Analysis on Metric Spaces Seminar 2025 | Periodic homogenization for non-local stable-like operators, Professor Takashi Kumagai, Waseda University

Periodic homogenization for non-local stable-like operators

Professor Takashi Kumagai Waseda University

Abstract :

Homogenization has been a very active area of research in both PDE and probability theory for many years. In this talk, we will first review classical results on periodic homogenization for divergence form operators. We will then present our recent results on periodic homogenization for non-local, stable-like operators. Both qualitative and quantitative results will be discussed. Moreover, we will explore quantitative periodic homogenization on bounded domains, where the rate of convergence near the boundary slows down

Analysis on Metric Spaces Seminar 2025 | Recent development in bilinear rough singular integrals, Professor Bae Jun Park Sungkyunkwan University, Suwon, South Korea

Analysis on Metric Spaces Seminar 2025

Speaker: Bae Jun Park, Sungkyunkwan University, Suwon, South Korea

Title: Recent development in bilinear rough singular integrals

Abstract:

In this talk we will study the Lp1 × Lp2 → Lp boundedness for bilinear rough singular
integral operators LΩ associated with Ω ∈ L(log L)α(S2n−1) with mean value zero. We
will first review classical linear singular integrals and then study how the results have been
extended and developed in the bilinear setting, presenting the most recent results. If time
permits, general multilinear problems will be also discussed

【Seminar】 The Brascamp-Lieb inequality | Professor Neal Bez, Saitama University

Speaker : Professor Neal Bez, Saitama University

Title : The Brascamp-Lieb Inequality

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[Seminar] Growth estimates for \(p-\)harmonic Green functions on weighted \(R^n\) and metric spaces | Professor Jana　Björn, Linköping University and OIST TSVP Visiting Scholar

【Seminar】The Dirichlet problem and boundary regularity for nonlinear parabolic equations | Prof. Professor Anders Björn, Linköping University and OIST TSVP Visiting Scholar

Speaker:  Professor Anders Björn, Linkoping University and OIST TSVP Visiting Scholar

Title: The Dirichlet problem and boundary regularity for nonlinear parabolic equations    

Abstract: The p-parabolic equation \[ \partial_t u = \Delta_p u := \dvg(|\nabla u|^{p-2}\nabla u) \] is a nonlinear cousin of the classical heat equation. As such, it offers both difficulties and advantages compared with the heat equation. In the talk, we consider the Perron method for solving the Dirichlet problem for the p-parabolic equation in general bounded domains in $R^{n+1}$. Compared to space-time cylinders, such domains allow the space domain to change in time. Of particular interest will be boundary regularity for such domains, i.e. whether solutions attain their boundary data in a continuous way. Relations between regular boundary points and barriers will be discussed, as well as some peculiar examples and surprising phenomena related to boundary regularity. Towards the end I will discuss the same type of questions for two other nonlinear cousins of the heat equation, the porous medium equation \[ \partial_t u = \dvg(u^m) \] and the so-called normalized p-parabolic equation \[ \partial_t u = |\nabla u|^{2-p}\Delta_p u. \] The talk is based on collaborations with Jana Bj\"orn (Link\"oping), Ugo Gianazza (Pavia), Mikko Parviainen (Jyv\"askyl\"a) and Juhana Siljander (Jyv\"askyl\"a).

[Mini-Course] Metric Embeddings - What, How and Why? Professor Sylvester Eriksson-Bique (University of Jyväskylä)

Title: Metric Embeddings - What, How and Why?
Speaker: Professor Sylvester Eriksson-Bique (University of Jyväskylä)

[Seminar] Different definitions of conformal dimension are (essentially) equal, Professor Sylvester Eriksson-Bique, University of Jyväskylä

Different definitions of conformal dimension are (essentially) equal,

Professor Sylvester Eriksson-Bique, University of Jyväskylä

Abstract:

I want to tell you about something that came out of OIST. Last May OIST hosted a work-
shop in analysis, random walks and potential theory on metric spaces, which showcased
some exciting developments in these areas. I was fortunate to participate, and the topics
of the conference prompted discussions with another participant Mathav Murugan, who
told about an open problem regarding the conformal dimension of the Sierpinski carpet.
The question asks, if different definitions of this notion are equal. Through discussions at
the workshop, I solved this question. I will explain the problem, the notions of conformal
dimension, and the crucial tool: a new notion of discrete modulus. There are roughly two
approaches classically to define discrete modulus, and this new approach lies roughly in
between the two — in such a way, that it can benefit from good estimates in both of the
worlds. The talk will be fairly mathematical, but I will try to give definitions of the main
concepts and some motivation.

[Mini-course]: Metric geometry on the configuration space | Professor Kohei Suzuki, Durham University

Title:  Metric geometry on the configuration space

Speaker: Professor Kohei Suzuki, Durham University

Abstract: The configuration space Y(X) over a base space X is the space of all Radon point measures on X. The space Y(X) has been studied in many fields such as algebraic geometry (e.g., the hyperplane arrangement with X=Grassmannian), algebraic topology (e.g., the braid group with X=Euclidean plane), representation theory (e.g., the L^2-representation of diffeomorphism groups on manifolds X), statistical physics (e.g., interacting particle diffusions with X=Euclidean space). In this series of lectures, I will focus on the metric geometry of Y(X) induced by the 2-Wasserstein distance. As Y(X) does not support the volume doubling property, the established theory of PI spaces does not apply. The goal of the series is to elaborate on

• Metric geometry on Y(X);
• Curvature analysis on Y(X);
• Applications to infinite particle diffusion processes (including e.g. infinite particle Dyson Brownian motion);
• Open questions.

Lecture 1 | L4E01 August 29, 2023  10:00 - 11:00

Lecture 2 | L4E01 August 30, 2023  10:00 - 11:00

Lecture 3 | L4F01 August 31, 2023  10:00 - 11:00

Lecture 4 |  B700 September 1, 2023  10:00 - 11:00

Register here

*After registering, you will receive a confirmation email containing information about joining the meeting.

This lecture be accessible to senior math undergraduate and anyone above the level. 

[Mini-course]: Lectures on Capacities | Professor Daniel Spector, National Taiwan Normal University

Title: Lectures on Capacities

Speaker: Professor Daniel Spector, National Taiwan Normal University

Zoom registration: https://oist.zoom.us/meeting/register/tJErce-tpj0jGNN5TM3gwMnRnGHaY5lNZ5Qk#/registration

Lecture 1   Tuesday, June 20    10 am

Lecture 2  Wednesday, June 21st 10 am

Lecture 3   Thursday, June 22nd 10 am

Zoom link: TBA

[Seminar] Critical Sobolev Spaces and Subspaces of BMO, Professor Daniel Spector, National Taiwan Normal University

Abstract: It is well-known that functions in critical Sobolev spaces embed into the space of functions of bounded mean oscillation (BMO) originating in the work of John and Nirenberg. Less well-known is the fact that they actually embed into BMO on subspaces of every smaller dimension. In this talk we introduce a class of spaces which are finer targets of these critical Sobolev embeddings than BMO that capture this phenomena, which we term beta-dimensional BMO. Interestingly, these spaces also gives an answer to the question of which BMO functions admit restrictions in BMO of subspaces. The key tool is a capacitary analogue of the John-Nirenberg inequality for the Hausdorff content, obtained recently in a joint work with You-Wei Chen.