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.

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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.

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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.

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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)

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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.

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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.

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Date/Time : 11 Sep 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Kihara Takayuki, Nagoya University

The theories of degrees of unsolvability and realizability interpretation both have long histories, having both been born in the 1940s. S. C. Kleene was a key figure who led the development of both theories. Despite having been developed by the same person, there seems to have been little deep mixing of these theories until recently. In this talk, we will reconstruct the theory of degrees of unsolvability from the perspective of realizability theory.

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Date/Time : 11 Sep 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Tran Chieu-Minh, National University of Singapore

A locally compact group is nonunimodular if it has no Haar measure (i.e. every left Haar measure m is not right invariant). Considering doubling ratio m(AA)/m(A) initially looks strange on such exotic setting but turns out to be crucial even for the purpose of studying doubling on unimodular locally compact groups. In this talk I will discuss this issue further including a recently published result in this direction. ( Based on joint works with Yifan Jing and Ruixiang Zhang).

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Date/Time : 11 Sep 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Binyong Sun, IASM, Zhejiang University

After introducing some basic notions in the representation theory of classical Lie groups, I will explain three results in this theory: the multiplicity one theorem for classical groups, the conservation relation for local theta correspondence, and the non-vanishing hypothesis at infinity for Rankin-Selberg convolutions. Some relevant or consequent results will also be explained.

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