[Catch-All Mathematical Colloquium] Scaling limits of random walks on random graphs: An electrical resistance approach, David Croydon (Kyoto University)2022-04-11
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).
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 University2022-03-10 to 2022-03-14
Speaker: Professor Jun Kitagawa, Michigan State University
Title: A brief introduction to branched optimal transport
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 University2022-03-08
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 University2022-03-08 to 2022-03-10
Speaker: Professor Asuka Takatsu, Tokyo Metropolitan University
Title: Curvature and Optimal transport
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 University2022-02-22
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!
Jun KitagawaMichigan State University
Speaker: Professor Mario Bonk, UCLA
Title: Fractals and the dynamics of Thurston maps