a parabolic subgroup $ Q \subset H $.

Let $ X = H/Q \times G/P $ be a real double flag variety on which $ H $ acts diagonally. We are interested in the geometry of $ X $. For example, there are only finitely many $ H $-orbits on $ X $, so that it is of finite type.

In this talk, we introduce integral kernel operators which intertwine degenerate principal series of $ G = Sp_{2n}(\R) $ and $ H = GL_n(\R) $. For the kernel of the operator, we use relative invariants associated with a prehomogeneous vector space naturally arising from the geometry of $ X $. The integrals have complex parameters, and at certain integral parameters we get finite dimensional representations as images.

Add to calendar :