[Seminar] Cantor Julia sets, Professor Vyron Vellis, University of Tennessee

Title: Cantor Julia sets

Speaker: Vyron Vellis,University of Tennessee

Abstract: The Julia set of a meromorphic function is the set of “chaotic points”, i.e., the set of all points that have no neighborhood under which the iterates of the function form a normal family. In absence of a rich holomorphic function theory in higher dimensions, the right substitute in complex dynamics are uniformly quasiregular (UQR) maps. Which Cantor sets can appear as Julia sets of UQR maps? How (topologically) complex can such sets be? This talk is based on joint works with Alastair Fletcher and Daniel Stoertz.

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

Speaker: 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 )

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

[Seminar] Conservation law for harmonic mappings in higher dimensions

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. 

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[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.

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[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.

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[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