### Past Events

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

2024-02-21 to 2024-03-01Title: Metric Embeddings - What, How and Why?

Speaker: Professor Sylvester Eriksson-Bique (University of Jyväskylä)

Lecture 1 | Feb 21, Wednesday | 10:00-12:00 | L4F01 |

Lecture 2 | Feb 26, Monday | 14:00-16:00 | L4F01 |

Lecture 3 | Feb 28, Wednesday | 14:00-16:00 | L4F01 |

Lecture 4 | Mar 1, Friday | 10:00-12:00 | L4F01 |

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

2024-01-22# 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

2023-08-29 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.

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

2023-06-20 to 2023-06-22### 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

2023-06-19

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.

### [Hybrid] OIST Workshop "Potential theory and random walks in metric spaces"

2023-05-30 to 2023-06-02OIST Workshop | Main organizer: Xiaodan Zhou (Analysis on Metric Spaces) | OIST members are welcome to attend all scientific sessions.

OIST campus will be closed on Thursday and Friday due to the typhoon. All talks will be online on Thursday and Friday. You can obtain a zoom link from here.

### [Seminar] Sharp Uncertainty Principles and their stability, Professor Nguyen Lam, Memorial University of Newfoundland

2022-12-02Abstract: 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.

### [Seminar] Poincare inequalities on the Vicsek set, Professor Chen Li, Louisiana State University

2022-11-18

Abstract:

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.

Zoom: https://oist.zoom.us/meeting/register/tJEpdu-uqTwrGdFV5IrA0woMhvhlxVa_5ttw

### [Seminar] Whitney Extension and Lusin Approximation for Horizontal Curves in the Heisenberg Group, Professor Andrea Pinamonti, University of Trento

2022-11-04

**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.

**Zoom:** https://oist.zoom.us/meeting/register/tJMpduGtqzooG9PA85LMRCwswpo5ODm2nU49

### [Seminar] Korevaar-Schoen-Sobolev spaces and critical exponents on metric measure spaces, Professor Fabrice Baudoin, University of Connecticut

2022-10-21**Title**: Korevaar-Schoen-Sobolev spaces and critical exponents on metric measure spaces

**Speaker:**Fabrice Baudoin, University of Connecticut

**Abstract:**We will review some of the recent developments in the theory of Korevaar-Schoen-Sobolev spaces. While this theory is equivalent to that of Cheeger and Shanmugalingam if the space supports a Poincare inequality, it offers new perspectives in situations, like fractals, where such inequalities are not available.