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 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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Date/Time : 11 Sep 2024 16:30
Venue : S17 #06-11 (SR6)
Speaker : Yang Liu, National University of Singapore

Can we build a single large model for a wide range of PDE-related scientific learning tasks? Can this model generalize to new PDEs without any fine-tuning? In-context operator learning and the corresponding model In-Context Operator Networks (ICON) represent an initial exploration of these questions. We present a detailed methodology for solving PDE problems with ICON, and show how a single ICON model can make forward and reverse predictions for different equations with different strides, provided with appropriately designed data prompts. We show positive evidence for the questions above, through a study on 1D scalar nonlinear conservation laws, a family of PDEs with temporal evolution. In particular, we show that an ICON model trained on conservation laws with cubic flux functions can generalize well to some other flux functions of more general forms, without fine-tuning. We will also discuss different theoretical interpretations of in-context learning.

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Date/Time : 11 Sep 2024 15:00
Venue : S17 #05-11 (SR5)
Speaker : Yuxuan Wang, Tsinghua University

First examined by Riemann in 1860 in one-dimensional space, the Riemann problem has since been extensively studied and advanced by numerous scholars. Our talk will investigate the interaction of two approaching shocks in the context of spherical symmetry in three-dimensional space, presuming the fluid to be ideally barotropic. We will outline the proof of the local existence, uniqueness, and higher regularity of the solution to the shock interaction problem.

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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 : 09 Sep 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Ichiro Shimada, Hiroshima University

The Ap\’ery-Fermi pencil is a certain one-dimensional family of complex affine surfaces related to a Fermi surface in solid state physics. This pencil is also connected to Ap\’ery’s proof of the irrationality of the zeta value at s=3.

A general member of the Ap\’ery-Fermi pencil is birational to a K3 surface with Picard number 19. We calculate the automorphism group of this K3 surface by means of Borcherds’ method.

I also present some general algorithms that may be useful in computational algebraic geometry.

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Date/Time : 09 Sep 2024 16:00
Venue : S17 #05-11 (SR5)
Speaker : Jean-Bernard Lasserre, LAAS-CNRS

We briefly describe the Christoffel-Darboux (CD) kernel and the Christoffel function (CF). While well-known tools in orthogonal polynomials and approximation theory, it is only recently that CF’s properties have been shown to be very useful and powerful (as well as easy to implement) to help solve some problems

— in data analysis and datra mining where the underlying (empirical) measure is usually discrete, e.g. for outlier detection, support inference, density approximation, and

— in approximation of discontinuous functions.

Finally we will also briefly describe some (in the author’s opinion surprising) links of the CF with seemingly less disconnected areas (Pell equation, equilibrium measure, convex optimization and positivity certificates).

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