Tue, 17 Jan 2023
16:00
C3

Partial Pontryagin duality for actions of quantum groups on C*-algebras

Kan Kitamura
(University of Tokyo)
Abstract

In view of Takesaki-Takai duality, we can go back and forth between C*-dynamical systems of an abelian group and ones of its Pontryagin dual by taking crossed products. In this talk, I present a similar duality between actions on C*-algebras of two constructions of locally compact quantum groups: one is the bicrossed product due to Vaes-Vainerman, and the other is the double crossed product due to Baaj-Vaes. I will explain the situation by illustrating the example coming from groups. If time permits, I will also discuss its consequences in the case of quantum doubles.

Tue, 10 May 2022

15:30 - 16:30
L4

Cohomological χ-independence for Higgs bundles and Gopakumar-Vafa invariants

Tasuki Kinjo
(University of Tokyo)
Abstract

In this talk, I will introduce the BPS cohomology of the moduli space of Higgs bundles on a smooth projective curve of rank r and degree d using cohomological Donaldson-Thomas theory. The BPS cohomology and the intersection cohomology coincide when r and d are coprime, but they are different in general. We will see that the BPS cohomology does not depend on d. This is a generalization of the Hausel-Thaddeus conjecture to non-coprime case. I will also explain that Toda's χ-independence conjecture (and hence the strong rationality conjecture) for local curves can be proved in the same manner. This talk is based on a joint work with Naoki Koseki and another joint work with Naruki Masuda.

Thu, 03 Jun 2021
14:00
Virtual

Distributing points by minimizing energy for constructing approximation formulas with variable transformation

Ken'ichiro Tanaka
(University of Tokyo)
Abstract


In this talk, we present some effective methods for distributing points for approximating analytic functions with prescribed decay on a strip region including the real axis. Such functions appear when we use numerical methods with variable transformations. Typical examples of such methods are provided by single-exponential (SE) or double-exponential (DE) sinc formulas, in which variable transformations yield single- or double-exponential decay of functions on the real axis. It has been known that the formulas are nearly optimal on a Hardy space with a single- or double-exponential weight on the strip region, which is regarded as a space of transformed functions by the variable transformations.

Recently, we have proposed new approximation formulas that outperform the sinc formulas. For constructing them, we use an expression of the error norm (a.k.a. worst-case error) of an n-point interpolation operator in the weighted Hardy space. The expression is closely related to potential theory, and optimal points for interpolation correspond to an equilibrium measure of an energy functional with an external field. Since a discrete version of the energy becomes convex in the points under a mild condition about the weight, we can obtain good points with a standard optimization technique. Furthermore, with the aid of the formulation with the energy, we can find approximate distributions of the points theoretically.

[References]
- K. Tanaka, T. Okayama, M. Sugihara: Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces, IMA Journal of Numerical Analysis Vol. 37 (2017), pp. 861-904.

- K. Tanaka, M. Sugihara: Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimization, IMA Journal of Numerical Analysis Vol. 39 (2019), pp. 1957-1984.

- S. Hayakawa, K. Tanaka: Convergence analysis of approximation formulas for analytic functions via duality for potential energy minimization, arXiv:1906.03133.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Tue, 03 Dec 2019
14:30
L1

Estimation of ODE models with discretization error quantification

Takeru Matsuda
(University of Tokyo)
Abstract

We consider estimation of ordinary differential equation (ODE) models from noisy observations. For this problem, one conventional approach is to fit numerical solutions (e.g., Euler, Runge–Kutta) of ODEs to data. However, such a method does not account for the discretization error in numerical solutions and has limited estimation accuracy. In this study, we develop an estimation method that quantifies the discretization error based on data. The key idea is to model the discretization error as random variables and estimate their variance simultaneously with the ODE parameter. The proposed method has the form of iteratively reweighted least squares, where the discretization error variance is updated with the isotonic regression algorithm and the ODE parameter is updated by solving a weighted least squares problem using the adjoint system. Experimental results demonstrate that the proposed method improves estimation accuracy by accounting for the discretization error in a data-driven manner. This is a joint work with Yuto Miyatake (Osaka University).

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