Analysis, PDE & Probability Seminar

2022 Seminars

2022. 2. 24. (Thu) 14:00-15:00  Room Online
Kunwoo Kim (POSTECH)

Positivity and compact support property

ABSTRACT. For the heat equation (without noise), the maximum principle says that if the initial data is non-negative, the solution is also non-negative. In fact, it is strictly positive for all t>0 (let's call this strict positivity). In this talk, we provide conditions on noise that guarantee strict positivity for stochastic heat equations (SHE). In addition, we also provide conditions on noise that guarantee the compact support property, which means that if the initial function has compact support, then the solution also has compact support with probability 1.

2022. 2. 22. (Tue) 14:00-15:00  Room Online
Kunwoo Kim (POSTECH)

Existence/Uniqueness/Regularity of stochastic heat equations

ABSTRACT. When noise is rough (i.e., a distribution-valued object), some care must be taken to define a solution to the stochastic heat equation (SHE). In this talk, we first provide what a solution to SHE means, and then show the existence and uniqueness of a random field solution under the condition on the correlation structure of noise. We also consider the regularity of a solution, depending on the correlation structure of noise.

2022. 2. 21. (Mon) 14:00-15:00  Room Online
Kunwoo Kim (POSTECH)

Introduction to stochastic heat equations

ABSTRACT. Stochastic heat equations usually refer to heat equations perturbed by noise, and the noise represents random external force or random potential. In this talk, we introduce the mathematical definition of noise and stochastic integration with respect to noise (so-called Walsh-Dalang integral). This stochastic integral provides a way to define a solution to stochastic heat equations.

2022. 2. 18. (Fri) 13:00-15:00  Room 1424
Jungin Lee (KIAS)

Joint distribution of the cokernels of random p-adic matrices

ABSTRACT. Let A be a random n by n matrix over Z_p with respect to the Haar measure. Friedman and Washington proved that the distribution of the cokernel of A follows the Cohen-Lenstra distribution. In this talk, we introduce two possible ways to generalize their work. In particular we calculate the joint distribution of the cokernels cok(P_1(A)), ... , cok(P_l(A)) for polynomials P_1(t), ... , P_l(t)∈Z_p[t] under some mild conditions. We also introduce the linearization of a random matrix model and its connection to our result.

2022. 2. 17. (Thu) 17:00-18:00  Room Online
Hardy Chan (Instituto de Ciencias Matemáticas)

Local and nonlocal ODEs in the singular fractional Yamabe problem

ABSTRACT. In conformal geometry, the Yamabe problem asks for Yamabe metrics, or conformal metrics of constant scalar curvature. In search of singular Yamabe metrics, one is led to the study of the Lane-Emden equation with a Sobolev-subcritical (or -critical) exponent that depends on the dimension of the singularity. The radial profile, which solves a classical ODE, is well-understood.

One could pose the same problem concerning the fractional curvature, a general notion that includes the scalar curvature, the curvatures associated with Paneitz and GJMS operators, as well as those with non-integer order. For the investigation of the corresponding radial profile, we discuss the development of the nonlocal ODE theory. Apart from the localizing Caffarelli-Silvestre extension, we show that nonlocal ODE can also be understood as a coupled infinite system of second order ODEs. Finally, we also mention a simple while surprising transformation that reduces the nonlocal ODE into almost a scalar first order ODE.

2022. 2. 15. (Tue) 17:10-17:50  Room 7323
Jaeseong Heo (Hanyang University)

SIC-POVM conjecture and condition for existence of conditional SIC-POVM

ABSTRACT. I examine the symmetric informationally complete POVMs and its generalization. We discuss SIC-POVM conjecture which is equivalent to Zauner's conjecture and the Weyl-Heisenberg group related to Zauner's strong conjecture. We introduce conditional SIC-POVM and discuss necessary conditions for the existence of conditional SIC-POVM.

2022. 2. 15. (Tue) 16:20-17:00  Room 7323
Jong-Do Park (Kyung Hee University)

Explicit computation of the Bergman kernel for some non-homogeneous domains

ABSTRACT. The Bergman kernel function is an important tool in several complex variables. It is defined for any bounded domains in C^n, but it is difficult to find the explicit form of the Bergman kernel function K_D(z, w) for any given domain D. It is known that each bounded homogeneous domain has an explicit form of the Bergman kernel function.
In this talk, we explain the methods of computing the Bergman kernel functions for some non-homogeneous domains. We focus on the Hartogs domains, Hartogs triangles, complex ellipsoids and intersection of cylindrical domains. The methods of computing the kernels for each domain will be explained using the hypergeometric series. Also we discuss the existence of zeros of the Bergman kernel functions.

2022. 1. 28. (Fri) 10:00-12:00  Room Online
Sang-Bum Yoo (Gongju National University of Education)

Invariant factors and conformal blocks

ABSTRACT. We study invariant factors and conformal blocks. We first see that the study on invariant factors is a classical motivation of that on the finite generation of the algebra of type A conformal blocks over a curve. We introduce the definition of the space of conformal blocks given by A. Beauville. Then we see that the conformal block is a quantum generalization of invariant factor.

2022. 1. 27. (Thu) 17:00-18:00  Room Online
Hong Chang Ji (IST Austria)

[GS_M_APP] Functional central limit theorem for non-Hermitian random matrices

ABSTRACT. In this talk, we discuss the fluctuation of f(X) as a matrix, where X is a large square random matrix with centered, independent, identically distributed entries. In particular, we show that for a generic deterministic matrix A of the same size as X, the trace of f(X)A is approximately Gaussian which decomposes into two different modes corresponding to tracial and traceless parts of A. The proof mainly relies on resolvents, in particular local laws for products of resolvents. This talk is based on a joint work with László Erdös.

2022. 1. 26. (Wed) 15:00-16:00  Room Online
Jongchon Kim (City University of Hong Kong)

Geometric maximal functions associated with spheres

ABSTRACT. This talk will be about maximal operators associated with a family of spheres, which include the classical spherical maximal function as a special case. We will study properties of sets, such as the Lebesgue measure and the Hausdorff dimension, containing a family of spheres which may be regarded as Nikodym sets for spheres. This talk will be based on a joint work with Alan Chang and Georgios Dosidis.

2022. 1. 13. (Thu) 15:00-17:00  Room Online
Sang-Bum Yoo (Gongju National University of Education)

An introduction to Mori's program

ABSTRACT. We introduce Mori's program which is related to the study of the finite generation of the algebra of type A conformal blocks over a curve. Log Fano variety, Mori Dream space and Cox ring are key ingredients of Mori's program in our work.

2022. 1. 12. (Wed) 16:00-18:00  Room 1424
Woocheol Choi (Sungkyunkwan University)

An introduction to the adaptive control theory

ABSTRACT. In this talk, I will introduce basic ideas and computations of the adaptive control theory.

2022. 1. 12. (Wed) 16:00-17:00  Room Online
Kiho Park (KIAS)

[GS_M_APP] Transfer operators and limit laws for typical cocycles

ABSTRACT. We consider a dynamical system f \colon X \to X and a matrix cocycle A \colon X \to GL_d(R), and study the product A(f^nx) \ldots A(x) of A along the orbit of x. In particular, we will show that for a class of dynamical systems f and a class of matrix cocycles A the product A(f^nx) \ldots A(x) satisfies various limit laws, such as the central limit theorem and the large deviation principle (with respect to certain invariant measures on X, including the measure of maximal entropy). We also establish the analytic dependence of the top Lyapunov exponent on the invariant measures. The transfer operator and its spectral properties play key roles in establishing these limit laws. This is joint work with Mark Piraino.

2022. 1. 12. (Wed) 14:00-15:00  Room Online
Minjae Park (MIT)

Large deviations of Schramm-Loewner evolutions

ABSTRACT. Schramm-Loewner evolution (SLE) is a central object in 2D conformal random geometry. Recently, several interesting objects in complex analysis have been derived from SLE large deviations. In this talk, we survey recent results on large deviation theory of SLE.

2022. 1. 10. (Mon) 10:00-11:00  Room 1424
Jinwan Park (Seoul National University)

The Regularity Theory for the Double Obstacle Problem

ABSTRACT. In this talk, I will introduce the regularity of the free boundary for the double obstacle problems.
First, I am going to introduce the proof of local $C^{1}$ regularity of free boundaries for the elliptic double obstacle problem with an upper obstacle $\psi$, \begin{align*} \Delta u &=f\chi_{\Omega(u) \cap\{ u< \psi\} }+ \Delta \psi \chi_{\Omega(u)\cap \{u=\psi\}}, \qquad u\le \psi \quad \text { in } B_1, \end{align*} where $\Omega(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$ under a thickness assumption for $u$ and $\psi$. The function $\psi$ satisfies $$\psi \in C^{1,1}(B_1) \cap C^{2,1}(\overline{\Omega(\psi)}),\quad \Omega(\psi)=B_1 \setminus \left( \{\psi=0\} \cap \{ \nabla \psi =0\}\right).$$ This is a joint work with Ki-ahm Lee and Henrik Shahgholian.
Next, I will talk about the parabolic problem. This is a joint work with Ki-ahm Lee.

2022. 1. 7. (Fri) 14:00-15:00  Room Online
Minhyun Kim (Universität Bielefeld)

Harnack inequality for nonlocal operators on Riemannian manifolds

ABSTRACT. We study the Krylov–Safonov theory for fully nonlinear nonlocal operators on Riemannian manifolds with nonnegative curvature and on hyperbolic spaces. The Harnack inequality and Holder estimates are established and these are robust in the sense that they recover the classical estimates for second order local operators as limits.

2022. 1. 6. (Thu) 16:00-18:00  Room Online
Sang-Bum Yoo (Gongju National University of Education)

Finite generation of the algebra of type A conformal blocks via birational geometry - basic terminologies and tools

ABSTRACT. We introduce a series of joint works with Han-Bom Moon on the finite generation of the algebra of type A conformal blocks over a curve. The study of the finite generation is closely related to a birational geometry of the moduli space of parabolic bundles over a smooth curve, in the framework of Mori's program. In the first lecture, we introduce basic useful terminologies and tools which appear in algebraic or complex geometry.

Past Seminars

2021 Seminars
2020 Seminars
2019 Seminars

ORGANIZERS	Sung-Soo Byun   -   Korea Institute for Advanced Study
	Sukjung Hwang   -   Chungbuk National University
	Nam-Gyu Kang   -   Korea Institute for Advanced Study
	Doheon Kim   -   Hanyang University
	Yehyun Kwon   -   Korea Institute for Advanced Study
	Jaehun Lee   -   Korea Institute for Advanced Study
	Taehun Lee   -   Korea Institute for Advanced Study
	Sung-Jin Oh   -   UC Berkeley & KIAS
	Bae Jun Park   -   Sungkyunkwan University
	Jung-Tae Park   -   Korea University of Technology and Education
	Sung Chul Park   -   Korea Institute for Advanced Study
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