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

Title: 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

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.

Please click here to register

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

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.

Please click here to register

[Catch-All Mathematical Colloquium] Scaling limits of random walks on random graphs: An electrical resistance approach, David Croydon (Kyoto University)

In describing properties of disordered media, physicists have long been interested in the behaviour of random walks on random graphs that arise in statistical mechanics, such as percolation clusters and various models of random trees. Random walks on random graphs are also of interest to computer scientists in studies of complex networks. In ‘critical’ regimes, many of the canonical models exhibit large-scale fractal properties, which means it is often a challenge to describe their geometry, let alone the associated random walks. In this talk, I will describe an approach suitable for understanding various ‘low-dimensional’ models of random walks on random graphs that builds on the deep connections that exist between electrical networks and stochastic processes.

Part II Discussion Theme:

Working in different places, and especially in different countries, naturally leads one to draw comparisons. Through such, one learns more about the working cultures of each. After some brief general musings on this topic, I plan to share some of my experiences from the UK of working on a departmental committee that was responsible for staff welfare (including work-life balance and gender equality).

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 April 7, 5 pm. Click here to register!

[Mini-course] A brief introduction to branched optimal transport | Professor Jun Kitagawa, Michigan State University

Speaker: Professor Jun Kitagawa, Michigan State University

Title: A brief introduction to branched optimal transport

Abstract:

The optimal transport (also known as Monge-Kantorovich) problem is a classical optimization problem which has recently become the focus of much research with connections to various fields such as PDEs, geometry, and applications. In particular, it provides an effective way to metrize the space of probability measures on a given metric space. However, there is an alternate approach to metrizing such spaces using so called branched optimal transport. Branched optimal transport is based on the classical Gilbert-Steiner problem, later adapted by Qinglan Xia, and in contrast to the Monge-Kantorovich approach tends to yield branching structures. In this series of lectures I will introduce the basics of branched optimal transport and discuss some of the known results in the literature.

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

This 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

[Mini-course] Curvature and Optimal transport | Professor Asuka Takatsu, Tokyo Metropolitan University

Speaker: Professor Asuka Takatsu, Tokyo Metropolitan University

Title: Curvature and Optimal transport

Abstract

In this series of lectures, I first review the notion of curvature (Gaussian curvature and Ricci curvature).
In particular, I recall some comparison theorems (Toponogov's triangle comparison theorem, Bishop--Gromov volume comparison theorem etc).
Then I introduce a generalized notion of curvature in non-smooth spaces.

[Catch-All Math Colloquium] Monge-Ampère equations related to optimal transport and geometric optics, Jun Kitagawa, Michigan State University

Monge-Ampère type equations are fully nonlinear elliptic partial differential equations that arise when considering the deformation of volume induced by some kind of transformation. In this talk I will discuss two such cases, the optimal transport problem, and geometric optics problems. The former discusses the most efficient way of transporting some resource to another location, and the second is a simplified model for optical instruments (such as lenses or mirrors) in which light is treated as a particle rather than a wave. I will attempt to focus more on heuristics rather than technical details; no knowledge of PDEs is assumed. A portion of this talk is based on joint work with N. Guillen.

Please register before February 17, 5 pm. Click here to register!

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. February speaker is Jun Kitagawa from Michigan State University.

Catch-All Mathematical Colloquium

The colloquium will be held once a month. It will be held online for the time being. Each event consists of a one-hour talk on mathematics followed by a one-hour diversity panel discussion session. Please register before January 14, 5 pm. Click here to register!

Analysis on Metric Spaces Seminar