In this talk, we consider the dispersion-managed nonlinear Schrödinger equation (DM NLS), which naturally arises in modeling of fiber-optic communication systems with periodically varying dispersion profiles. We discuss the well-posedness of the DM NLS and the threshold phenomenon related to the existence of minimizers for its ground states.

The talk is divided into two parts. In the first part, we review the concept of phase transition in probability theory and mathematical physics, focusing on the standard +/- Ising model. In the second part, we discover why one may expect metastability in the low-temperature regime, and look at some concrete examples that exhibit this phenomenon.

What causes a graph to have high chromatic number? One obvious reason is containing a large clique (a set of pairwise adjacent vertices). This naturally leads to investigation of \(\chi\)-bounded classes of graphs — classes where a large clique is essentially the only reason for large chromatic number. Unfortunately, many interesting graph classes are not \(\chi\)-bounded. An eerily common obstruction to being \(\chi\)-bounded are the Burling graphs — a family of triangle-free graphs with unbounded chromatic number. These graphs have served as counterexamples in many settings: demonstrating that graphs excluding an induced subdivision of \(K_{5}\) are not \(\chi\)-bounded, that string graphs are not \(\chi\)-bounded, that intersection graphs of boxes in \({\mathbb{R}}^{3}\) are not \(\chi\)-bounded, and many others. In many of these cases, this sequence is the only known obstruction to \(\chi\)-boundedness. This led Chudnovsky, Scott, and Seymour to conjecture that any graph of sufficiently high chromatic number must either contain a large clique, an induced proper subdivision of a clique, or a large Burling graph as an induced subgraph. The prevailing belief was that this conjecture should be false. Somewhat surprisingly, we did manage to prove it under an extra assumption on the “locality” of the chromatic number — that the input graph belongs to a \(2\)-controlled family of graphs, where a high chromatic number is always certified by a ball of radius \(2\) with large chromatic number. In this talk, I will present this result and discuss its implications in structural graph theory, and algorithmic implications to colouring problems in specific graph families. This talk is based on joint work with Tara Abrishami, James Davies, Xiying Du, Jana Masaříková, Paweł Rzążewski, and Bartosz Walczak conducted during the STWOR2 workshop in Chęciny Poland.

In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).

In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).

We discuss the fine gradient regularity of nonlinear kinetic Fokker-Planck equations in divergence form. In particular, we present gradient pointwise estimates in terms of a Riesz potential of the right-hand side, which leads to the gradient regularity results under borderline assumptions on the right-hand side. The talk is based on a joint work with Ho-Sik Lee (Bielefeld) and Simon Nowak (Bielefeld).

In this talk, we will discuss some global regularity results for weak solutions to fractional Laplacian type equations. In particular, the operator under consideration involves a weight function satisfying appropriate ellipticity conditions. Under suitable assumptions on the weight function and the right hand side, we show some sharp global regularity results for the function u/d^s in the sense of Lebesgue, Sobolev and H¨older, where d(x) = dist(x, ∂Ω) is the distance to the boundary function. This talk is based on a joint work with S.-S. Byun and K. Kim.

In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).

In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).

In this talk, we discuss the paper “Boolean modelling as a logic-based dynamic approach in systems medicine” by Ahmed Abdelmonem Hemedan et al., Computational and Structural biotechnology journal (2022).

This is a reading seminar presented by the graduate student, Mr. Taeyoon Woo. Following the lecture note of Yuri Manin, he will study K_0 of schemes, and its essential properties, such as functoriality, projective bundle formula, filtrations, relationship to Picard group, blow-up squares, Chern classes, Todd classes and the Grothendieck-Riemann-Roch theorem.

Let S and T be two sets of points in a metric space with a total of n points. Each point in S and T has an associated value that specifies an upper limit on how many points it can be matched with from the other set. A multimatching between S and T is a way of pairing points such that each point in S is matched with at least as many points in T as its assigned value, and vice versa for each point in T. The cost of a multimatching is defined as the sum of the distances between all matched pairs of points. The geometric multimatching problem seeks to find a multimatching that minimizes this cost. A special case where each point is matched to at most one other point is known as the geometric many-to-many matching problem. We present two results for these problems when the underlying metric space has a bounded doubling dimension. Specifically, we provide the first near-linear-time approximation scheme for the geometric multimatching problem in terms of the output size. Additionally, we improve upon the best-known approximation algorithm for the geometric many-to-many matching problem, previously introduced by Bandyapadhyay and Xue (SoCG 2024), which won the best paper award at SoCG 2024. This is joint work with Shinwoo An and Jie Xue.

De novo mutations provide a powerful source of information for identifying risk genes associated with phenotypes under selection, such as autism spectrum disorder (ASD), obsessive-compulsive disorder (OCD), congenital heart disease, and schizophrenia (SCZ). However, identifying de novo variants is costly, as it requires trio-based sequencing to obtain parental genotypes. To address this limitation, we propose a method to infer inheritance class using only offspring genetic data. In our new integrated model, we evaluate variation in case and control samples, attempt to distinguish de novo mutations from inherited variation, and incorporate this information into a gene-based association framework. We validate our method through ASD gene identification, demonstrating that it provides a robust and powerful approach for identifying risk genes.

This is a reading seminar presented by the graduate student, Mr. Taeyoon Woo. Following the lecture note of Yuri Manin, he will study K_0 of schemes, and its essential properties, such as functoriality, projective bundle formula, filtrations, relationship to Picard group, blow-up squares, Chern classes, Todd classes and the Grothendieck-Riemann-Roch theorem.

We prove global well-posedness and scattering for the massive Dirac-Klein-Gordon system with small and low regularity initial data in dimension two, under non-resonance condition. We introduce new resolutions spaces which act as an effective replacement of the normal form transformation.

Given a tournament $S$, a tournament is $S$-free if it has no subtournament isomorphic to $S$. Until now, there have been only a small number of tournaments $S$ such that the complete structure of $S$-free tournaments is known. Let $\triangle(1, 2, 2)$ be a tournament obtained from the cyclic triangle by substituting two-vertex tournaments for two of its vertices. In this talk, we present a structure theorem for $\triangle(1, 2, 2)$-free tournaments, which was previously unknown. As an application, we provide tight bounds for the chromatic number as well as the size of the largest transitive subtournament for such tournaments. This talk is based on joint work with Taite LaGrange, Mathieu Rundström, Arpan Sadhukhan, and Sophie Spirkl.

A knot bounds an oriented compact connected surface in the 3-sphere, and consequently in the 4-ball. The 4-genus of a knot is the minimal genus among all such surfaces in the 4-ball, and the 4-genus of a link is defined analogously. In this talk, I will discuss lower bounds on the 4-genus derived from Cheeger-Gromov-von Neumann rho-invariants. This is joint work with Jae Choon Cha and Min Hoon Kim.

TBD

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PPT Slide: English

Abstract: In this talk, we discuss the global-in-time existence of strong solutions to the one-dimensional compressible Navier-Stokes system. Classical results establish only local-in-time existence under the assumption that the initial data are smooth and the initial density remains uniformly positive. These results can be extended to global-in-time existence using the relative entropy and Bresch-Desjardins entropy under the same hypotheses. This approach allows for possibly different end states and degenerate viscosity. Reference: A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal., 39(4):1344–1365, 2007/08.

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심사위원장: 권순식, 심사위원: 김용정, 변재형, 배명진, 패트릭 제라드(Paris-Saclay)

In this talk, we will discuss the current state and future prospects of multimodal AI. In particular, we will focus on the key challenges in ensuring reliability and efficiency in multimodal AI, explaining why addressing these factors is crucial for the successful real-world deployment of next-generation intelligent systems.

Many natural systems exhibit oscillations that show sizeable fluctuations in frequency and amplitude. This variability can arise from a wide variety of physical mechanisms. Phase descriptions that work for deterministic oscillators have a limited applicability for stochastic oscillators. In my talk I review attempts to generalize the phase concept to stochastic oscillations, specifically, the mean-return-time phase and the asymptotic phase. For stochastic systems described by Fokker-Planck and Kolmogorov-backward equations, I introduce a mapping of the system’s variables to a complex pointer (instead of a real-valued phase) that is based on the eigenfunction of the Kolmogorov equation. Under the new (complex-valued) description, the statistics of the oscillator’s spontaneous activity, of its response to external perturbations, and of the coordinated activity of (weakly) coupled oscillators, is brought into a universal and greatly simplified form. The theory is tested for three theoretical models of noisy oscillators arising from fundamentally different mechanisms: a damped harmonic oscillator with dynamical noise, a fluctuation-perturbed limit-cycle system, and an excitable system in which oscillations require noise to occur.

TBD

Abstract : When a plane shock hits a wedge head on, it experiences a reflection diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. In particular, the C^{1,1}-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock where the wedge has large-angle. Also, one can obtain the C^{2,\alpha} regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. Reference : Myoungjean Bae, Gui-Qiang Chen, and Mikhail Feldman. "Regularity of solutions to regular shock reflection for potential flow." (2008) Gui-Qiang Chen and Mikhail Feldman. "Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow"

This talk concerns the classification problem of long-term dynamics for critical evolutionary PDEs. I will first discuss critical PDEs and soliton resolution for these equations. Building upon soliton resolution, I will further introduce the classification problem. Finally, I will also touch on a potential instability mechanism of finite-time singularities for some critical PDEs, suggesting the global existence of generic solutions.

Spontaneous rhythmic oscillations are widely observed in real-world systems. Synchronized rhythmic oscillations often provide important functions for biological or engineered systems. One of the useful theoretical methods for analyzing rhythmic oscillations is the phase reduction theory for weakly perturbed limit-cycle oscillators, which systematically gives a low-dimensional description of the oscillatory dynamics using only the asymptotic phase of the oscillator. Recent advances in Koopman operator theory provide a new viewpoint on phase reduction, yielding an operator-theoretic definition of the classical notion of the asymptotic phase and, moreover, of the amplitudes, which characterize distances from the limit cycle. This led to the generalization of classical phase reduction to phase-amplitude reduction, which can characterize amplitude deviations of the oscillator from the unperturbed limit cycle in addition to the phase along the cycle in a systematic manner. In the talk, these theories are briefly reviewed and then applied to several examples of synchronizing rhythmic systems, including biological oscillators, networked dynamical systems, and rhythmic spatiotemporal patterns.

Spontaneous rhythmic oscillations are widely observed in real-world systems. Synchronized rhythmic oscillations often provide important functions for biological or engineered systems. One of the useful theoretical methods for analyzing rhythmic oscillations is the phase reduction theory for weakly perturbed limit-cycle oscillators, which systematically gives a low-dimensional description of the oscillatory dynamics using only the asymptotic phase of the oscillator. Recent advances in Koopman operator theory provide a new viewpoint on phase reduction, yielding an operator-theoretic definition of the classical notion of the asymptotic phase and, moreover, of the amplitudes, which characterize distances from the limit cycle. This led to the generalization of classical phase reduction to phase-amplitude reduction, which can characterize amplitude deviations of the oscillator from the unperturbed limit cycle in addition to the phase along the cycle in a systematic manner. In the talk, these theories are briefly reviewed and then applied to several examples of synchronizing rhythmic systems, including biological oscillators, networked dynamical systems, and rhythmic spatiotemporal patterns.

In this talk, we discuss the paper “Identifying key drivers in a stochastic dynamical system through estimation of transfer entropy between univariate and multivariate time series” by Julian Lee, Physical Review E, 2025.

The Lipshitz-Ozsvath-Thurston correspondence is a combinatorial way to describe the bordered Floer homology of a knot complement from the UV=0 coefficient knot Floer homology of the given knot. This is then used to compute the knot Floer homology of satellite knots. In this talk, we show that there is a "relative" version of this correspondence, between homotopy classes of type D morphisms of bordered Floer homology and locally symmetric chain maps of knot Floer complexes, modulo the "canonical negative class". This gives us a fully combinatorial process to compute knot Floer cobordism maps of satellite concordances in the UV=0 knot Floer homology.

A rooted spanning tree of a graph $G$ is called normal if the endvertices of all edges of $G$ are comparable in the tree order. It is well known that every finite connected graph has a normal spanning tree (also known as depth-first search tree). Also, all countable graphs have normal spanning trees, but uncountable complete graphs for example do not. In 2021, Pitz proved the following characterisation for graphs with normal spanning trees, which had been conjectured by Halin: A connected graph $G$ has a normal spanning tree if and only if every minor of $G$ has countable colouring number, i.e. there is a well-order of the vertices such that every vertex is preceded by only finitely many of its neighbours. More generally, a not necessarily spanning tree in $G$ is called normal if for every path $P$ in $G$ with both endvertices in $T$ but no inner vertices in $T$, the endvertices of $P$ are comparable in the tree order. We establish a local version of Pitz’s theorem by characterising for which sets $U$ of vertices of $G$ there is a normal tree in $G$ covering $U$. The results are joint work with Max Pitz.

Abstract: In this talk, we consider the Navier-Stokes-Poisson (NSP) system which describes the dynamics of positive ions in a collision-dominated plasma. The NSP system admits a one-parameter family of smooth traveling waves, known as shock profiles. I will present my research on the stability of the shock profiles. Our analysis is based on the pointwise semigroup method, a spectral approach. We first establish spectral stability. Based on this, we obtain pointwise bounds on the Green's function for the associated linearized problem, which yield linear and nonlinear asymptotic orbital stability.

In this talk, we discuss the paper, “Entrainment and multi-stability of the p53 oscillator in human cells” by Alba Jiménez et al., Cell Systems, 2024.

Serrin’s overdetermined problem is a famous result in mathematics that deals with the uniqueness and symmetry of solutions to certain boundary value problems. It is called "overdetermined" because it has more boundary conditions than usually required to determine a solution, which leads to strong restrictions on the shape of the domain. In this talk, we discuss overdetermined boundary value problems in a Riemannian manifold and discuss a Serrin-type symmetry result to the solution to an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature and it will be discussed about an overdetermined problems with a nonconstant Neumann boundary condition in a warped product manifold.

We present recent developments on the quantitative stability of the Sobolev inequalities, as well as the stability of critical points of their Euler–Lagrange equations.  In particular, we introduce our recent joint work with H. Chen (Hanyang University) and J. Wei (The Chinese University of Hong Kong) on the stability of the Yamabe problem, the fractional Lane–Emden equation for all possible orders, and the Brezis-Nirenberg problem.

Abstract:The logistic diffusive model provides the population distribution of a species according to time under a fixed open domain in R^n, a dispersal rate, and a given resource distribution. In this talk, we discuss the solution of the model and its equilibrium. First, we show the existence, uniqueness, and regularity results of the solution and the equilibrium. Then, we investigate two contrasting behaviors of the equilibrium with respect to the dispersal rate by applying two methods for each case: sub-super solution method and asymptotic expansion. Finally, we introduce an optimizing problem of a total population of the equilibrium with respect to resource distribution and prove a significant property of an optimal control called bang-bang. References: [1] Cantrell, R.S., Cosner, C. Spatial ecology via reaction-diffusion equation. Wiley series in mathematical and computational biology, John Wiley & Sons Ltd (2003) [2] I. Mazari, G. Nadin, Y. Privat, Optimization of the total population size for logistic diffusive equations: Bang-bang property and fragmentation rate, Communications in Partial Differential Equation 47 (4) (Dec 2021) 797-828

The advent of single-cell transcriptomics has brought a greatly improved understanding of the heterogeneity of gene expression across cell types, with important applications in developmental biology and cancer research. Single-cell RNA sequencing datasets, which are based on tags called universal molecular identifiers, typically include a large number of zeroes. For such datasets, genes with even moderate expression may be poorly represented in sequencing count matrices. Standard pipelines often retain only a small subset of genes for further analysis, but we address the problem of estimating relative expression across the entire transcriptome by adopting a multivariate lognormal Poisson count model. We propose empirical Bayes estimation procedures to estimate latent cell-cell correlations, and to recover meaningful estimates for genes with low expression. For small groups of cells, an important sampling procedure uses the full cell-cell correlation structure and is computationally feasible. For larger datasets, we propose a gene-level shrinkage procedure that has favorable performance for datasets with approximately compound symmetric cell-cell correlation. A fast procedure that incorporates matrix approximations is also promising, and extensible to very large datasets. We apply our approaches to simulated and real datasets, and demonstrate favorable performance in comparisons to competing normalization approaches. We further illustrate the applications of our approach in downstream analyses, including cell-type clustering and identification.

In this talk, we discuss the paper “Accurate predictions on small data with a tabular foundation model” by Noah Hollmann et al., Nature (2025).

Graph coloring is one of the central topics in graph theory, and there have been extensive studies about graph coloring and its variants. In this talk, we focus on the structural and algorithmic aspects of graph coloring together with their interplay. Specifically, we explain how local information on graphs can be transformed into global properties and how these can be used to investigate coloring problems from structural and algorithmic perspectives. We also introduce the notion of dicoloring, a variant of coloring defined for directed graphs, and present our recent work on dicoloring for a special type of directed graph called tournaments.

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심사위원장: 이용남, 심사위원: 곽시종, 박진형, 홍재현(기초과학연구원), 황준묵(기초과학연구원)

TBD

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심사위원장: 이용남, 심사위원: 곽시종, 박진형, 박의성(고려대학교), 한강진(DGIST)

Modular forms continue to attract attention for decades with many different application areas. To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to be able to compute a large number of Fourier coefficients. In this talk, firstly, we will show that this can be achieved in level 4 for a large range of half-integral weights by making use of one of three explicit bases, the elements of which can be calculated via fast power series operations. After having "many" Fourier coefficients, it is time to ask the following question: Can the dis- tribution of normalised Fourier coefficients of half-integral weight level 4 Hecke eigenforms with bounded indices be approximated by a distribution? We will suggest that they follow the generalised Gaussian distribution and give some numerical evidence for that. Finally, we will see that the appar- ent symmetry around zero of the data lends strong evidence to the Bruinier- Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute values are distributed independently. This is joint work with Gabor Wiese (Luxembourg), Zeynep Demirkol Ozkaya (Van) and Elif Tercan (Bilecik).

Diophantine equations involving specific number sequences have attracted considerable attention. For instance, studying when a Tribonacci number can be expressed as the product of two Fibonacci numbers is an interesting problem. In this case, the corresponding Diophantine equation has only two nontrivial integer solutions. While finding these solutions is relatively straightforward, proving that no further solutions exist requires a rigorous argument-this is where Baker’s method plays a crucial role. After conducting a comprehensive literature review on the topic, we present our recent results on Diophantine equations involving Fibonacci, Tribonacci, Jacobsthal, and Perrin numbers. Furthermore, as an application of Baker’s method, we will briefly demonstrate how linear forms in logarithms can be effectively applied to Diophantine equations involving Fibonacci-like sequences. This is joint work with Zeynep Demirkol Ozkaya (Van), Zekiye Pinar Cihan (Bilecik) and Meltem Senadim (Bilecik).

By utilizing the recently developed hypergraph analogue of Godsil’s identity by the second author, we prove that for all $n \geq k \geq 2$, one can reconstruct the matching polynomial of an $n$-vertex $k$-uniform hypergraph from the multiset of all induced sub-hypergraphs on $\lfloor \frac{k-1}{k}n \rfloor + 1$ vertices. This generalizes the well-known result of Godsil on graphs in 1981 to every uniform hypergraph. As a corollary, we show that for every graph $F$, one can reconstruct the number of $F$-factors in a graph under analogous conditions. We also constructed examples that imply the number $\lfloor \frac{k-1}{k}n \rfloor + 1$ is the best possible for all $n\geq k \geq 2$ with $n$ divisible by $k$. This is joint work Donggyu Kim.