Given a manifold, the vertices of a geometric intersection graph are defined as a class of submanifolds. Whether there is an edge between two vertices depends on their geometric intersection numbers. The geometric intersection complex is the clique complex induced by the geometric intersection graph. Common examples include the curve (arc) complex and the Kakimizu complex. Curve complexes and arc complexes are used to understand mapping class groups and Teichmüller spaces, while Kakimizu complexes are primarily used to study hyperbolic knots. We can study these geometric intersection complexes from various perspectives, including topology (e.g., homotopy type), geometry (e.g., dimension, diameter, hyperbolicity), and number-theoretic connections (e.g., trace formulas of maximal systems). In this talk, we will mainly explain how to determine the dimension of the (complete) $1$-curve (or arc) complex on a non-orientable surface and examine the transitivity of maximal complete $1$-systems of loops on a punctured projective plane.

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.

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심사위원장: 곽시종, 심사위원: 이용남, 박진형, 박의성(고려대학교), 김영락(부산대학교)

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

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심사위원장: 임보해, 심사위원: 김완수, 박진형, 백형렬, Tuan Ngo Dac(Université de Caen Normandie, CNRS)

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심사위원장: 정연승, 심사위원:박철우, 하우석, 황강욱, 문진영(한국전자통신연구원)

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).

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심사위원장: 홍영준, 심사위원: 김동환, 박노성(전산학부), 박은병(성균관대학교), 윤석배(성균관대학교)

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).