### Past Events

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

2022年11月18日 (金) 10:00

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年11月4日 (金) 17:00

**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年10月21日 (金) 9:00 〜 10:00**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.

### [Mini-course] BV functions in Carnot groups | Speaker Dr. Sebastiano Nicolussi Golo, University of Jyväskylä

2022年8月22日 (月) 15:00 〜 2022年8月24日 (水) 17:00Speaker: Dr. Sebastiano Nicolussi Golo, University of Jyväskylä

Title: BV functions in Carnot groups

Language: English, no interpretation.

### [Catch-All Mathematical Colloquium] Professor Motoko Kotani (Tohoku University )

2022年7月22日 (金) 15:00 〜 17:00### The colloquium will be held once a month online. Each event consists of a one-hour talk on mathematics followed by a one-hour diversity panel discussion session. Please register before July 18, 5 pm. Click here to register!

**Speaker**: Professor Motoko Kotani, Tohoku University

**Title**: Discrete geometric analysis and its application

**Abstract:**Discrete geometric analysis is an attempt to discretize geometric analysis. Mathematics is often said “a common language of science”. As our world consists of atoms, which we consider as discrete objects, developing language to describe discrete objects, their geometric structures in particular, is important. I would like to discuss our challenge to establish discrete geometric analysis and its application to other science.

Part II

Have fun in interacting with people from different interests.

### [Seminar] Conservation law for harmonic mappings in higher dimensions

2022年7月1日 (金) 10:00 〜 11:00### Speaker: Professor Changyu Guo, Shangdong University

**Title**: Conservation law for harmonic mappings in higher dimensions

**Abstract**:

It has been a longstanding open problem to find a direct conservation law for harmonic mappings into manifolds. In the late 1980s, Chen and Shatah independently found a conservation law for weakly harmonic maps into spheres, which can be interpreted by Noether's theorem. This leads to Helein's celebrated regularity theorem on weakly harmonic maps from surfaces. For general target manifolds, Riviere discovered a direct conservation law in two dimension in 2007, allowing him to solve two well known conjectures of Hildebrandt and Heinz. As observed by Riviere-Struwe in 2008, due to lack of Wente's lemma, Riviere's approach does not extend to higher dimensions. In a recent joint work with Chang-Lin Xiang, we successfully found a conservation law, in the spirit of Riviere, for a class of weakly harmonic maps (around regular points) into general closed manifolds in higher dimensions.

### [Catch-All Mathematical Colloquium] The level-set mean curvature flow equation versus the total variation flow equation,Yoshikazu Giga (University of Tokyo )

2022年5月12日 (木) 16:00 〜 18:00Title: The level-set mean curvature flow equation versus the total variation flow equation

Abstract: The level-set mean curvature flow equation has been introduced to track an evolving hypersurface by its mean curvature after it develops singularities. A level-set of a solution of the level-set mean curvature flow equation moves its mean curvature. The total variation flow equation is often used to remove noise from images. Although these two equations look similar, analytic properties are quite different; the former equation is a local equation while the latter is a nonlocal equation. In this talk, we compare these two equations as well as a few applications.

**Discussion Theme (for the 2nd part of the event) : **

How to collaborate with researchers other than mathematicians

The colloquium will be held once a month online. Each event consists of a one-hour talk on mathematics followed by a one-hour diversity panel discussion session. Please register before May , 5 pm. Click here to register!

### [Seminar] Variational problems with gradient constraint, Professor Xiao Zhong, University of Helsinki

2022年5月10日 (火) 16:00 〜 17:00### Speaker: Professor Xiao Zhong, University of Helsinki

Title: Variational problems with gradient constraint

### Abstract:

#### I will talk about three different classes of variational problems with gradient constraint.

They arise from elastic-plastic torsion, hypersurfaces in the Lorentz-Minkowski spaces with

given mean curvature and dimer models.

### [Seminar] Sub-Gaussian heat kernel bounds and singularity of energy measures for symmetric diffusions,Professor Naotaka Kajino (Kyoto University)

2022年4月22日 (金) 10:00 〜 11:00### Abstract:

This talk will present the result of a joint work with Mathav Murugan(University of British Columbia) that, for a symmetric diffusion on a complete locally compact separable metric space, two-sided sub-Gaussian heat kernel bounds imply the singularity of the energy measures with respect to the reference measure.

For self-similar (scale-invariant) diffusions on self-similar fractals, the singularity of the energy measures is known to hold in many cases by Kusuoka (1989, 1993), Ben-Bassat, Strichartz and Teplyaev (1999),

Hino (2005), and Hino and Nakahara (2006), but these results heavily relied on the self-similarity of the space.

It was conjectured, and had remained open for the last two decades to prove, that the singularity of the energy measures should follow, without assuming the self-similarity, just from two-sided sub-Gaussian

heat kernel bounds of the same form as those for diffusions on typical self-similar fractals. The main result of this talk answers this conjecture affirmatively.

The first half of the talk will be devoted to a brief introduction to self-similar diffusions (and their associated Dirichlet forms) on self-similar fractals and to sub-Gaussian heat kernel bounds for symmetric diffusions, so that the talk will (hopefully) be accessible even to those without prior knowledge about diffusions on fractals.

### [Seminar] Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case, Juha Kinnunen, Aalto University

2022年3月8日 (火) 16:00This talk discusses a generalized class of supersolutions, so-called \(p\)-supercaloric functions, to the parabolic \(p\)-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood in the slow diffusion case \(p>2\), but little is known in the fast diffusion case \(1