학과 세미나 및 콜로퀴엄
In recent years, machine learning techniques based on neural networks have achieved remarkable success across various fields, and they have demonstrated a notable ability to represent solutions to inverse problems. From a mathematical perspective, the core aspect of this success lies in their strong approximation ability to target functions, underscoring the importance of understanding their approximation properties. As wavelet systems offer notable advantages in approximation, this talk focuses on neural network approximations that employ such systems. We will begin by studying wavelet systems' fundamental structures and basic properties, then introduce main approximation theories using wavelet frames. Finally, we will explore recent studies on neural networks that incorporate these wavelet systems.
In this talk, we study the scattering problem for the initial value problem
of the generalized Korteweg-de Vries (gKdV) equation. The purpose of
this talk is to achieve two primary goals. Firstly, we show small data scattering
for (gKdV) in the weighted Sobolev space, ensuring the initial and
the asymptotic states belong to the same class. Secondly, we introduce
two equivalent characterizations of scattering in the weighted Sobolev space.
In particular, this involves the so-called conditional scattering in the weighted
Sobolev space. This talk is based on a joint work with Satoshi Masaki (Hokkaido University)
We present HINTS, a Hybrid, Iterative, Numerical, and Transferable Solver that combines Deep Operator Networks (DeepONet) with classical numerical methods to efficiently solve partial differential equations (PDEs). By leveraging the complementary strengths of DeepONet’s spectral bias for representing low-frequency components and relaxation or Krylov methods’ efficiency at resolving high-frequency modes, HINTS balances convergence rates across eigenmodes. The HINTS is highly flexible, supporting large-scale, multidimensional systems with arbitrary discretizations, computational domains, and boundary conditions, and can also serve as a preconditioner for Krylov methods. To demonstrate the effectiveness of HINTS, we present numerical experiments on parametric PDEs in both two and three dimensions.
We present scEGOT, a comprehensive single-cell trajectory inference framework based on entropic Gaussian mixture optimal transport. The main advantage of scEGOT allows us to go back and forth between continuous and discrete problems, and it provides a versatile trajectory inference method including reconstructions of the underlying vector fields at a low computational cost. Applied to the human primordial germ cell-like cell (PGCLC) induction system, scEGOT identified the PGCLC progenitor population and bifurcation time of segregation. Our analysis shows TFAP2A is insufficient for identifying PGCLC progenitors, requiring NKX1-2.
Title: On a polynomial basis for MZV’s in positive characteristic
Abstract: We recall the notion of the stuffle algebra and review known results for this algebra in characteristic 0. Then, we construct a polynomial basis for the stuffle algebra over a field in positive characteristic. As an application, we determine the transcendence degree for multiple zeta values in positive characteristic for small weights. This is joint work with Nguyen Chu Gia Vuong and Pham Lan Huong
Topological data analysis (TDA) is an emerging concept in applied mathematics, by which we can characterize shapes of massive and complex data using topological methods. In particular, the persistent homology and persistence diagrams are nowadays applied to a wide variety of scientific and engineering problems. In this talk, I will survey our recent research on persistent homology from three interrelated perspectives; quiver representation theory, random topology, and applications on materials science. First, on the subject of quiver representation theory, I will talk about our recent challenges to develop a theory of multiparameter persistent homology on commutative ladders. By applying interval decompositions/approximations on multiparameter persistent homology (Asashiba et al, 2022) to our setting, I will introduce a new concept called connected persistence diagrams, which properly possess information of multiparameter persistence, and show some properties of connected persistence diagrams. Next, about random topology, I will show our recent results on limit theorems (law of large numbers, central limit theorem, and large deviation principles) of persistent Betti numbers and persistence diagrams defined on several stochastic models such as random cubical sets and random point processes in a Euclidean space. Furthermore, I will also explain a preliminary work on how random topology can contribute to understand the decomposition of multiparameter persistent homology discussed in the first part. Finally, about applications, I will explain our recent activity on materials TDA project. By applying several new mathematical tools introduced above, we can explicitly characterize significant geometric and topological hierarchical features embedded in the materials (glass, granular systems, iron ore sinters etc), which are practically important for controlling materials funct
Recently, Bowden-Hensel-Webb introduced the notion of fine curve graph as an analogue of the classical curve graph. They used this to construct nontrivial quasi-morphisms (in fact, infinitely many independent ones) on Homeo_0(S). Their method crucially uses independent pseudo-Anosov conjugacy classes, whose existence follows from the WPD-ness of pseudo-Anosov mapping classes on the curve graph. Meanwhile, the WPD-ness of pseudo-Anosov maps on the fine curve graph is not achievable, as Homeo_0(S) is a simple group.
In this talk, I will explain my ongoing regarding an analogue of WPD-ness for point-pushing pseudo-Anosov maps on the fine curve graph. If time allows, I will explain how this is related to the construction of independent pseudo-Anosov conjugacy classes in Homeo_0(S).
(E6-1) Room 1501
편미분방정식
Professor Sir John Macleod Ball (Heriot-Watt University)
Distinguished Lectures
(E6-1) Room 1501
편미분방정식
Ist lecture: Understanding material microstructure Abstract Under temperature changes or loading, alloys can form beautiful patterns of microstructure that largely determine their macroscopic behaviour. These patterns result from phase transformations involving a change of shape of the underlying crystal lattice, together with the requirement that such changes in different parts of the crystal fit together geometrically. Similar considerations apply to plastic slip. The lecture will explain both successes in explaining such microstructure mathematically, and how resolving deep open questions of the calculus of variations could lead to a better understanding. 2nd lecture: Monodromy and nondegeneracy conditions in viscoelasticity Abstract For certain models of one-dimensional viscoelasticity, there are infinitely many equilibria representing phase mixtures. In order to prove convergence as time tends to infinity of solutions to a single equilibrium, it is necessary to impose a nondegeneracy condition on the constitutive equation for the stress, which has been shown in interesting recent work of Park and Pego to be necessary. The talk will explain this, and show how in some cases the nondegeneracy condition can be proved using the monodromy group of a holomorphic function. This is joint work with Inna Capdeboscq and Yasemin Şengül.
In this talk, we will discuss Leray-Hopf solutions to the two-dimensional Navier-Stokes equations with vanishing viscosity. We aim to demonstrate that when the initial vorticity is only integrable, the Leray-Hopf solutions in the vanishing viscosity limit do not exhibit anomalous dissipation. Moreover, we extend this result to the case where the initial vorticity is merely a Radon measure, assuming its singular part maintains a fixed sign. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. This is a joint work with Luigi De Rosa (University of Basel).
Dimensionality reduction represents the process of generating a low dimensional representation of high dimensional data. In this talk, I explain what dimensionality reduction is and shortly mention formation control. After that, I will introduce a nonlinear dynamical system designed for dimensionality reduction. I briefly discuss mathematical properties of the model and demonstrate numerical experiments on both synthetic and real datasets.
This is a reading seminar to be given by Mr. Jaehong Kim (a graduate student in the department) on foundations of the intersection theory and the classification theory of complex algebraic surfaces. He will give three 2-hour long talks.
Let (R,m_R) be a d-dimensional, excellent, normal local ring. A divisorial filtration {I_n} is determined by a divisor D on a normal scheme X determined by blowing up an ideal on R, so that I_n are the global sections of nD. Associated to an m_R-primary divisorial filtration, we have the Hilbert function f(n)=\lambda_R(R/I_n), where \lambda_R is the length of an R-module. We discuss how close or far this function is from being a polynomial, focusing on examples which are constructed and analyzed geometrically.
In this talk, we consider a finite rational map determined by a linear system with base locus. The degree of such map has been studied in many situation, for instance, the degree of Gauss map of theta divisors. In principal, this degree can be computed by Segre class of the base locus. In practice, one can use Vogel's cycle to give an estimation. Associated to the base locus, one can define distinguished subvarieties, which has been used to the study of geometric Nullstellensatz by Ein-Lazarsfeld. We discuss how distinguished subvarieties and their coefficients can be used to estimate the degree of finite rational map. This is a joint work with Yilong Zhang.
Given two relatively prime positive integers, p < q, Kunz and Waldi defined a class of numerical semigroups which we denote by KW(p, q) consisting of semigroups of embedding dimension n and type n−1 and multiplicity p by filling in the gaps of the semigroup < a, b >. We study these semigroups, give a criterion for these in terms of principal matrices or their critical binomials and generalize the notion to KW(p, q, w) and prove
some results and questions. We will discuss their resolutions and Betti Numbers. Most of this is a joint work with Srishti Singh.
This is a presentation by Mr. Taeyoon Woo, a graduate student in the department, after his reading course on basics on compact Riemann surfaces.
He will concentrate on topics such as degree theory of holomorphic maps, Riemann-Roch theorem, residue theorem, Serre duality, Riemann-Hurtiwz theorem, Hodge decomposition, etc. on the compact Riemann surfaces. If time permits, he will discuss its connections to smooth manifolds and algebraic curves.
Modern datasets are often characterized by high-dimensionality and heterogenous environments under distribution shifts, posing significant challenges in terms of signal recovery, robustness, and interpretability. In this talk, I will present three research contributions to address these challenges. First, I will introduce the notion of local concavity coefficients, a novel tool for quantifying the concavity of a set. I will demonstrate its effectiveness in analyzing optimization problems for signal recovery in high-dimensional settings. Second, I will discuss recent advancements in machine learning to handle distribution shifts and emphasize the critical role of invariant features in achieving robust predictions. Finally, I will show how wavelets allow for interpreting feature spaces learned by deep neural networks. Motivated by applications in cosmology, I'll showcase how this tool can be applied to the problem of cosmological parameter inference.
This is a reading seminar to be given by Mr. Jaehong Kim (a graduate student in the department) on foundations of the intersection theory and the classification theory of complex algebraic surfaces. He will give three 2-hour long talks.
A perfect field is said to be Kummer-faithful if the Kummer maps for semiabelian varieties over the field are injective. This notion originates in the study of anabeian geometry. At the same time, our study is also motivated by a conjecture of Frey and Jarden on the structure of Mordell-Weil groups over large algebraic extensions of a number field.
I will begin with a review of known results in this direction, as well as a brief discussion on anabelian geometry. Then I will introduce some recent results on the construction of "large" Kummer faithful fields. This is a joint work with Takuya Asayama.
I will begin by a brief introduction to anabelian geometry.
In particular, I will try to explain the distinction between "bi-" and "mono-anabelian" reconstruction.
Then I review some of the known (elementary) mono-anabelian reconstruction of invariants of mixed characteristic local fields.
Finally, I will explain my (on-going) trial of the mono-anabelian reconstruction of fundamental character and Lubin-Tate character.
In this talk, we consider the self-dual O(3) Maxwell–Chern–Simons-Higgs equation, a semilinear elliptic system, defined on a flat two torus. We discuss about pointwise convergence behavior, which represents the Chern-Simons limit behavior of our system. Building upon this observation, we study the existence, stability, and asymptomatic behavior of solutions.
Virtual element method (VEM) is a generalization of the finite element method to general polygonal (or polyhedral) meshes. The term ‘virtual’ reflects that no explicit form of the shape function is required. The discrete space on each element is implicitly defined by the solution of certain boundary value problem. As a result, the basis functions include non-polynomials whose explicit evaluations are not available. In implementation, these basis functions are projected to polynomial spaces. In this talk, we briefly introduce the basic concepts of VEM. Next, we introduce mixed virtual volume methods (MVVM) for elliptic problems. MVVM is formulated by multiplying judiciously chosen test functions to mixed form of elliptic equations. We show that MVVM can be converted to SPD system for the pressure variable. Once the primary variable is obtained, the Darcy velocity can be computed locally on each element.
In this talk, I will discuss the expansion of the free energy of two-dimensional Coulomb gases as the size of the system increases. This expansion plays a central role in proving the law of large numbers and central limit theorems. In particular, I will explain how potential theoretic, topological, and conformal geometric information of the model arises in this expansion and present recent developments.
The Gromov-Wasserstein (GW) distance is a generalization of the standard Wasserstein distance between two probability measures on a given ambient metric space. The GW distance assumes that these two probability measures might live on different ambient metric spaces and therefore implements an actual comparison of pairs of metric measure spaces. A metric-measure space is a triple (X,dX,muX) where (X,dX) is a metric space and muX is a fully supported Borel probability measure over X.
In Machine Learning and Data Science applications, this distance is estimated either directly via gradient based optimization approaches, or through the computation of lower bounds which arise from distributional invariants of metric-measure spaces. One particular such invariant is the so-called ‘global distance distribution’ which precisely encodes the distribution of pairwise distances between points in a given metric measure space. This invariant has been used in many applications yet its classificatory power is not yet well understood.
This talk will overview the construction of the GW distance, the stability of distributional invariants, and will also discuss some results regarding the injectivity of the global distribution of distances for smooth planar curves, hypersurfaces, and metric trees.
This is a reading seminar to be given by Mr. Jaehong Kim (a graduate student in the department) on foundations of the intersection theory and the classification theory of complex algebraic surfaces. He will give three 2-hour long talks.