Title: A BRIEF INTRODUCTION TO UNIFORMIZATION THEOREM AND DEFORMATION THEORY IN COMPLEX GEOMETRY 

Abstract: In this presentation, we will discuss a brief introduction to two significant theories in complex geometry: the uniformization theorem and deformation theory. 

To begin, we revisit some fundamental concepts from one-dimensional complex analysis, including holomorphic functions and their properties, along with the Riemann mapping theorem. Next, we discuss the uniformization theorem for Riemann surfaces, which says that every Riemann surface is covered by one of three spaces: the Riemann sphere, 1- dimensional complex Euclidean space, the unit disc. Lastly, we explore deformation theory, which says how complex structures can change under a smooth deformation of a given manifold. Deformation theory is foundational to the study of moduli spaces of Riemann surfaces, more generally compact complex manifolds with certain topological conditions.

Title: Hecke algebras, Whittaker functions and quantum groups 

Abstract: : I will give a brief overview of some topics related to the representation theory of p-adic groups and how one can use certain algebraic and combinatorial tools in their study (ex: Satake isomorphism, Casselman-Shalika formula, Hecke algebras, Kazhdan-Lusztig polynomials and symmetric functions). Time permitting, I will explain recent work by Williamson et. al. that uses "machine learning" to study Kazhdan-Lusztig polynomials.

Title : A survey of scalar curvature rigidity of compact domains

Abstract: Recently, because of the influence of Gromov, there is a lot of progress in scalar curvature geometry. I will survey a few results from others and report some of my findings in scalar curvature rigidity of compact domains: in particular, Gromov dihedral rigidity conjecture in hyperbolic space, Gromov's conjecture on the rigidity of three-dimensional balls, warped products.

Title: Game-theoretic approaches to partial differential equations and related problems 

Abstract: Probabilistic methodology has been used as a useful tool to study PDEs in recent decades. This can provide a different perspective on comprehending PDEs and contribute to new mathematical discoveries. It is well-known about the relation between the Laplacian and random walk processes. The mean value property of harmonic functions plays a key role in the discussion. We can also consider similar approaches to more general equations. For nonlinear cases, 'tug-of-war' is a representative example in this respect, which is a discretized scheme for the normalized p-Laplace operator. In this talk, we are concerned with game-theoretic approaches to PDEs and their applications. We will give several examples of stochastic games relevant to various PDE problems.

Title: Optimal breakpoints of continuous piecewise polynomial regression with Wasserstein distance 

Abstract: The continuous piecewise regression is a powerful tool for identifying trend in datasets. The aim of this study is to simultaneously optimize two key elements of the piecewise polynomial regression: the number and the locations of breakpoints. In this study, we use the Wasserstein distance as a criterion to determine the optimal number of breakpoints. Compared to traditional MSE-based methods, the method proposed can effectively determine the optimal number of breakpoints. Numerical experiments are conducted using both synthetic and real datasets to demonstrate the effectiveness of the proposed method. 

Title: A hyperbolic one-relator group that is a variant of the Baumslag-Gersten group 

Abstract: In geometric group theory, Gromov's longstanding open question asks whether every hyperbolic group is residually finite. Many geometric group theorists conjecture the existence of non-residually finite hyperbolic groups. One potential candidate is the Heineken group, which is known to be hyperbolic, though its residual finiteness remains unclear. On the other hand, the Baumslag-Gersten group is a well-known example of a non-hyperbolic, non-residually finite, but Hopfian group. We construct a hyperbolic one-relator group as a variant of the Baumslag-Gersten group, which is, in fact, derived from observations of the Heineken group. Through this hyperbolic one-relator group, we present a new candidate for answering Gromov's open question. 

Title: Dynamical Billiard and a long-time behavior of the Boltzmann equation in general 3D toroidal domains  

Abstract: In this talk, we consider the Boltzmann equation in general 3D toroidal domains with a specular reflection boundary condition. So far, it is a well-known open problem to obtain the low-regularity solution for the Boltzmann equation in general non-convex domains because there are grazing cases, such as inflection grazing. Thus, it is important to analyze trajectories which cause grazing. We will provide new analysis to handle these trajectories in general 3D toroidal domains.

Title: Advances in Physics-Informed Neural Networks for Plasma Simulations and Stiff Differential Equations

Abstract: Scientific machine learning (SciML) combines machine learning with traditional scientific computing to tackle complex, high-dimensional problems often formulated as differential equations. Physics-informed neural networks (PINNs) have emerged as a powerful tool in this area, offering a flexible framework that incorporates physical principles directly into neural network training.

In this presentation, we introduce two contributions to PINNs. First, we propose Plasma-Simulation PINNs (PS-PINNs), an approach tailored for plasma discharge simulations. By employing logarithmic transformations when necessary to handle stiffness and applying a self-adaptive loss balancing (SALB) method to manage multiple physics-based loss terms, PS-PINNs effectively solve both forward and inverse problems in plasma physics. This framework proves robust and versatile for analyzing parameter dependencies under various discharge conditions.

Second, we address the challenges of solving stiff differential equations, which often exhibit rapid changes over certain regions. We introduce the Re-spacing layer (RS-layer), a pre-trained encoding layer that tackles the data imbalance caused by the skewness in point distributions. By transforming skewed sampling distributions into uniform ones, the RS-layer helps stabilize training, accelerate convergence, and improve accuracy across stiff problems.

Title: Orbital integrals and ideal class monoids for a Bass order

Abstract: A Bass order is a certain subring of a number field. The majority of number fields contain infinitely many Bass orders. For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is 4th-power-free in $\mathbb{Z}$, is a Bass order.

In this talk, I will propose a closed formula for the number of fractional ideals of a Bass order $R$, up to its invertible ideals, using the conductor of $R$. I will also explain explicit enumeration of all orders containing $R$. Our method is based on local-global argument and exhaustion argument, by using orbital integrals for $\mathfrak{gl}_n$ as a mass formula. This is joint work with Sungmun Cho and Yuchan Lee.

Title: Compressible Stokes flows in the domain with cut boundary

Abstract: In this talk we study a compressible Stokes flows in a domain with cut boundary. The cut is a non-Lipschitz boundary whose tip has 2π-opening angle. The divergence of the leading corner singularity vector function has different trace values on either side of the cut. In consequence the pressure solution derived by the continuity equation must have a jump across the streamline emanating from the cut tip and the pressure gradient in the momentum equation is not well-defined.

To handle this, we construct an auxiliary vector function called lifting function. It lifts the pressure jump value on the curve into the region. We split the velocity solution into the lifting function plus the singular one plus a smoother one. Similarly, we split the pressure solution. With these results, we establish piecewise regularity of the solutions of compressible Stokes system.

Title: Persistent Homology with Path-Representable Distances on Graph Data

Abstract: Topological data analysis over graph has been actively studied to understand the underlying topological structure of data. However, limited research has been conducted on how different distance definitions impact persistent homology and the corresponding topological inference. To address this, we introduce the concept of path-representable distance in a general form and prove the main theorem for the case of cost-dominated distances.We found that a particular injection exists among the $1$-dimensional persistence barcodes of these distances with a certain condition. We prove that such an injection relation exists for $0$- and $1$-dimensional homology. For higher dimensions, we provide the counterexamples that show such a relation does not exist.

Title: Topological rigidities of translating solitons for the mean curvature flow 

Abstract: In this talk, we study the topology of a complete translating soliton for the mean curvature flow in Euclidean space. First, we give basic preliminaries for translating solitons. Afterwords, based on the theory of minimal surfaces, we show that if a translating soliton satisfies that the L^m norm of the second fundamental form is smaller than an explicit constant, then it has no non-trivial f-harmonic 1-form of L^2_f. We also prove that the same assumption yields one end of a translating soliton. This is a joint work with Juncheol Pyo.

Title: Liouville type theorems for a quasilinear elliptic differential inequality 

Abstract: This talk is concerned with the nonexistence of nontrivial nonnegative weak solutions for a strongly p-coercive elliptic differential inequality with weighted nonlocal source and gradient absorption terms in the whole space. Under the condition that the positive weight in the absorption term is either a sufficiently small constant or more general, we establish new Liouville type results containing the critical case. The key ingredient in the proof is the rescaled test function method developed by Mitidieri and Pohozaev. This talk is based on a joint work with Zhong Bo Fang. 

Title: Eigenvalue estimates on λ-hypersurfaces 

Abstract: Self-shrinkers are important in the study of the mean curvature flow as Type I singularity model. In addition, self-shrinkers are considered as (weighted) minimal hypersurfaces in a certain weighted manifold. In the same weighted manifold, it is natural to consider hypersurfaces having constant (weighted) mean curvature, which are called λ-hypersurfaces. In this talk, we discuss the first eigenvalue estimate on complete non-compact λ-hypersurfaces. This estimate for the first eigenvalue generalizes the work of Brendle and Tsiamis on complete non-compact self-shrinkers, and Ding and Xin on closed self-shrinkers.