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.

---

심사위원장: 권순식, 심사위원: 강문진, 변재형, 배명진, 조용근(전북대학교)

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.

In 2005, Bollobás, Janson and Riordan introduced and extensively studied a general model of inhomogeneous random graphs parametrised by graphons. In particular, they studied the emergence of a giant component in these inhomogeneous random graphs by relating them to a broad collection of inhomogeneous Galton-Watson branching processes. Fractional isomorphism of finite graphs is an important and well-studied concept at the intersection of graph theory and combinatorial optimisation. It has many different characterizations that involve a range of very different and seemingly unrelated properties of graphs. Recently, Grebík and Rocha developed a theory of fractional isomorphism for graphons. In our work, we characterise inhomogeneous random graphs that yield the same inhomogeneous Galton-Watson branching process (and hence have a similar component structure). This is joint work with Jan Hladký and Anna Margarethe Limbach.

---

심사위원장: 김재경, 심사위원: 김용정, 정연승, 김진수(포항공과대학교), 이승규(고려대학교)

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.

Malignant gliomas are highly invasive brain tumors. Recent attention has focused on their capacity for network-driven invasion, whereby mitotic events can be followed by the migration of nuclei along long thin cellular protrusions, termed tumour microtubes (TM). Here I develop a mathematical model that describes this microtube-driven invasion of gliomas. I show that scaling limits lead to well known glioma models as special cases such as go-or-grow models, the PI model of Swanson, and the anisotropic model of Swan. I compute the invasion speed and I use the model to fit experiments of cancer resection and regrowth in the mouse brain. (Joint work with N. Loy, K.J. Painter, R. Thiessen, A. Shyntar).

We discuss on fractional weighted Sobolev spaces with degenerate weights and related weighted nonlocal integrodifferential equations. We provide embeddings and Poincare inequalities for these spaces and show robust convergence when the parameter of fractional differentiability goes to $1$. Moreover, we prove local H\"older continuity and Harnack inequalities for solutions to the corresponding nonlocal equations. The regularity results naturally extend those for degenerate linear elliptic equations presented in [Comm. Partial Differential Equations 7 (1982); no. 1; 77?116] by Fabes, Kenig, and Serapioni to the nonlocal setting. This is a joint work with Linus Behn, Lars Diening and Julian Rolfes from Bielefeld.

---

심사위원장: 안드레아스 홈슨, 심사위원: 박진형, 김은정(전산학부), 엄상일(ibs), 권오정(한양대학교)

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).

---

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.

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.

An $r$-graph is an $r$-regular graph in which every odd set of vertices is connected to its complement by at least $r$ edges. A central question regarding $r$-graphs is determining the maximum number of pairwise disjoint perfect matchings they can contain. This talk explores how edge connectivity influences this parameter. For ${0 \leq \lambda \leq r}$, let $m(\lambda,r)$ denote the maximum number $s$ such that every $\lambda$-edge-connected $r$-graph contains $s$ pairwise disjoint perfect matchings. The values of $m(\lambda,r)$ are known only in limited cases; for example, $m(3,3)=m(4,r)=1$, and $m(r,r) \leq r-2$ for all $r \not = 5$, with $m(r,r) \leq r-3$ when $r$ is a multiple of $4$. In this talk, we present new upper bounds for $m(\lambda,r)$ and examine connections between $m(5,5)$ and several well-known conjectures for cubic graphs. This is joint work with Davide Mattiolo, Eckhard Steffen, and Isaak H. Wolf.

TBA

Semi-supervised domain adaptation (SSDA) is a statistical learning problem that involves learning from a small portion of labeled target data and a large portion of unlabeled target data, together with many labeled source data, to achieve strong predictive performance on the target domain. Since the source and target domains exhibit distribution shifts, the effectiveness of SSDA methods relies on assumptions that relate the source and target distributions. In this talk, we develop a theoretical framework based on structural causal models to analyze and compare the performance of SSDA methods. We introduce fine-tuning algorithms under various assumptions about the relationship between source and target distributions and show how these algorithms enable models trained on source and unlabeled target data to perform well on the target domain with low target sample complexity. When such relationships are unknown, as is often the case in practice, we propose the Multi-Start Fine-Tuning (MSFT) algorithm, which selects the best-performing model from fine-tuning with multiple initializations. Our analysis shows that MSFT achieves optimal target prediction performance with significantly fewer labeled target samples compared to target-only approaches, demonstrating its effectiveness in scenarios with limited target labels.

---

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

TBA

For a positive real number $p$, the $p$-norm $\|G\|_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices, focusing on the case where $F$ is a bipartite graph. It is natural to conjecture that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $\mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions. We determine the exact value of $p_{F}$, under a mild assumption on the growth rate of $\mathrm{ex}(n,F)$. Our results extend to $r$-uniform hypergraphs as well. We also prove a general upper bound that is tight up to a $\log n$ factor for $\mathrm{ex}_{p}(n,F)$ when $p = p_{F}$. We conjecture that this $\log n$ factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles. This is a joint work with Xizhi Liu, Jie Ma and Oleg Pikhurko.

An edge-weighted graph $G$, possibly with loops, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory, including the number of independent sets and proper vertex colourings. We obtain a number of new homomorphism inequalities for antiferromagnetic target graphs $G$. In particular, we prove that, for any antiferromagnetic $G$, $|\mathrm{Hom}(K_d, G)|^{1/d} ≤ |\mathrm{Hom}(K_{d,d} \setminus M, G)|^{1/(2d)}$ holds, where $K_{d,d} \setminus M$ denotes the complete bipartite graph $K_{d,d}$ minus a perfect matching $M$. This confirms a conjecture of Sah, Sawhney, Stoner and Zhao for complete graphs $K_d$. Our method uses the emerging theory of Lorentzian polynomials due to Brändén and Huh, which may be of independent interest. Joint work with Jaeseong Oh and Jaehyeon Seo.

Isotropy irreducible spaces are first introduced by Riemannian geometers, as homogeneous real manifolds carrying a canonical invariant metric. Such spaces are classified by Manturov (1960s), Wolf (1968) and Krämer (1975), and their classification provides a number of interesting new examples, for example satisfying the Einstein condition. In this talk, I will introduce a complexified version of isotropy irreducible space, which is called isotropy irreducible variety. In the first half, I will explain geometric properties of isotropy irreducible varieties, and give several non-classical examples belonging to algebraic geometry. Next, I will present a connection between isotropy irreducible varieties and complex contact geometry, which has not been observed in the real setting.

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 presentation will consider the Sobolev regularity for solutions to space-time non-local equations. Spatial non-local operators in this presentation are the infinitesimal generator of Levy processes. The relation between the fundamental solution and the transition density of corresponding processes which generates the operator allows one to obtain the estimation of solutions in Lp-spaces. Several results will be introduced with assumptions. The main ingredients are a representation of solutions, heat kernel estimations, and some properties of (singular) integral operators.

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.

I will give a report on the applications of free convolution theory to the singular behavior of random matrix perturbations. This is a joint work with Hari Bercovici and Ping Zhong.

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.

---

심사위원장: 황강욱, 심사위원: 강완모, 정연승, 하우석, 안정연(산업및시스템공학과)

The effect of malaria on the developing world is devastating. Each year there are more than 200 million cases and over 400,000 deaths, with children under the age of five the most vulnerable. Ambitious malaria elimination targets have been set by the World Health Organization for 2030. These involve the elimination of the disease in at least 35 countries. However, these malaria elimination targets rest precariously on being able to treat the disease appropriately; a difficult feat with the emergence and spread of antimalarial drug resistance, along with many other challenges. In this talk, I will introduce several statistical and mathematical models that can be used to monitor malaria transmission and to support malaria elimination. For example, I’ll present mechanistic models of disease transmission, statistical models that allow the emergence and spread of antimalarial drug resistance to be monitored, mechanistic models that capture the role of bioclimatic factors on the risk of malaria and optimal geospatial sampling schemes for future malaria surveillance. I will discuss how the results of these models have been used to inform public health policy and support ongoing malaria elimination efforts.

Ehrhart theory is the study of lattice polytopes, specifically aimed at understanding how many lattice points are inside dilates of a given lattice polytope, and the study has a wide range of connections ranging from coloring graphs to mirror symmetry and representation theory. Recently, we introduced new algebraic tools to understand this theory, and resolve some classical conjectures. I will explain the combinatorial underpinnings behind two of the key techniques: Parseval identities for semigroup algebras, and the character algebra of a semigroup.

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.

TBA

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.

This lecture explores the mathematical foundations underlying neural network approximation, focusing on the development of rigorous theories that explain how and why neural networks approximate functions effectively. We talk about key topics such as error estimation, convergence analysis, and the role of activation functions in enhancing network performance. Additionally, the lecture will demonstrate convergence analysis in the context of scientific machine learning, further bridging the gap between empirical success and theoretical understanding. Our goal is to provide deeper insights into the mechanisms driving neural network efficiency, reliability, and their applications in scientific computing.

We will give an overview of the recent attempts of building a structure theory for graphs centered around First-Order transductions: a notion of containment inspired by finite model theory. Particularly, we will speak about the notions of monadic dependence and monadic stability, their combinatorial characterizations, and the developments on the algorithmic front.

---

심사위원장: 백형렬, 심사위원: 박정환, 박지원, 김상현(고등과학원), 임선희(서울대학교)

The objective of the tutorial Computer-Assisted Proofs in Nonlinear Analysis is to introduce participants to fundamental concepts of a posteriori validation techniques. This mini-course will cover topics ranging from finite-dimensional problems, such as finding periodic orbits of maps, to infinite-dimensional problems, including solving the Cauchy problem, proving the existence of periodic orbits, and computing invariant manifolds of equilibria. Each session will last 2 hours: 1 hour of theory followed by 1 hour of hands-on practical applications. The practical exercises will focus on implementing on implementing computer-assisted proofs using the Julia programming language.

---

Please bring your laptop to the tutorial and find the attached syllabus for more information. (to download the syllabus) https://saarc.kaist.ac.kr/boards/view/seminars/32

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.