[Mini-Course] Metric Embeddings - What, How and Why? Professor Sylvester Eriksson-Bique (University of Jyväskylä)2024-02-21 to 2024-03-01
Title: Metric Embeddings - What, How and Why?
Speaker: Professor Sylvester Eriksson-Bique (University of Jyväskylä)
|Feb 21, Wednesday
|Feb 26, Monday
|Feb 28, Wednesday
|Mar 1, Friday
[Seminar] Different definitions of conformal dimension are (essentially) equal, Professor Sylvester Eriksson-Bique, University of Jyväskylä2024-01-22
Different definitions of conformal dimension are (essentially) equal,
Professor Sylvester Eriksson-Bique, University of Jyväskylä
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 University2023-08-29 to 2023-09-01
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.
*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.
Title: Lectures on Capacities
Speaker: Professor Daniel Spector, National Taiwan Normal University
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 University2023-06-19
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.
OIST Workshop | Main organizer: Xiaodan Zhou (Analysis on Metric Spaces) | OIST members are welcome to attend all scientific sessions.
[Seminar] Sharp Uncertainty Principles and their stability, Professor Nguyen Lam, Memorial University of Newfoundland2022-12-02
Abstract: The Heisenberg uncertainty principle, which is a fundamental result in quantum mechanics, and related inequalities such as the hydrogen and Hardy uncertainty principles, belong to the family of geometric inequalities known as the Caffarelli-Kohn-Nirenberg inequalities. In this talk, we discuss some recent results about the optimal uncertainty principles, Caffarelli-Kohn-Nirenberg inequalities, and their quantitative stability. The talk is based on recent joint works with C. Cazacu, J. Flynn and G. Lu.
The Vicsek set is a tree-like fractal on which neither analog of curvature nor differential structure exists, whereas the heat kernel satisfies sub-Gaussian estimates. I will talk about Sobolev spaces and scale invariant $L^p$ Poincar\'e inequalities on the Vicsek set. Several approaches will be discussed, including the metric approach of Korevaar-Schoen and the approach by limit approximation of discrete p-energies.
[Seminar] Whitney Extension and Lusin Approximation for Horizontal Curves in the Heisenberg Group, Professor Andrea Pinamonti, University of Trento2022-11-04
Abstract: Whitney extension results characterize when one can extend a mapping from a compact subset to a smooth mapping on a larger space. Lusin approximation results give conditions under which one can approximate a rough map by a smoother map after discarding a set of small measure. We first recall relevant results in the Euclidean setting, then describe recent work extending them to horizontal curves in the Heisenberg group. We focus on C^m curves.