Analysis, PDE & Probability Seminar

2022 Seminars

2022. 12. 28. (Wed) 11:00-12:00  Room 1424
Jinwoo Sung (University of Chicago)

Liouville quantum gravity metric and the mating-of-trees theory

ABSTRACT. Liouville quantum gravity (LQG) is a theory of surfaces with random geometry, which is described and studied in several different manners. The LQG metric rigorously defines distances on a surface whose curvature is given formally by an exponential of the Gaussian free field. It is conjectured to describe scaling limits of graph distances in various random planar map models. On the other hand, the mating-of-trees theory constructs an LQG surface by welding two continuum random trees along a Schramm–Loewner evolution (SLE) curve.

In this talk, I will survey recent results that bridge the LQG metric and the mating-of-trees theory. In joint work with Ewain Gwynne, I showed that the Minkowski content of the LQG metric gives the LQG measure. The proof of this fact hinges on reformulating the mating-of-trees theory in terms of the LQG metric.

Recently, Hughes and Miller showed that the LQG metric is compatible with the welding of quantum surfaces in the mating-of-trees theory. I will discuss its implications on the global metric structure of thin quantum wedges. This is an ongoing joint project with Yuyang Feng (University of Chicago).

2022. 12. 22. (Thu) 17:00-18:00  Room 1424
Xiaocong Xu (HKUST)

Extreme eigenvalue of log-concave ensemble

ABSTRACT. In this talk, we consider the log-concave ensemble of random matrices. A main example is the covariance estimator of the uniform measure on isotropic convex body. Non-asymptoptic estimates and first order asymptotic limits for the extreme eigenvalues have been obtained in the literature. In this talk, we discuss the spectral rigidity property of this ensemble. Based on the spectral rigidity and some additional assumptions, we further derive the Tracy-Wisdom law for the extreme eigenvalue of this ensemble, and the Gaussian law for the extreme eigenvalues in case strong spikes are present. This talk is based on a joint work with Zhigang Bao.

2022. 12. 15. (Thu) 10:00-11:00  Room Online
Tom Alberts (University of Utah)

Random Geometry in the Spectral Measure of the Circular Beta Ensemble

ABSTRACT. The Circular Beta Ensemble is a family of random unitary matrices whose eigenvalue distribution plays an important role in statistical physics. The spectral measure is a canonical way of describing the unitary matrix that takes into account the full operator, not just its eigenvalues. When the matrix is infinitely large (i.e. an operator on some infinite-dimensional Hilbert space) the spectral measure is supported on a fractal set and has a rough geometry on all scales. This talk will describe the analysis of these fractal properties. Joint work with Raoul Normand.

2022. 12. 14. (Wed) 14:00-15:00  Room 1424
Noda Kohei (Kyushu University)

Zeros of random power series with dependent Gaussian coefficients

ABSTRACT. Peres and Virág showed that zeros point process of random power series with independent, identically distributed (i.i.d.) standard complex Gaussian coefficients is the determinantal point process on the unit disk. In this talk, if we replace i.i.d. complex Gaussian coefficients with stationary complex Gaussian coefficients, we show that the expected number of zeros is sensitively affected by the zeros of spectral density of stationary Gaussian process of coefficients. This is joint work with Tomoyuki Shirai (Kyushu University).

2022. 12. 13. (Tue) 17:00-18:00  Room Online
Subhro Ghosh (National University of Singapore)

The unreasonable effectiveness of determinantal processes

ABSTRACT. In 1960, Wigner published an article famously titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. In this talk we will, in a small way, follow the spirit of Wigner’s coinage, and explore the unreasonable effectiveness of determinantal processes (a.k.a. DPPs) far beyond their context of origin. DPPs originated in quantum and statistical physics, but have emerged in recent years to be a powerful toolbox for many fundamental learning problems. In this talk, we aim to explore the breadth and depth of these applications. On one hand, we will explore a class of Gaussian Determinantal Processes (GDP) and the novel stochastic geometry of their parameter modulation, and their applications to the study of directionality in data and dimension reduction. At the other end, we will consider the fundamental paradigm of stochastic gradient descent, where we leverage connections with orthogonal polynomials to design a minibatch sampling technique based on data-sensitive DPPs; with provable guarantees for a faster convergence exponent compared to traditional sampling. Based on the following works.

[1] Gaussian determinantal processes: A new model for directionality in data, with P. Rigollet, Proceedings of the National Academy of Sciences, vol. 117, no. 24 (2020), pp. 13207--13213 (PNAS Direct Submission)

[2] Determinantal point processes based on orthogonal polynomials for sampling minibatches in SGD, with R. Bardenet and M. Lin, Spotlight at NeurIPS 2021

2022. 12. 12. (Mon) 16:00-18:00  Room 1423
Soojung Kim (Soongsil University)

Field-induced instabilities of smectic liquid crystals

ABSTRACT. In this talk, we discuss liquid crystal flows in an applied magnetic field. In the nematic phase of a liquid crystal, molecules tend to align along a preferred direction but have no positional order. When lowering the temperature, layered molecular arrangement of the smectic phase occurs. Considering the de Gennes model for smectic liquid crystals, we study dynamical instabilities of pure smectic states and pure nematic states induced by magnetic fields. This is joint work with Xing-Bin Pan.

2022. 12. 8. (Thu) 17:00-18:00  Room Online
Yacin Ameur (Lund)

Two-dimensional Coulomb gas: fluctuations through a spectral gap

ABSTRACT. The talk will focus around some new results relating to fluctuations of smooth linear statistics of two-dimensional Coulomb gas ensembles, where the droplet is disconnected, i.e., it has some "spectral gaps". The particles which fall near the boundary of the gap come with an additional uncertainty, since there are several disjoint boundary components to choose from. For a class of radially symmetric model ensembles, we quantify this uncertainty in terms of the Jacobi theta function as well as a discrete Gaussian distribution. (Joint work with Christophe Charlier and Joakim Cronvall.)

2022. 12. 2. (Fri) 15:00-16:00  Room 8309
Jinwook Jung (Junbook University)

On the Cauchy problem for the pressureless Euler-Navier-Stokes system in the whole space

ABSTRACT. In this talk, we discuss the global Cauchy problem for a two-phase fluid model consisting of the pressureless Euler equations and the incompressible Navier-Stokes equations where the coupling of two equations is through the drag force. We establish the global-in-time existence and uniqueness of classical solutions for that system when the initial data are sufficiently small and regular. Main difficulties arise in the absence of pressure in the Euler equations. In order to resolve it, we properly combine the large-time behavior of classical solutions and the bootstrapping argument to construct the global-in-time unique classical solutions. Moreover, under a suitable assumption on the density of the pressureless Euler fluid flow, we show that the decay rate of the higher-order derivatives of fluid velocities, whose order is smaller than the dimension, is faster than that of lower-order derivatives. This talk is based on the joint work with Y.-P. Choi (Yonsei University).

2022. 12. 2. (Fri) 10:00-11:00  Room Online
Giorgio Cipolloni (Princeton)

How do the eigenvalues of a large non-Hermitian random matrix behave?

ABSTRACT. We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. Then, motivated by the long time behaviour of the ODE \dot{u}=Xu, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of X.

2022. 12. 1. (Thu) 17:00-18:00  Room Online
Felipe Marceca (University of Vienna)

Perfect freezing and discrepancy estimates for Gaussian beta ensembles at the microscopic scale

ABSTRACT. The points from a Gaussian beta ensemble tend to distribute according to the semicircular law. In this talk, we characterize which betas guarantee that this behavior holds even at microscopic scales. More precisely, we show when there is an almost sure low discrepancy at the microscale as the number of points goes to infinity. This happens exactly under the perfect-freezing regime, where beta grows at least logarithmically with the number of points. Joint work with Yacin Ameur and José Luis Romero.

2022. 11. 28. (Mon) 17:00-18:00  Room Online
Jinyeop Lee (Ludwig Maximilian University of Munich)

On the characterisation of fragmented Bose-Einstein condensation and its emergent effective evolution

ABSTRACT. Fragmented Bose-Einstein condensates are large systems of identical bosons displaying multiple macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states.

In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.

2022. 11. 24. (Thu) 17:00-18:00  Room Online
Lun Zhang (Fudan University)

Gap probability for the hard edge Pearcey process

ABSTRACT. The hard edge Pearcey process is universal in random matrix theory and many other stochastic models. In this talk, we consider gap probabilityfor the thinned/unthinned hard edge Pearcey process over the interval (0,s). By working on the relevant Fredholm determinants, we obtain an integral representation of the gap probability via a Hamiltonian related a system of coupled differential equations and the large gap asymptotics. Moreover, we also establish asymptotic statistical properties of the counting function for the hard edge Pearcey process. This talk is based on joint works with Dan Dai, Shuai-Xia Xu and Luming Yao.

2022. 11. 21. (Mon) 17:00-18:00  Room Online
Minhyun Kim (Bielefeld University)

Wiener criterion for nonlocal Dirichlet problems

ABSTRACT. In this talk, we study the boundary behavior of solutions to the Dirichlet problems for nonlocal nonlinear operators. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory. This talk is based on a joint work with Ki-Ahm Lee and Se-Chan Lee.

2022. 11. 18. (Fri) 12:00-13:00  Room 8101
Soobin Cho (Seoul National University)

Heat kernel estimates for Dirichlet forms degenerate at the boundary

ABSTRACT. In this talk, I will consider the existence and two-sided estimates of heat kernels for two types of purely discontinuous symmetric Markov processes in the upper half-space of $\mathbb R^d$ with jump kernels degenerate at the boundary. The first type of processes are conservative Markov processes on the closed half-space, and the second type of processes are the processes aforesaid killed either by a critical potential or upon hitting the boundary of the half-space. The approximate factorization property holds for these types of processes. I will explain a probabilistic viewpoint on the result.

Joint work with Panki Kim (SNU), Renming Song (Univ. Of Illinois) and Zoran Vondraček (Univ. Of Zagreb)

2022. 11. 18. (Fri) 11:00-12:00  Room 8101
Zoran Vondraček (University of Zagreb)

Positive self-similar Markov processes obtained by resurrection

ABSTRACT. In this talk I will consider positive self-similar Markov processes obtained by resurrecting a strictly $\alpha$-stable process at their first exit time from $(0,\infty)$. These processes are constructed by using the Lamperti transform.

I will explain their long term behavior and give conditions for absorption at 0 at finite time. In case the process is absorbed at 0, I will give a necessary and sufficient condition for the existence of a recurrent extension. The interest in such resurrected processes comes from the fact that their jump kernel may explode at 0.

Joint work with Panki Kim (SNU) and Renming Song (Univ. Of Illinois)

2022. 11. 17. (Thu) 14:00-15:00  Room 1424
Sunghan Kim (Uppsala University)

On vectorial obstacle problems

ABSTRACT. In this talk, I will discuss some new observations on vectorial obstacle problems. The problem itself has a long history, which began around late 70s and continued through 80s. However, the problem was studied from the standpoint of the theory of harmonic maps, and the main focus was laid upon the partial regularity of solutions, rather than their behavior near the free boundaries. About three decades later, the extension of free boundary problems into vectorial settings has regained its attention from the free boundary community. Very recently, we observed some interesting issues regarding the behavior of the vectorial solutions. More specifically, when we decompose the solutions into tangential and normal components, the optimal regularity of the tangential component has been untouched, and turned out to be highly tantalizing. The study of the optimal regularity for the tangential component seems to open up some new type of PDE in connection with the free boundary problems, which is interesting of its own. In this talk, I will present some partial result obtained in this direction. This talk will be based on a recent collaboration with A. Figalli and H. Shahgholian.

2022. 11. 9. (Wed) 16:00-17:00  Room 1424
Garca Martnez, Xabier (Vrije Universiteit Brussel)

Characterising the variety of Lie algebras.

ABSTRACT. The variety of Lie algebras is the central example of a non-associative algebra. Many properties, studied from a universal algebra point of view, were firstly introduced to Lie algebras and then generalised to other different structures. One of the many aims of categorical algebra is to give general definitions that can help to understand these notions and their generalisations.

The aim of this talk is to discuss two categorical algebraic ideas, representability of actions and the existence of algebraic exponents. They are both interesting in the following way: we can characterise the variety of Lie algebras amongst all varieties of non-associative algebras over an infinite field. Moreover, these characterisations are categorical; in the sense that no elements are used, only morphisms and universal properties.

We will discuss the algebraic meanings of these properties, the computational methods used to obtain the characterisations, and what can be done beyond them.

Joint work with Vladimir Dotsenko, Corentin Vienne and Tim Van der Linden.

2022. 8. 29. (Mon) 14:00-15:00  Room 1424
Jinwoo Sung (University of Chicago)

Equivalence of Liouville quantum gravity measure and metric

ABSTRACT. A Liouville quantum gravity (LQG) surface is a “canonical” random two-dimensional surface. In a geometric approach to the LQG, it has initially been formulated as a random measure space, and later as a random metric space. In this talk, we show that the LQG volume measure can be recovered as a Minkowski content measure for the LQG metric. Our primary tool is the continuum mating-of-trees theory for space-filling SLE. Joint work with Ewain Gwynne (University of Chicago).

2022. 7. 28. (Thu) 11:00-12:00  Room 1424
Jiewon Park (Yale)

Lojasiewicz inequalities and their implications

ABSTRACT. We will discuss two instances of how the Lojasiewicz inequality is used to prove uniqueness of limits: for tangent cones of Ricci-flat manifolds, and for tangent flows for mean curvature flow. In the former situation, the Lojasiewicz inequality follows from a careful analysis of the Green function, which we will examine in detail and survey some related results.

2022. 7. 14. (Thu) 17:00-18:00  Room 8101
Seoyeon Yang (Princeton)

Cutoff and Dynamical Phase Transition for the General Multi-component Ising Model

ABSTRACT. The phase transition between cutoff and metastability is observed universally in many cases of statistical physics, typically when the system is determined by parameters, such as in the Ising/Potts models. Several rigorous studies have been conducted including the classical Curie--Weiss Ising model and the Ising model on a regular tree or lattice. Meanwhile, The multi-component model also has received considerable attention as an approximation of lattice models or as a block networking system. Despite the growing interest and significant research on the multi-component model, coupled with the widespread belief that the dynamics can exhibit cutoff–metastability transition, relatively few cases have been introduced to rigorously verify this conjecture. In this talk, the overall proof flow is a coupling argument, which is the universally accepted method, the detailed content is sufficiently delicate and sophisticated to deal with the general interaction strength.

2022. 7. 14. (Thu) 16:00-17:00  Room 8101
Seonwoo Kim (Seoul National University)

Metastability of Ising/Potts models in large volumes in the zero-temperature limit

ABSTRACT. In this talk, we consider Ising and Potts models on finite Euclidean lattices in the zero-temperature limit, which are classical spin systems studied from the early 20th century. In the case where the lattice is fixed, there are a lot of references starting from the fundamental work [Neves and Schonmann, CMP 1991]. However in the case where the lattice size also diverges, nothing much is obtained up to this point, [Bovier, den Hollander and Spitoni, AOP 2010] being the only remarkable result so far. We introduce our recent research on this large-volume case under zero external field. There is an interesting phenomenon of phase transition regarding the competition of divergence rates of the inverse temperature and the lattice size. Moreover, in the regime where the inverse temperature diverges sufficiently faster, we formulate the metastable behavior using the Markov chain model reduction method. Joint work with Insuk Seo.

2022. 6. 13. (Mon) 16:00-17:30  Room 8101
Hanbaek Lyu (University of Wisconsin - Madison)

Convergence and Complexity of Stochastic Block Majorization-Minimization

ABSTRACT. Stochastic majorization-minimization (SMM) is an online extension of the classical principle of majorization-minimization, which consists of sampling i.i.d. data points from a fixed data distribution and minimizing a recursively defined majorizing surrogate of an objective function. In this talk, we introduce stochastic block majorization-minimization, where the surrogates can now be only block multi-convex and a single block is optimized at a time within a diminishing radius or with a proximal regularization. Relaxing the standard strong convexity requirements for surrogates in SMM, our framework gives wider applicability including online CANDECOMP/PARAFAC (CP) dictionary learning and yields greater computational efficiency especially when the problem dimension is large. We provide an extensive convergence analysis on the proposed algorithm, which we derive under possibly dependent data streams, relaxing the standard i.i.d. assumption on data samples. We show that the proposed algorithm converges almost surely to the set of stationary points of a nonconvex objective under constraints at a rate $O((\log n)^{1+\epsilon}/n^{1/2})$ for the empirical loss function and $O((\log n)^{1+\epsilon}/n^{1/4})$ for the expected loss function, where n denotes the number of data samples processed. Under some additional assumption, the latter convergence rate can be improved to $O((\log n)^{1+\epsilon}/n^{1/2})$. Our results provide first convergence rate bounds for various online matrix and tensor decomposition algorithms under a general Markovian data setting.

2022. 6. 9. (Thu) 15:00-16:00  Room 1423
Daesung Kim (University of Illinois at Urbana-Champaign)

Probabilistic approach to discrete singular integrals on $\mathbb{Z}^d$

ABSTRACT. Gundy and Varopoulos introduced the probabilistic representation of singular integrals and Fourier mulitpliers such as Hilbert transforms and Riesz transforms as conditional expectations of some stochastic integrals. Combining with the sharp martingale inequalities by Burkholder and Banuelos-Wang, the representations have played a crucial role in finding the sharp, or nearly sharp, $L^p$-bounds for these operators in a variety of geometric settings. Motivated by a recent breakthrough by Banuelos and Kwasnicki on the sharp $\ell^p$-norm of the discrete Hilbert transform, we construct a natural collection of discrete operators in $\mathbb{Z}^{d}$ which have $\ell^p$-norms independent of the dimension. This collection of discrete operators include the probabilistic discrete Riesz transforms, which is the analogues of the probabilistic discrete Hilbert transform used in the paper by Banuelos-Kwasnicki. We also discuss related open problems for the $\ell^p$-norm of discrete operators. This is based on joint work with Rodrigo Banuelos and Mateusz Kwasnicki.

2022. 5. 20. (Fri) 16:50-17:50  Room 1424
Juyoung Lee (Seoul National University)

$L^p$-$L^q$ estimates for the circular maximal operator on Heisenberg radial functions

ABSTRACT. $L^p$ boundedness of various maximal functions have received considerable attentions recently. One of the interesting open problem is the circular maximal function on Heisenberg group H^1. While it remains open, Beltran, Guo, Hickman, Seeger obtained sharp L^p boundedness when the function is Heisenberg radial. Recently, we obtained the improved result on Heisenberg radial functions. Also, our method significantly simplify the proof of Beltran et al. In this talk, we first compare Euclidean problems and Heisenbreg problems. Then, we focus on the major difference between the proof of Beltran et al and ours.

2022. 5. 20. (Fri) 15:40-16:40  Room 1424
Sewook Oh (Seoul National University)

Optimal Sobolev regularity estimate for averages over curves

ABSTRACT. The regularity property of averages over curves is closely related to boundedness of maximal averages. In this talk, we are interested in Lp Sobolev regularity estimate for averages over curves. We briefly overview previous results when d=2,3,4 and show the main idea to prove optimal Sobolev regularity estimate in higher dimensions. This is based on the joint work with Hyerim Ko and Sanghyuk Lee.

2022. 5. 20. (Fri) 14:00-15:30  Room 1424
Seheon Ham (Seoul National University)

Circular average relative to fractal measures

ABSTRACT. In this talk, we consider $L^p-L^q$ estimates for averages over dilates of the unit circle with respect to fractal measures, which unify different type of maximal functions for the circular average. We show the estimates can be applied to the problem of determining Hausdorff dimension of union of circles. We also discuss similar results concerning the spherical average. This talk is based on recent joint works with Hyerim Ko and Sanghyuk Lee.

2022. 5. 12. (Thu) 16:30-18:00  Room 1424
Jin Bong Lee (Seoul National University)

Multilinear Fourier multipliers on Hardy spaces

ABSTRACT. A study on multilinear operators is motivated by certain functionals arising in theory of partial differential equations or operator theory. For a general frame work of the study, it is required that quantitative estimates for multilinear operators of abstract forms. In this talk, the speaker introduces recent results on boundedness of multilinear operators associated with Fourier multipliers on Hardy spaces. The Fourier multipliers of our particular interest satisfy multilinear analogues of the H\"ormander condition, which is a sufficient condition for the $L^p$ multiplier problem in linear theory.

2022. 5. 12. (Thu) 14:40-16:10  Room 1424
Hyerim Ko (Seoul National University)

Boundedness of maximal averages over curves

ABSTRACT. The study of maximal function has been of interest in harmonic analysis. In this talk, we are interested in Lp boundedness of the maximal function which takes average over space curve in R3. We first give brief overview of earlier works of Pramanik and Seeger and show the main idea to obtain the sharp maximal bound. This is based on the joint work with Sanghyuk Lee and Sewook Oh.

2022. 5. 12. (Thu) 12:00-13:00  Room Online
Kang-Ju Lee (Seoul National University)

Simplicial networks and Laplacians for higher-order interactions

ABSTRACT. Pairwise interactions between two vertices are encoded in edges of graph networks. However, these edges are not sufficient to describe a wide variety of interactions arising from the world around us. Recently, faces in simplicial complexes have been employed to represent interactions among more than two vertices, which have shown various applications in several areas. In this talk, I will introduce where higher-order interactions emerge and explain a simplicial generalization of graph Laplacians as a tool for analyzing simplicial networks.

2022. 5. 12. (Thu) 13:00-14:30  Room 1424
Chuhee Cho (Seoul National University)

Pointwise convergence of sequences of Schrodinger means

ABSTRACT. We study pointwise convergence of the generalized Schr\"odinger mean $e^{it_n(-\Delta)^{\alpha/2}}f$ for a sequence $\{t_n\}$ which converges to zero. Our results extend the recent work of Dimou and Seeger, and Li, Wang and Yan to higher dimensions. The main novelty of the proof is to showing that the pointwise convergence of the sequential Schr\"odinger means follows automatically from the classical results on convergence of the Schr\"odinger operators. This strategy also works for other dispersive operators, which in turn provides alternative proof of the previous results on the sequential convergence of various operators. This talk is based on the joint work with Hyerim Ko, Youngwoo Koh and Sanghyuk Lee.

2022. 3. 15. (Tue) 15:00-17:00  Room Online
Jeongho Kim (Hanyang University)

Numerical methods for gauged Schr\"odinger equations

ABSTRACT. Developing the numerical schemes for PDE is essential to understand the nature of physical phenomena. However, there has been a limited literature on the gauged Schr\"odinger equations, describing the dynamics of quantum particle under the electromagnetic fields. We suggest numerical schemes for gauged Schr\"odinger equations in 1D and 2D, based on the finite difference method and semi-Lagrangian method. For the finite difference method, we prove and validate that the total charge of the system is exactly conserved, while we present a convergence analysis for the semi-Lagrangian method.

2022. 3. 11. (Fri) 16:00-18:00  Room Online
Jeongho Kim (Hanyang University)

Hydrodynamic limits of the Schr\"odinger-type equations

ABSTRACT. A hydrodynamic formulation of the quantum system, so-called "quantum hydrodynamics" has been interesting research area from 1930s. However, a rigorous convergence analysis of the Schr\"odinger-type equation to the hydrodynamic system, such as compressible Euler equations, is justified recently. In this talk, we use the modulated energy methods to obtain hydrodynamic limits for the Schr\"odinger-type equations, including the gauged Schr\"odinger equation. We also combine the theory of Wasserstein distance estimate for the continuity equation with the modulated energy method, to derive the hydrodynamic limit in the absence of nonlinear potential.

2022. 3. 8. (Tue) 16:00-18:00  Room Online
Jeongho Kim (Hanyang University)

Asymptotic analysis of the Fokker-Planck-Alignment equation to the Euler-Alignment system with the singular interaction.

ABSTRACT. A Fokker-Planck-Alignment system is recently introduced to model the collective behaviors of multi-agent system, in particular when the number of agents is large enough. We present a rigorous, and quantified hydrodynamic limit from the Fokker-Planck-Alignment equation to the Euler-Alignment system, which is macroscopic description of the multi-agent alignment system. When the interaction weight is bounded and regular, hydrodynamic limit problems are already well-known. However, when the interaction is singular, few results are known and in particular, asymptotic convergence toward the isothermal Euler-Alignment system is completely open. We use the relative entropy method, together with the careful estimate on the singular interaction between agents, to derive a quantified estimate for the hydrodynamic limit of the Fokker-Planck-Alignment equation.

2022. 3. 4. (Fri) 16:00-18:00  Room Online
Jeongho Kim (Hanyang University)

Cauchy problems for generalized BGK model for Boltzmann equation.

ABSTRACT. In this talk, we suggest a generalization of the celebrated BGK model for Boltzmann equation in kinetic theory of gases, referred as Tsallis-BGK model. We consider the one-parameter family of equilibrium, so-called q-Maxwellian, as a relaxation distribution for the BGK-type equations. For 1 < q < 7/5, we present a global existence of smooth solution to the Tsallis-BGK equation near the global q-Maxwellian, using the linearization technique. On the other hand, when 0 < q < 1, we study a global existence of weak solution to the Tsallis-BGK equation, based on the careful estimate of newly introduced entropy functional.

2022. 2. 25. (Fri) 14:00-15:00  Room Online
Kunwoo Kim (POSTECH)

Intermittency and related results

ABSTRACT. Intermittency is usually referred to as a phenomenon that shows very tall peaks in small regions, and a large family of stochastic heat equations exhibits those intermittent phenomena. In this talk, we first introduce the definition of mathematical intermittency by Carmona-Molchanov, who define full intermittency in terms of the moment Lyapunov exponents. After that, we consider stochastic heat equation with space-time white noise on the real line and study limit theorems of time-dependent averages of a solution to stochastic heat equation with space-time white noise. We provide the explicit conditions that guarantee the law of large numbers and the central limit theorem. The conditions are related to the moment Lyapunov exponents and the full intermittency condition.

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

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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
	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
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