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.
[Seminar] Whitney Extension and Lusin Approximation for Horizontal Curves in the Heisenberg Group, Professor Andrea Pinamonti, University of Trento2022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.
[Seminar] Korevaar-Schoen-Sobolev spaces and critical exponents on metric measure spaces, Professor Fabrice Baudoin, University of Connecticut2022年10月21日 (金) 9:00 〜 10:00
[Mini-course] BV functions in Carnot groups | Speaker Dr. Sebastiano Nicolussi Golo, University of Jyväskylä2022年8月22日 (月) 15:00 〜 2022年8月24日 (水) 17:00
Speaker: Dr. Sebastiano Nicolussi Golo, University of Jyväskylä
Title: BV functions in Carnot groups
Language: English, no interpretation.
Please register before July 18, 5 pm. Click here to register!
Speaker: Professor Motoko Kotani, Tohoku University
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.
Have fun in interacting with people from different interests.
Speaker: Professor Changyu Guo, Shangdong University
Title: Conservation law for harmonic mappings in higher dimensions
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 〜 18:00
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 May , 5 pm. Click here to register!
[Seminar] Variational problems with gradient constraint, Professor Xiao Zhong, University of Helsinki2022年5月10日 (火) 16:00 〜 17:00
Speaker: Professor Xiao Zhong, University of Helsinki
Title: Variational problems with gradient constraint
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
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 University2022年3月8日 (火) 16:00
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