Many mathematical and computational analyses have been performed for deterministic partial differential equations (PDEs) that have perfectly known input data. However, in reality, many physical and engineering problems involve some level of ...

In the early 90's, physicists Bershadsky-Cecotti-Ooguri-Vafa conjectured that the analytic torsion was the counterpart in complex geometry of the counting problem of elliptic curves in Calabi-Yau threefolds. It seems that this conjecture is ...

Anomalous diffusion phenomenon has been observed in many natural systems, from the signalling of biological cells, to the foraging behaviour of animals, to the travel times of contaminants in groundwater. In this talk, I will first discuss t...

Elliptic curves defined over the rationals satisfy two finiteness properties; its group of rational points is a finitely generated abelian group and it has only finitely many points with integral coordinates. Bhargava and his collaborators e...

We will show that the averaging formula for Nielsen numbers holds for continuous maps on infra-nilmanifolds: Let M be an infra-nilmanifold with a holonomy group Phi and f : M -> M be a continuous map. Then N(f ) = 1/| Phi | Sum_{A in Phi} | ...

Fano varieties are fundamental objects in algebraic geometry. These can be considered as the unique output of the -K -minimal model program on the varieties with effective anticanonical divisors. Thus the initial models should encode the in...

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Brownian motion with darning (BMD) is a diffusion process obtained from Brownian motion by shorting each hole in the space into one point. In this talk, I will present a quick introduction to BMD and its basic properties including the zero p...

Representation theory is to study the actions of groups or algebras on vector spaces. Recently, its categorical version, categorical representation theory, attracts researchers in representation theory. In this theory we replace "vector spac...

In 1980s, Donaldson discovered his famous invariant of 4-manifolds which was subsequently proved to be an integral on the moduli space of semistable sheaves when the 4-manifold is an algebraic surface. In 1994, the Seiberg-Witten invariant w...

In this talk, I will explain a relationship of the Chern-Simons invariant and the eta invariant for Schottky hyperbolic manifolds. The relating formula involves a defect term given by the Bergman tau function over the conformal boundary Riem...

The spherical average has been a source of many problems in harmonic analysis. Since late 90's, the study of the maximal spherical means on the Heisenberg group $mathbb{H}^n$ has been started to show the pointwise ergodic theorems on the gro...

I will talk about arithmetic topology, in particular, some issues related to class field theory for 3-dimensional foliated dynamical systems.

We start with the famous Heisenberg uncertainty principle to give the idea of the probability in quantum mechanics. The Heisenberg uncertainty principle states by precise inequalities that the product of uncertainties of two physical quantit...

The field of operator algebras deals with suitable closed subalgebras of the algebra of bounded linear operators on a Hilbert space. There are two types of operator algebras: C*-algebras and von Neumann algebras. In the 1970s A. Connes obtai...

We consider the problem of identifying the material properties from boundary measurements. For the conductivity case, this is known as Calderon problem: “Is it possible to determine the electrical conductivity inside a domain from the bounda...

We proved the codimension three conjecture that says the micro-local perverse sheaves extend if it is defined outside odimension three (counting from Lagrangian subvarity). It is a joint work with Kari Vilonen.

The main topic of the talk is a determinantal formula for high dimensional tree numbers of acyclic complexes via combinatorial Laplace operators . This result is a generalization of Temperley's tree number formula for graphs, motivated by a ...

I will tell two interrelated stories illustrating fruitful interactions between combinatorics and Hodge theory. The first is that of Lorentzian polynomials, based on my joint work with Petter Brändén. They link continuous convex...

In this talk I will talk about existence and regularity for solutions to the compressible viscous Navier-Stokes equations on nonsmooth domains, especially with corners. The solution is constructed by the decomposition of the corner singulari...