Analysis, PDE & Probability Seminar

2023 Seminars

2023. 12. 28. (Thu) 17:00-18:00  Room Online
Alicia Quero (University of Granada)

Generating operators between Banach spaces

ABSTRACT. Motivated by the notion of spear operator, we introduce the weaker concept of generating operator. Given two (real or complex) Banach spaces $X$ and $Y$, we say that a norm-one operator $G\in\mathcal{L}(X,Y)$ is generating if the equality $$\|T\|=\inf_{\delta>0} \sup\{\|Tx\|:x\in X,\ \|x\|=1,\ \|Gx\|>1−\delta\}$$ holds for every $T\in\mathcal{L}(X,Y)$. Equivalently, $G$ is generating if (and only if), for every $0<\delta<1$, the set $\{x\in X:\|x\|=1,\ \|Gx\|>1−\delta\}$ generates the unit ball of $X$ by closed convex hull. This class of operators includes isometric embeddings, spear operators, and other examples such as the natural inclusion of $\ell_1$ into $c_0$. Along this talk, we present different charaterizations of this type of operators, discuss when they are norm-attaining, and analyze the set of all generating operators between two Banach spaces. This is based on a joint work with Vladimir Kadets, Miguel Martín, and Javier Merí.

2023. 12. 28. (Thu) 16:00-18:00  Room 1424
Eunhee Jeong (Jeonbuk National University)

Localisation for the Bochner-Riesz means

ABSTRACT. I introduce the localisation property of the classical Bochner–Riesz means, which is based on the paper of K. Jotsaroop entitled "Localisation of Bochner-Riesz means on sets of positive Hausdorff dimension in $\mathbb{R}^d$". The result was obtained by using the decay of the spherical means of Fourier transform of fractal measure.

2023. 12. 21. (Thu) 17:00-18:00  Room Online
Audrey Fovelle (University of Granada)

Asymptotic smoothness, concentration properties in Banach spaces and applications

ABSTRACT. In this talk, we will be interested in concentration properties for Lipschitz maps defined on Hamming graphs. After introducing all the objects at stake and explaining their interest, we will see how one can construct the first example of a Banach space that has some concentration without any smoothness property.

2023. 12. 15. (Fri) 15:00-17:00  Room 8101
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #10: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 12. 15. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #10: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 12. 14. (Thu) 17:00-18:00  Room Online
Daniel L. Rodríguez-Vidanes (Universidad Complutense de Madrid)

Algebraic structures of non-norm-attaining operators

ABSTRACT. The field of lineability seeks to identify algebraic structures, like vector spaces, within non-linear subsets of linear sets. This presentation shows classical and contemporary lineability findings within norm-attaining theory, exploring their relevance in Banach space theory. The talk will encompass recent advancements, focusing specifically on the lineability properties observed in families of non-norm-attaining operators. Joint work with S. Dantas, J. Falco ́, and M. Jung.

2023. 12. 14. (Thu) 16:00-17:00  Room 1423
Panki Kim (Seoul National University)

Markov processes with jump kernel decaying at the boundary

ABSTRACT. In this talk, we discuss pure-jump Markov processes on smooth open sets whose jumping kernels vanishing at the boundary and part processes obtained by killing at the boundary or (and) by killing via the killing potential. The killing potential may be subcritical or critical. This work can be viewed as developing a general theory for non-local singular operators whose kernel vanishing at the boundary. Due to the possible degeneracy at the boundary, such operators are, in a certain sense, not uniformly elliptic. These operators cover the restricted, censored and spectral Laplacians in smooth open sets and much more. The main results are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates.

2023. 12. 14. (Thu) 15:00-17:00  Room 8101
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #9: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 12. 14. (Thu) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #9: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 12. 8. (Fri) 17:00-18:00  Room Online
Gerardo Barrera Vargas (University of Helsinki)

Gradual convergence in total variation distance for Langevin dynamics with degenerate fixed point

ABSTRACT. In this talk we study an ordinary differential equation such that the vector field has a unique fixed degenerate point at the origin. We then add a Brownian perturbation with a small parameter that controls the magnitude of the noise. Under general growth conditions on the vector field, for any fixed magnitude of the noise, as time goes by, the solution of the preceding SDE converges exponentially fast in total variation distance to its unique equilibrium distribution. We show that such convergence occurs in a gradual form, that is, the total variation distance of the random dynamics and its equilibrium distribution tends to a non-degenerate limiting profile, which corresponds to the total variation distance between the marginal of a suitable stochastic differential equation that comes down from infinity and its corresponding equilibrium distribution. The limiting profile that arises from this convergence is not a step profile that appears in the context of cut-off phenomenon for random processes. Joint work Conrado da Costa (Durham University, UK) and Milton Jara (IMPA, Brazil).

2023. 12. 8. (Fri) 15:00-17:00  Room 8101
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #8: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 12. 8. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #8: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 12. 8. (Fri) 10:00-11:00  Room 1423
Ja A Jeong (Seoul National University)

C*-algebras associated to homeomorphisms twisted by vector bundles

ABSTRACT. One of the most pleasing aspects of C*-algebra theory is the ability to study various mathematical objects using a C*-algebra toolkit. Well known examples include studying dynamical systems (via the crossed product construction C(X) ⋊α Z), directed graphs-shift of finite type (via Cuntz-Krieger algebras and more general graph C*-algebras), Topological groups (via their group C*-algebras C*(G)), and many more. In this talk, from the point of view of C*-classification programme, we consider C*-algebras O(E) associated to Hilbert bimodules E arising from a minimal dynamical system (X,f) of a compact metric space X twisted by a vector bundle and show that O(E) are all classifiable.

2023. 12. 1. (Fri) 17:00-18:00  Room Online
Gianmarco Bet (University of Florence)

Detecting a late changepoint in the preferential attachment model

ABSTRACT. Motivated by the problem of detecting a change in the evolution of a network, we consider the preferential attachment random graph model with a time-dependent attachment function. Our goal is to detect whether the attachment mechanism changed over time, based on a single snapshot of the network at time $n$ and without directly observable information about the dynamics. We cast this question as a hypothesis testing problem, where the null hypothesis is a preferential attachment model with a constant affine attachment parameter $\delta_0$, and the alternative hypothesis is a preferential attachment model where the affine attachment parameter changes from $\delta_0$ to $\delta_1$ at an unknown changepoint time $\tau_n$. For our analysis we assume that the changepoint occurs close to the observation time of the network, which corresponds to the relevant scenario where we aim to detect the changepoint shortly after it has happened. We present two tests based on the number of vertices with minimal degree. The first test assumes knowledge of $\delta_0$, while the second does not. We show that these are asymptotically powerful when $n-\tau_n \gg n^{1/2}$. Furthermore, we prove that the test statistics for both tests are asymptotically normal, allowing for accurate calibration of the tests. If time allows, we will present numerical experiments to illustrate the finite sample test properties.

2023. 12. 1. (Fri) 15:00-17:00  Room 8101
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #7: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 12. 1. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #7: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 11. 30. (Thu) 17:00-18:00  Room Online
Ruben Medina (University of Granada, Czech Technical University in Prague)

Radon and Nikodym versus Krein and Milman: A clash of titans

ABSTRACT. In this talk we are going to go through some of the most important results on the Radon-Nikodym and Krein-Milman properties. In particular, we will exhibit a way to generalize Schachermayer-Rosenthal theorem, which shows the equivalence between the latter properties in the case the space is strongly regular, that is, every convex subset has convex combinations of weakly open sets of arbitrarily small diameter.

2023. 11. 29. (Wed) 11:00-12:00  Room 1423
Yongwoo Lee (Seoul National University)

Finite size corrections for the elliptic Ginibre ensembles

ABSTRACT. The ellipitic Ginibre ensemble belongs to a random matrix ensemble which interpolates between the Ginibre and Gaussian orthogonal ensembles. A common characteristic of the matrix models is a presence of the real eigenvalues whose distributions depend on the non-Hermiticity parameter. In this talk, we consider the real eigenvalue densities of the elliptic Ginibre ensembles for two different regimes which are called strong and weak non-Hermiticity regimes. We present the finite size corrections for the densities for the both regimes. This is based on a joint work with Sung-Soo Byun.

2023. 11. 29. (Wed) 10:00-11:00  Room 1423
Jeonghyun Ahn (Seoul National University)

Metastability of the three-state Potts model with asymmetrical external field

ABSTRACT. In this talk, we explore the metastability of the three-state Potts model with an asymmetrical external field in a low-temperature regime. First, we analyze the energy landscape of the model and its analogy to the original Ising model. This analysis leads to the identification of two metastable states and a unique stable state. Due to the presence of two metastable states with different stability levels, the model exhibits a non-metastable transition. We elaborate this behavior by showing the Eyring-Kramers law. Finally, we illustrate the Markov chain model reduction, which indicates the convergence of the model as the temperature goes to zero. We establish this convergence for two different speed-up scales.

2023. 11. 28. (Tue) 16:00-17:30  Room 1424
Sanggyun Youn (Seoul National University)

Fourier analysis on groups

ABSTRACT. One of the central topics in abstract harmonic analysis is Fourier analysis. The Fourier transforms on groups can be understood using convolution operators, and their analytic properties are heavily influenced by the underlying spaces and their algebraic structures. For example, the Fourier coefficients of integrable functions on non-abelian compact groups are no longer numbers, but they are matrices. The main aim of this talk is to provide an introduction to abstract harmonic analysis on groups, including the Plancherel theorem, the Hausdorff-Young inequality, Hardy-Littlewood inequalities, and Sobolev embedding properties.

2023. 11. 24. (Fri) 15:00-17:00  Room 8101
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #6: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 11. 24. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #6: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 11. 17. (Fri) 15:00-17:00  Room 8101
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #5: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 11. 17. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #5: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 11. 15. (Wed) 16:30-18:00  Room 1423
Chulkwang Kwak (Ewha Womans University)

Long-time dynamics of Breather solutions for the fifth-order Gardner equation

ABSTRACT. In this talk, we deal with the initial value problem of the 5th-order Gardner equation in $\mathbb{R}$. First, we briefly discuss the global well-posedness. Second, special solutions, the so-called Breathers, are introduced, and we also discuss spectral properties of a particular fourth-order elliptic operator, which ensure that Breather solutions of 5th order Gardner equation are $H^2(\mathbb{R})$ stable.

2023. 11. 10. (Fri) 15:00-17:00  Room 1424
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #4: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 11. 10. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #4: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 11. 3. (Fri) 15:00-17:00  Room 1424
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #3: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 11. 3. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #3: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 10. 31. (Tue) 17:00-18:00  Room 1424
Jinsol Seo (KIAS)

$L^p$ theory for the Poisson equation in non-smooth domains

ABSTRACT. The Poisson equation ($\Delta(u)=f$) is one of the most fundamental and classical PDE, and its $L^p$ theory is important for the regularity of solutions. In this presentation, we discuss a general $L^p$ theory for the Poisson equation in non-smooth domains, together with its applications based on a relation between the Hardy inequality, superharmonic functions, and various domain conditions.

2023. 10. 25. (Wed) 16:30-17:30  Room 1423
Youngran Lee (Sogang University)

Global existence versus finite time blowup dichotomy for the dispersion managed NLS

ABSTRACT. In this talk, we consider the dispersion managed nonlinear Schrödinger equations(NLS) which arise naturally in modeling fiber-optics communication systems with periodically varying dispersion profiles. We discuss the global existence versus finite time blowup dichotomy for the dispersion managed NLS with mass critical and supercritical nonlinearities.

2023. 10. 25. (Wed) 10:00-11:30  Room 1423
Sungsoo Byun (Seoul National University)

Harer-Zagier type recursion formula for the elliptic GinOE

ABSTRACT. Random matrix theory enjoys an intimate connection with various branches of mathematics. One prominent illustration of this relationship is the Harer-Zagier formula, which serves as a well-known example demonstrating the combinatorial and topological significance inherent in random matrix statistics. While the Harer-Zagier formula originates from the study of the moduli space of curves, it also gives rise to a fundamental formula in the study of spectral moments of classical random matrices. In this talk, I will introduce the Harer-Zagier type formulas for the classical Hermitian Gaussian random matrix ensembles and present recent results on these formulas for the non-Hermitian random matrix model called the elliptic Ginibre ensemble.

2023. 10. 20. (Fri) 10:00-11:00  Room 1423
Sungsoo Byun (Seoul National University)

The product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

ABSTRACT. The study of products of random matrices was proposed many decades ago by Bellman and by Furstenberg and Kesten, with the motivation to understand the properties of the Lyapunov exponents in this toy model for chaotic dynamical systems. In this talk, I will discuss the real eigenvalues of products of random matrices with i.i.d. Gaussian entries. In the critical regime where the size of matrices and the number of products are proportional in the large system, I will present the mean and variance of the number of real eigenvalues. Furthermore, in the Lyapunov scaling, I will introduce the densities of real eigenvalues, which interpolate Ginibre's circular law with Newman's triangular law. This is based on joint work with Gernot Akemann.

2023. 10. 17. (Tue) 16:00-17:30  Room 1424
Jani Virtanen (University of Reading)

Asymptotics of block Toeplitz determinants with piecewise continuous functions

ABSTRACT. I discuss recent results on the asymptotics of the block Toeplitz determinants generated by matrix-valued piecewise continuous functions with finitely many jumps under mild additional conditions. The approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix-valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models. This is joint work with Estelle Basor and Torsten Ehrhardt.

2023. 10. 13. (Fri) 15:00-17:00  Room 8101
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #2: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 10. 13. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #2: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 10. 6. (Fri) 15:00-17:00  Room 8101
Caixing Gu (California Polytechnic State University)

Focused Lecture Series #1: De Branges-Rovynak spaces and their connections with sub-Hardy and sub-Bergman spaces

ABSTRACT. De Branges-Rovynak spaces played an important role in the solution of Bieberbach conjecture by de Branges. They now find applications in function theory, operator theory and system theory. The aim of these lectures is to lay a solid function on these spaces and bring the audience to the frontier work of active research in several directions, for example, the more recent work of high order De Branges-Rovynak spaces, sub-Bergman spaces, to further develop the operator theory on these spaces.

2023. 10. 6. (Fri) 13:40-14:50  Room 8309
Jaehui Park (Seoul National University)

Tutorial Series #1: The theory of H(b) spaces

ABSTRACT. The theory of H(b) spaces is an emerging important issue recently. The H(b) spaces have an elegant structure, but the essence of H(b) spaces still remains shrouded in mystery and many questions still remain unsolved. The aim of this tutorial series is to provide the opportunity for intensive open debate and interact on the theory of H(b) spaces.

2023. 9. 22. (Fri) 10:30-12:00  Room 1424
Jinyeop Lee (Ludwig Maximilian University of Munich)

Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction

ABSTRACT. In the talk, we first review nonlinear delta model, so called NLS with point nonlinearity, widely investigated in the last decades. Then we derive the NLS with point nonlinearity from the dynamics of N identical bosons in dimension one, in the presence of a tiny impurity at the origin. We assume that the interaction between every pair of bosons is mediated by the impurity through a three-body interaction. From a simultaneous mean-field and short-range scaling and choosing an initial data as a fully factorized state, we prove propagation of chaos and obtain an effective one-particle Schrödinger equation with a nonlinearity concentrated at a point. More precisely, we prove convergence of one-particle density operators in the trace-class topology, and estimate the fluctuations as superexponential. This is the first derivation of the so-called nonlinear delta model, widely investigated in the last decades. This is a joint work with Riccardo Adami.

2023. 9. 21. (Thu) 15:30-17:00  Room 1424
Juyoung Lee (Seoul National University)

Critical weighted inequalities of the spherical maximal function

ABSTRACT. Weighted inequality on the Hardy-Littlewood maximal function is completely understood while it is not well understood for the spherical maximal function. For the power weight $|x|^\alpha$, it is known that the spherical maximal operator on Rd is bounded on $L^p(|x|^\alpha)$ only if $1−d≤\alpha<(d−1)(p−1)−1$ and under this condition, it is known to be bounded except $\alpha=1−d$ . In this talk, we prove the case of the critical order, α=1−d.

2023. 9. 19. (Tue) 10:30-11:30  Room 1424
Seonwoo Kim (KIAS)

Spectral gap of the symmetric inclusion process

ABSTRACT. In this talk, we consider the symmetric inclusion process of intensity $\alpha$ on a general finite graph. We establish universal upper and lower bounds for the spectral gap of this process in terms of the spectral gap of the random walk of a single particle on the same graph. In particular, in the regime $\alpha\ge1$, our bounds imply a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture originally formulated for the interchange process.

2023. 9. 15. (Fri) 16:00-17:00  Room 1423
Woo Yong Lee (HCMC)

Describing backward shift-invariant subspaces

ABSTRACT. The backward shift in the vector-valued world is the model of operators in the scalar-valued world. In this talk we examine the problem of describing the set of vectors generating backward shift-invariant subspaces in the context of an interplay between function theory and operator theory.

2023. 9. 5. (Tue) 16:00-17:00  Room 1432
Il Doo Kim (Korea University)

A well-posedness theory of stochastic partial differential equations with time-measurable pseudo-differential operators

ABSTRACT. In this talk, we introduce some recent results showing well-posedness of stochastic partial differential equations with time-measurable pseudo-differential operators in $L_p$ -spaces.

2023. 8. 28. (Mon) 14:00-15:30  Room 1423
Jinbong Lee (Seoul National University)

Lp estimates for maximal functions associated with fractal dimensional dilation

ABSTRACT. In this talk, we consider maximal operators given by Fourier multipliers where the supremum of $t$ is taken over $E \subset \mathbb{R}$ with certain covering condition. By the covering condition, one can assume that $E$ is a set of the Assouad dimension $\alpha \in [0,1]$. Thus one may expect that $L_p$ estimates of the maximal operators depend on both Fourier multipliers and the dimension of $E$. We present a criterion for the Fourier multipliers to be an $L_p$ multipliers of the maximal operators for each $p$, where the criterion is written in terms of egularity of the multipliers and the dimension of $E$.

2023. 8. 9. (Wed) 10:00-11:00  Room 1423
Heejong Bong (Carnegie Mellon University)

Dual Induction CLT for High-dimensional m-dependent Data

ABSTRACT. In this work, we provide a $1/\sqrt{n}$ rate finite sample Berry-Esseen bound for $m$-dependent high-dimensional random vectors over the class of hyper-rectangles. This bound imposes minimal assumptions on the random vectors such as nondegenerate covariances and finite third moments. The proof uses inductive relationships between anti-concentration inequalities and Berry-Esseen bounds, which are inspired by the classical Lindeberg swapping method and the concentration inequality approach for dependent data. Performing a dual induction based on the relationships, we obtain tight Berry-Esseen bounds for dependent samples.

2023. 7. 31. (Mon) 10:30-11:30  Room 1423
Robert Schippa (KIAS)

Oscillatory integral operators with homogeneous phase functions

ABSTRACT. We prove $L^p$-$L^p$-estimates for oscillatory integral operators with 1-homogeneous phase functions. The phase functions are required to satisfy a non-degeneracy and convexity assumptions. Making use of polynomial partitioning and decoupling, we recover the estimates for the cone extension operator due to Ou-Wang. Secondly, we illustrate how $L^p$-estimates can be used to solve nonlinear wave equations with slowly decaying initial data. The latter part of the talk is joint work with Jan Rozendaal (IMPAN).

2023. 7. 20. (Thu) 10:00-11:00  Room Online
Qiang Wu (UIUC)

Cluster expansion approach to the mean field spin glass theory

ABSTRACT. Cluster expansion, as a powerful combinatorial scheme, has been widely used in many statistical mechanics problems. However, most of those focused on the non-disordered spin system. In disordered system, such as spin glasses, Aizenman-Lebowitz-Ruelle first utilized the cluster expansion idea to obtain some rigorous results in the Sherrington-Kirkpatrick (SK) model at zero external field. Nevertheless, since then, it was believed in the literature that this approach does not work once the external is present. In this talk, we will show that under some “weak” external field, the cluster expansion still works, but a new cluster emerges. This implies some transition results for the free energy in the SK model. Time permits, we will also discuss the extension of this approach to a generic setting, the mixed $p$-spin models, where a new multiple transition phenomenon was shown along with the discovery of a collection of fruitful cluster structures. This is based on joint works with Partha S. Dey.

2023. 7. 18. (Tue) 17:00-18:00  Room Online
Alex Little (University of Bristol)

A Riemann-Hilbert approach to quaternion random matrices

ABSTRACT. Self-dual random matrices with quaternion matrix elements were first considered by Dyson & Mehta because of their application to quantum mechanical systems with time-reversal invariance and odd spin. The eigenvalues of these random matrices can be viewed as particles in a log-gas with inverse temperature beta=4.

In this talk I will present work (arXiv:2306.14107) that allows one to directly study these ensembles via Riemann-Hilbert problems (RHPs). An RHP involves representing a quantity of interest as the solution of a boundary value problem in the complex plane. An RHP can be viewed as a generalisation of the notion of a contour integral. RHPs are central to the modern theory of integrable systems and appear in the study of integrable PDEs (e.g. the Korteweg-de Vries equation), orthogonal polynomials, the 2D Ising model, Painlevé equations, amongst others. In our work we construct a Riemann-Hilbert representation of "skew orthogonal polynomials of symplectic type," and prove an analogue of the Christoffel-Darboux formula. This allows all eigenvalue correlations to be expressed in terms of our RHP.

2023. 7. 5. (Wed) 15:30-16:30  Room Online
Oscar Roldan (Dongguk University)

An overview on the Bishop-Phelps-Bollobás property

ABSTRACT. In 2008, in order to quantitatively study the density of norm-attaining operators, Acosta, Aron, García, and Maestre introduced and studied the Bishop Phelps Bollobás property: a pair of Banach spaces has the Bishop-Phelps-Bollobás property if whenever an operator almost attains its norm at some point, a nearby operator attains its norm at a nearby point, and all the approximations are uniform in some sense. Ever since then, many researches have made contributions to this topic, and nowadays the research on related questions is vast and fruitful. In this talk, we will see some of the main known results and open questions about the Bishop-Phelps-Bollobás property.

2023. 6. 20. (Tue) 16:00-17:00  Room 1423
Minsuk Yang (Yonsei University)

Liouville-type theorems for the MHD equations

ABSTRACT. We review famous results on the Liouville-type theorems for NSE equations and then introduce our recent progress.

2023. 6. 14. (Wed) 16:00-17:30  Room 8101
Sun-Sig Byun (SNU)

Local, nonlocal and mixed local-nonlocal elliptic and parabolic problems

ABSTRACT. I present some regularity results for solutions of local, nonlocal and mixed local-nonlocal elliptic and parabolic problems.

2023. 6. 14. (Wed) 10:30-11:30  Room 1423
Steffen Rohde (University of Washington)

The infinite trivalent tree and the developed deltoid

ABSTRACT. Every finite tree has an essentially unique conformally natural embedding into the complex plane, by means of "Shabat polynomials". Numerical computations show that large trivalent trees approximate the "developed deltoid", a fascinating quadrature domain introduced and studied by Lee, Ljubich, Makarov and Mukherjee through repeated reflection of the deltoid curve over its three sides. In my talk, I will carefully describe these objects, and discuss a form of convergence of the trees to the deltoid.

2023. 6. 13. (Tue) 17:00-18:00  Room Online
Klara Courteaut (KTH Royal Institute of Technology)

The planar Coulomb gas on a Jordan curve

ABSTRACT. The eigenvalues of a Haar distributed unitary matrix (CUE) have the physical interpretation of a system of particles subject to a logarithmic pair interaction, restricted to lie on the unit circle, and at inverse temperature 2. In his thesis, Kurt Johansson generalized this model by replacing the unit circle with a sufficiently regular Jordan curve. He obtained the asymptotics of the Laplace transform of linear statistics. In this talk I will present a recent paper with Johansson, in which we adapt the proof to any positive temperature and obtain the asymptotic partition function.

2023. 6. 12. (Mon) 16:00-17:00  Room 1424
Minjae Park (University of Chicago)

Yang-Mills theory and random surfaces

ABSTRACT. I will explain how to understand Wilson loop expectations as sums over surfaces in the 2D continuum Euclidean Yang-Mills theory (Park-Pfeffer-Sheffield-Yu). Building on the same techniques, we provide alternate derivations and generalizations of several lattice string trajectory theorems (Chatterjee-Jafarov) and surface sum theorems (Magee-Puder). Based on the joint work with Sky Cao and Scott Sheffield.

2023. 6. 1. (Thu) 10:30-11:30  Room 1424
Yuyang Feng (University of Chicago)

Triviality of critical Fortuin-Kasteleyn decorated planar maps for $q>4$

ABSTRACT. We consider infinite random planar maps decorated by the critical Fortuin-Kasteleyn model with parameter $q>4$. I will demonstrate that when appropriately rescaled, these maps, equipped with graph distance and counting measure on edges, converge in law to the infinite continuum random tree, with respect to the local Gromov-Hausdorff-Prokhorov topology. Furthermore, we also show that these maps do not admit any Fortuin-Kasteleyn loops with a macroscopic graph distance diameter. Our proof is based on Scott Sheffield's hamburger-cheeseburger bijection.

2023. 6. 1. (Thu) 9:30-10:30  Room 1424
Jinwoo Sung (University of Chicago)

Quasiconformal deformation of Loewner chains

ABSTRACT. Given a simple planar curve, we can describe it using a family of uniformizing maps called the Loewner chain, which is, in turn, encoded by a real-valued continuous function called the Loewner driving function. In this talk, I will derive the variational formula of the Loewner driving function of a simple chord in a planar domain under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we find a new explanation for the identity between the Loewner energy, which was identified by Y. Wang as the large deviation rate function for Schramm–Loewner evolution, and the universal Liouville action, which was introduced by Takhtajan and Teo as the Kähler potential of the Weil–Petersson metric on the universal Teichmüller space. This is joint work with Yilin Wang (IHES).

2023. 5. 25. (Thu) 17:00-18:00  Room Online
Aron Wennman (KTH Royal Institute of Technology)

Random Weierstrass zeta functions

ABSTRACT. In this talk I will describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary (i.e. translation invariant) point process in the plane. These functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions, or equivalently as the electric fields generated by a random configuration of point charges.

I plan to describe those point processes for which the electric field, after subtracting the mean, becomes translation invariant in law. If time permits we will discuss how these invariant fields may be used to study the growth of the number variance (“charge fluctuations”) in large Jordan domains with quite irregular boundaries.

This talk is based on partially ongoing joint work with Kostya Slutsky (Iowa State), Misha Sodin and Oren Yakir (Tel Aviv) (see https://arxiv.org/abs/2210.09882 and https://arxiv.org/abs/2211.01312)

2023. 5. 4. (Thu) 17:00-18:00  Room Online
Nedialko Bradinoff (KTH Royal Institute of Technology)

Benford’s law and the Circular $\beta$-Ensemble

ABSTRACT. Benford's law is a remarkable phenomenon that governs the digital expansion of both deterministic and random real quantities, coming from a broad variety of contexts, including terms of geometric progressions, stock prises, population numbers… Roughly, the leading digits of such quantities are likely to be heavily skewed towards smaller values. The focus of this talk is to describe a connection between Benford's law and another beautiful object, the Circular $\beta$-Ensemble. The C$\beta$E(N) is classical in random matrix theory and has been studied extensively to exhibit universal properties. A natural object that is studied in the context of the C$\beta$E(N) is its characteristic polynomial. In this talk I will describe how its absolute value obeys Benford's law in a strong sense as $N$ tends to infinity. Along the way I will discuss sufficient conditions for such a result and illustrate their necessity with some examples. Key to the discussed result is a central limit theorem in total variation norm of the logarithm of the absolute value of the characteristic polynomial. The talk is based on joint work with Maurice Duits.

2023. 5. 2. (Tue) 10:00-11:00  Room Online
Peter J. Forrester (University of Melbourne)

[KIAS Springer Series in Mathematics Distinguished Lectures] Progress on the study of the Ginibre ensembles

ABSTRACT. By their very definition, the Ginibre ensembles are a class of non-Hermitian random matrix ensembles. At the same time, the statistical states formed by the eigenvalues in the complex plane permit interpretations in terms of the equilibrium statistical mechanics of particular two-dimensional Coulomb gases. Ginibre’s work dates back to 1965. Come the decade of the 2020’s and its lead up, more and more researchers in mathematics and theoretical physics have been drawn to the field of non-Hermitian quantum mechanics on one front, and that of higher dimensional point processes relating to long range potentials on another. With the Ginibre ensembles impacting on both of these broad themes, their relevance is on the increase, and moreover research on these themes has contributed to the development of the theory and applications of Ginibre ensembles. The two lectures I will present on this topic will in the first lecture introduce the Ginibre ensembles in a random matrix context, and also in the context of Coulomb gas theory, and in the second lecture develop themes relating to the boundary.

2023. 4. 26. (Wed) 16:00-17:00  Room Online
Sheldon Dantas (University of Valencia)

Smoothness in normed spaces

ABSTRACT. Consider the following problem (at a first sight very general and ambitious): let $X$ be a Banach space; is there a dense subspace $Y$ of $X$ that admits a $C^k$-smooth norm? In this talk, we discuss general results related to this problem and compare them to (already known) results in the literature on renorming theory.

2023. 4. 25. (Tue) 10:00-11:00  Room Online
Peter J. Forrester (University of Melbourne)

[KIAS Springer Series in Mathematics Distinguished Lectures] Progress on the study of the Ginibre ensembles

ABSTRACT. By their very definition, the Ginibre ensembles are a class of non-Hermitian random matrix ensembles. At the same time, the statistical states formed by the eigenvalues in the complex plane permit interpretations in terms of the equilibrium statistical mechanics of particular two-dimensional Coulomb gases. Ginibre’s work dates back to 1965. Come the decade of the 2020’s and its lead up, more and more researchers in mathematics and theoretical physics have been drawn to the field of non-Hermitian quantum mechanics on one front, and that of higher dimensional point processes relating to long range potentials on another. With the Ginibre ensembles impacting on both of these broad themes, their relevance is on the increase, and moreover research on these themes has contributed to the development of the theory and applications of Ginibre ensembles. The two lectures I will present on this topic will in the first lecture introduce the Ginibre ensembles in a random matrix context, and also in the context of Coulomb gas theory, and in the second lecture develop themes relating to the boundary.

2023. 4. 24. (Mon) 10:30-11:30  Room 1423
Mingu Jung (KIAS)

On the Daugavet property and holomorphic functions on a Banach space

ABSTRACT. In 1963, I. K. Daugavet proved a remarkable result that $\|Id+T\|=1+\|T\|$ holds for every compact operator $T$ on $C[0,1]$. Afterwards, the same equation (so called the Daugavet equation) was discovered for compact operators on $L^1[0,1]$ and some $C(K)$-spaces. When all compact operators on a Banach space $X$ satisfy the Daugavet equation, we say that X has the Daugavet property. It turned out that the Daugavet property used to have a significant influence on the isomorphic structure of a Banach space. A more systemic study on the Daugavet property was initated by Kadets, Shvidkoy, Sirotkin and Werner in 1997, and it has since become a very active area of research in Banach space theory. In this talk, we review some basic facts on the Daugavet property and investigate the Daugavet property of algebras of continuous functions, and holomorphic functions.

2023. 4. 17. (Mon) 10:30-11:30  Room 1423
Sewook Oh (KIAS)

Maximal averages over curves in higher dimensions

ABSTRACT. Maximal average over submanifold is a fundamental subject which has been extensively studied since 1970s. Classical example is Hardy-Littlewood maximal function and its Lp boundedness. After Stein's celebrated theorem, which tells that Lp boundedness of maximal average over sphere, maximal averages over various submanifolds including curve have been considered by numerous authors. Even though curve has simple geometric structure, boundedness of maximal average over curve in higher dimensions is far-less understood. In this talk, I will explain recent results and some ideas of the proofs.

2023. 4. 10. (Mon) 16:00-17:00  Room 1424
Sang-Jun Park (Seoul National University)

Tensor Norm approach in Quantum Information Theory

ABSTRACT. Functional analysis has recently become a crucial tool in modern quantum information theory (QIT). For example, the various tensor product structures arising in QIT have led to the application of tensor norm theory in operator algebra and Banach space theory. In this talk, we explore the use of tensor norms in two key areas of QIT: compatibility of measurements and non-locality. We discuss several applications and insights gained from this approach.

2023. 4. 6. (Thu) 16:40-17:40  Room 1424
Hyung-Joon Tag (Sejong University)

Daugavet property and geometrical points on the unit sphere of a Banach space

ABSTRACT. In the theory of Banach spaces, there have been fruitful results concerning specific behaviors of certain operators on a Banach space in connection to the geometrical structures of the unit ball. Among various properties investigated in this manner, this talk will focus on the Daugavet property with which Banach space satisfies ∥I +T∥ = 1+∥T∥ for every rank-one operator T : X → X and exhibits the big slice phenomenon in Banach spaces, i.e. every slice of the unit ball has diameter two. In this talk, we will look at weaker notions than the Daugavet property and specific geometrical points on the unit sphere related to those properties, which have been active research topics in the last decade.

2023. 4. 6. (Thu) 15:30-16:30  Room 1424
Geunsu Choi (Dongguk University )

Orthogonality in Banach spaces and its applications

ABSTRACT. In this talk, we review previously defined orthogonalities in Banach spaces and their related properties. Next, we study so-called the Bhatia-Semrl property of a bounded linear operator in view of norm-attaining theory. This property extends one of the introduced orthogonality concepts from an operator to an image produced by the operator. We investigate quantity problems on those operators.

2023. 3. 16. (Thu) 10:30-11:30  Room 1423
Daehan Park (KIAS)

An Lp-theory for time-fractional stochastic partial differential equations driven by Levy processes

ABSTRACT. In this presentation, we discuss about an Lp-theory for time-fractional stochastic partial differential equations driven by Levy processes. In particular, we consider the estimation of the second order derivative of solution in terms of free terms. To obtain this, we use Littlewood-Paley theory in Harmonic analysis, quadratic variation of stochastic integrals.

2023. 3. 15. (Wed) 10:30-11:30  Room 1423
Jaeyun Yi (KIAS)

Fractal geometry of the Parabolic Anderson model with spatial white noise

ABSTRACT. In this talk, we discuss the fractal geometry of the Parabolic Anderson model (PAM) with spatial white noise. We show that tall peaks of the solution to the PAM in dimensions 2 and 3 are macroscopically multifractal. We also obtain the spatial asymptotics of the solution as a byproduct. In order to analyze the solution, we use the theory of paracontrolled distribution, a modern tool for studying singular SPDEs.

2023. 2. 22. (Wed) 16:00-17:00  Room 1424
Yoonjun Lee (Pusan National University)

Weighted Hessian estimates in Orlicz spaces for nondivergence elliptic equations

ABSTRACT. To the scope of the Schrodinger operator, we are interested in nondivergence elliptic operator L:=-a_{ij}D_{ij}+V with V satisfying a reverse Holder condition and small-BMO coefficient a_{ij}. To investigate the solvability of Lu=f with an adequate integrability condition of f, Hessian estimates of the solution u play a crucial role. I n this talk, we review the known results for such Lp type estimates and we discuss a weighted version of Hessian estimates in Orlicz spaces which is a generalization of Lp spaces.

2023. 2. 20. (Mon) 10:30-11:30  Room 1423
Woo Yong Lee (Seoul National University)

Halmos' Problem 5 and Abrahamse's Theorem

ABSTRACT. In 1970, P.R. Halmos posed the following problem, listed as Problem 5 in his lecture ``Ten problems in Hilbert space": Is every subnormal Toeplitz operator either normal or analytic? In 1984, Halmos' Problem 5 was answered in the negative by C. Cowen and J. Long. However, Cowen and Long's construction does not provide an intrinsic connection between subnormality and the theory of Toeplitz operators. Until now researchers have been unable to characterize subnormal Toeplitz operators in terms of their symbols. Thus, we reformulate Halmos' Problem 5: Which Toeplitz operators are subnormal ? The most interesting partial answer to Halmos' Problem 5 was given by M.B. Abrahamse (in 1976), who gave a general sufficient condition for the answer to Halmos' Problem 5 to be affirmative. More concretely, every subnormal Toeplitz operator with a symbol in the Nevanlinna class must be either normal or analytic. On the other hand, block Toeplitz operators have many applications to several other fields. In this sense and as the first inquiry in the reformulation of Halmos' Problem 5 the following question can be raised: Is Abrahamse's Theorem valid for block Toeplitz operators? In general, a straightforward block version of Abrahamse's Theorem is doomed to fail. In this talk we examine Abrahmase's Theorem for block Toeplitz operators.

2023. 2. 16. (Thu) 10:30-11:30  Room 1423
Jinyeop Lee (Ludwig Maximilian University of Munich)

Deriving effective PDE from many-body Schrödinger equations

ABSTRACT. In this talk, we derive effective dynamics governed by PDEs (e.g. Vlasov-Poisson equation) from the N-body Schrödinger equation with interactions in the large N limit. For the derivation, we first visit quantum mechanics. Then, using the knowledge of many-body quantum mechanics given in the talk and with the suitable conditions for each situation, we derive PDEs describing the effective motion of the system.

2023. 2. 2. (Thu) 13:00-14:00  Room 1423
Kyung-Youn Kim (National Chung Hsing University)

Estimates on transition densities for Markov processes with singular jumps

ABSTRACT. We consider a non-local operator with a singular kernel. Corresponding to the non-local operator, there exists a discontinuous Markov process with this operator as infinitesimal generator, and the heat kernel of this operator is the transition density of the process.

In this talk, we discuss the heat kernel bounds for the anisotropic discontinuous Markov process. Let $L_i, i = 1~d$, be identical and independent 1-dimensional symmetric Levy processes whose characteristic functions satisfy the weakly scaling condition. Define a Markov process $M := (M_1,...,M_d)$ whose jumping kernel is comparable to that of $L := (L_1,...,L_d).$ Then M is a pure jump process that jumps parallel to the coordinate axes. We discuss the sharp two-sided heat kernel estimates on $R_d$ and $C^{1,1}$-open set $D$ in $R_d$.

This is the joint work with Lidan Wang.

2023. 1. 25. (Wed) 11:00-12:00  Room 1424
Mouad Ramil (Seoul National University)

Langevin process in bounded-in-position domains. Application to quasi-stationary distributions

ABSTRACT. Quasi-stationary distributions can be seen as the first eigenvector associated with the generator of the stochastic differential equation at hand, on a domain with Dirichlet boundary conditions (which corresponds to >absorbing boundary conditions at the level of the underlying stochastic processes). Many results on the quasi-stationary distribution hold for non degenerate stochastic dynamics, whose associated generator is elliptic. The case of degenerate dynamics is less clear. In this work, together with T. Lelièvre and J. Reygner (Ecole des Ponts, France) we generalize well-known results on the probabilistic representation of solutions to parabolic equations on bounded domains to the so-called kinetic Fokker Planck equation on bounded domains in positions, with absorbing boundary conditions. Furthermore, a Harnack inequality, as well as a maximum principle, is provided for solutions to this kinetic Fokker-Planck equation, as well as the existence of a smooth transition density for the associated absorbed Langevin dynamics. The continuity of this transition density at the boundary is studied as well as the compactness, in various functional spaces, of the associated semigroup. Following these results we obtain the existence of a unique quasi-stationary distribution as well as a convergence result of the Langevin process conditioned on non-exiting the domain towards its quasi-stationary distribution. Using estimates obtained in the particular one-dimensional case, this result can also be sharpened to a uniform convergence result with respect to its initial distribution. This work is a cornerstone to prove the consistency of some algorithms used to simulate metastable trajectories of the Langevin dynamics, for example the Parallel Replica algorithm.

2023. 1. 20. (Fri) 10:30-11:30  Room 8101
Jiwoon Park (University of Cambridge)

2D Discrete Gaussian model and its high temperature phase

ABSTRACT. The Discrete Gaussian model is a type of integer-valued random height function. In the 2D setting, it exhibits a phase transition between a localised phase and a delocalised phase. This phenomenon is also called the Kosterlitz-Thouless phase transition, whose terminology originates from its dual counterpart, the planar XY model.

Motivation for studying the Discrete Gaussian model is multifold. Due to its duality relations with a number of 2D mathematical physics models, such as the XY model or the Coulomb gas, studies on integer-value height functions are capable of proving a number of conjectures usually not accessible using classical methods. Other discrete height functions also have dualities with a number of different interesting models, so it will be of vast interest to develop a general framework that deals with discrete height functions.

Also, discrete height functions are considered to be appropriate test cases for recently developed techniques from probability theory.

In this talk, we discuss a particular method called the renormalisation group method, which is believed to serve as a general framework for studying random fields. We also discuss briefly how the renormalisation group method can be used to prove that the scaling limit of the 2D Discrete Gaussian model is a 2D Gaussian free field.

2023. 1. 12. (Thu) 16:20-17:00  Room 1423
Myeongju Kang (Seoul National University)

Phase-locking in a stochastic Winfree model with inertia

ABSTRACT. In this talk, we study the emergence of phase-locking for Winfree oscillators under the effect of inertia and multiplicative noise. It is known that in a large coupling regime, oscillators governed by the deterministic Winfree model with inertia converge to a unique equilibrium. In contrast, in this talk, we observe the asymptotic emergence of non-trivial synchronization in a suitably small coupling regime. Moreover, we study the effect of multiplicative noise and obtain lower estimates in probability for the pathwise emergence of such a synchronizing pattern, provided the noise is sufficiently small.

2023. 1. 12. (Thu) 15:40-16:20  Room 1423
Jaeyoung Yoon (Seoul National University)

Emerging asymptotic patterns in a Winfree ensemble with higher-order couplings

ABSTRACT. To describe the synchronization in real neurons, some pulse-coupled model such as Peskin model, were designed in the past. However, the lack of smoothness of models, it is very difficult to perform rigorous mathematical analyses, which yields very few studies. For that, the Winfree model, a kind of phase-coupled models, was built to approximate pulse-coupled synchronization phenomenon using a smoothly approximated influence function for the Dirac-delta function. We studied the phenomenon for the case of Dirac-delta function by using the sequence of smooth influence functions. In this talk, we introduce the detail about it and also, asymptotic patterns: incoherence, locking, death.

2023. 1. 12. (Thu) 15:00-15:40  Room 1423
Minhyun Kim (Bielefeld University)

Nonlocal problems with non-standard growth

ABSTRACT. In the Calculus of Variations, functionals with non-standard growth have been studied extensively since the late 1980s. We introduce several nonlocal analogues of these classical models, attempting to develop a unified theory. We study local regularity properties such as local boundedness, weak Harnack inequality, local Hölder regularity and Harnack inequality.

2023. 1. 10. (Tue) 11:00-12:00  Room 1424
Seoyeon Yang (Princeton)

Conjecture on the sharp phase transition for the interchange process on $\mathbb{Z}^d$

ABSTRACT. The interchange process, which is also called the random stirring process, is a random permutation on a given graph with a Poisson clock of rate one associated with each edge independently until time $\beta$. We obtain the random permutation $\pi_\beta$ by the interchange process. Introduced as a probabilistic representation of the Quantum Heisenberg Ferromagnet by Toth, various research on the interchange process and the random permutation have been conducted. Especially, the cycle structure and the existence of the infinite cycle of the permutation $\pi_\beta$ have been actively studied. Angel showed the existence of the infinite cycle on the tree for some $\beta$. Hammond has improved it by showing the existence of the infinite cycle for all sufficiently large $\beta$. Elboim and Sly demonstrated the existence of an infinite cycle on $\mathbb{Z}^d$. Alon and Kozma developed a formula for the probability of the event that the permutation contains only one cycle. Despite the growing interest and significant research, it is not easy to prove phase transition in the interchange process due to the lack of monotonicity. Recently, Hammond showed the phase transition on the tree. We suggest there also will be a sharp phase transition on lattice $\mathbb{Z}^d$ based on Elboim and Sly and Hammond’s study.

Past Seminars

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ORGANIZERS	Sung-Soo Byun   -   Seoul National University
	Yong-Gwan Ji   -   Korea Institute for Advanced Study
	Mingu Jung   -   Korea Institute for Advanced Study
	Myeongju Kang   -   Korea Institute for Advanced Study
	Nam-Gyu Kang   -   Korea Institute for Advanced Study
	Junha Kim   -   Korea Institute for Advanced Study
	Seonwoo Kim   -   Korea Institute for Advanced Study
	Joonhyun La   -   Korea Institute for Advanced Study
	Jaehun Lee   -   Korea Institute for Advanced Study
	Jung-Kyoung Lee   -   Korea Institute for Advanced Study
	Taehun Lee   -   Korea Institute for Advanced Study
	Sewook Oh   -   Korea Institute for Advanced Study
	Sung-Jin Oh   -   UC Berkeley & Korea Institute for Advanced Study
	Daehan Park   -   Korea Institute for Advanced Study
	Hyangdong Park   -   Korea Institute for Advanced Study
	Jaehyeon Ryu   -   Korea Institute for Advanced Study
	Jinsol Seo   -   Korea Institute for Advanced Study
	Jaeyun Yi   -   École Polytechnique Fédérale de Lausanne
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