In this talk, we discuss the paper, “Data splitting to avoid information leakage with DataSAIL” by Roman Joeres, et al., Nature Communications, 2025.

The derivation of fluid equations from the Boltzmann equation is one of the most important problems in kinetic theory. In this talk, we investigate several results concerning the diffusive limit of the Boltzmann equation within the L²–L^∞ framework. Based on this framework, we present two results: (1) a global diffusive expansion in an exterior domain, and (2) the global hydrodynamic limit with Maxwell boundary conditions in a bounded domain.

Let $\mathcal F$ be a fixed, finite family of graphs. In the $\mathcal F$-Deletion problem, the input is a graph G and a positive integer k, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of $\mathcal F$ as a minor. The $\mathcal F$-Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover, Feedback Vertex Set, Treewidth-$\eta$ Deletion, Treedepth-$\eta$ Deletion, Pathwidth-$\eta$ Deletion, Outerplanar Deletion, Vertex Planarization and many more. We study the $\mathcal F$-Deletion problem from the kernelization perspective. In a seminal work, Fomin, Lokshtanov, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family $\mathcal F$: that is, the size of the kernel is $k^{f(\mathcal F)}$, for some function f that depends only on $\mathcal F$. Also the size of the kernel depends on non-constructive constants. Later Giannopoulou, Jansen, Lokshtanov & Saurabh [TALG 2017] showed that the non-uniformity in the kernel size bound of Fomin et al. is unavoidable as Treewidth-$\eta$ Deletion, cannot admit a kernel of size $O(k^{(\eta +1)/2 -\epsilon)}$, for any $\epsilon >0$, unless NP $\subseteq$ coNP/poly. On the other hand it was also shown that Treedepth-$\eta$ Deletion, admits a uniform kernel of size $f(\mathcal F) k^6$, showcasing that there are subclasses of $\mathcal F$ where the asymptotic kernel sizes do not grow as a function of the family $\mathcal F$. This work leads to the natural question of determining classes of $\mathcal F$ where the problem admits uniform polynomial kernels. In this work, we show that if all the graphs in $\mathcal F$ are connected and $\mathcal F$ contains $K_{2,p}$ (a bipartite graph with 2 vertices on one side and p vertices on the other), then the problem admits a uniform kernel of size $f(\mathcal F) k^{10}$ where the constants in the size bound are also constructive. The graph $K_{2,p}$ is one natural extension of the graph $\theta_p$, where $\theta_p$ is a graph on two vertices and p parallel edges. The case when $\mathcal F$ contains $\theta_p$ has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel. This is joint work with William Lochet.

Recent advances in artificial intelligence and computational technology have begun to reshape the landscape of mathematical research. In this talk, I will discuss how modern computational techniques such as machine learning, reinforcement learning, local search algorithms, and gradient descent can be applied to problems in extremal combinatorics. In particular, I will share my own experience using these tools, while reflecting on both the opportunities and limitations of using such methods.

(This is a reading seminar talk by a graduate student, Mr. Jaehong Kim.) This talk is a reading seminar about basic intersection theory, following chapter 1 to 6 of the book of William Fulton. The main objects to be dealt with are Chow groups, pullback/pushforward, pseudo-divisors, divisor intersection, Chern/Segre classes, deformation to the normal cone and intersection products.

In this talk, we discuss the paper “Large language models for scientific discovery in molecular property prediction” by Yizhen Zheng et.al., nature machine intelligence, 2025.

General linear model concerns the statistical problem of estimating a vector x from the vector of measurements y=Ax+e, where A is a given design matrix whose rows correspond to individual measurements and e represents errors in measurements. Popular iterative algorithms, e.g. message passing, used in this context requires a "warm start", meaning they must be initialized better than a random guess. In practice, it is often the case that a spectral estimator, i.e. the principal component of a certain matrix built from Y, serves as such an initialization. In this talk, we discuss the theoretical aspect of the spectral estimator and present a theorem on its performance guarantee. Our result gives a threshold for the sample complexity, that is, how many measurements are needed for a warm start to be obtainable, as well as a concrete estimator. If time permits, we will also discuss our method based on (vector-valued) Approximate Message Passing.

An n-dimensional k-handlebody is an n-manifold obtained from an n-ball by attaching handles of index up to k, where n ≥ k. We will discuss that for any n ≥ 2k + 1, any n-dimensional k-handlebody is diffeomorphic to the product of a 2k-dimensional k-handlebody and an (n − 2k)-ball. For example, a 2025-dimensional 6-handlebody is the product of an 12-dimensional 6-handlebody and a 2013-ball. We also introduce (n,k)-Kirby diagrams for some n-dimensional k-handlebodies. Here (4,2)-Kirby diagrams correspond to the classical Kirby diagrams for 4-dimensional 2-handlebodies.

In this talk, we study several computational problems related to knots and links. We investigate lower bounds on the computational complexity of theoretical knot theory problems. Unknotting number is one of the most interesting knot invariants, and various research has been done to find unknotting numbers of knots. However, compared to its simple definition, it is generally hard to find the unknotting number of a knot, and it is known for only some knots. There is no algorithm for determining unknotting numbers yet. First, we show that for an arbitrary positive integer n, a non-torus knot exists with the unknotting number n. Second, we show that the computational complexity of the diagrammatic un-knotting number problem is NP-hard. We construct a Karp reduction from 3-SAT to the diagrammatic unknotting number problem. Third, we also prove that the prime sublink problem is NP-hard by making a Karp reduction from the known NP-complete problem, the non-tautology problem.

Two celebrated extensions of Helly’s theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Barany, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family $F$ of $n$ convex sets in $\mathbb{R}^d$ such that at least $\alpha \binom{n}{d+1}$ of the $(d+1)$-tuples of $F$ have an intersection of volume at least 1, then one can select $\Omega_{d,\alpha}(n)$ members of $F$ whose intersection has volume at least $\Omega_d(1)$. Joint work with Nora Frankl and Istvan Tomon.

Computing obstructions is a useful tool for determining the dimension and singularity of a Hilbert scheme at a given point. However, this task can be quite challenging when the obstruction space is nonzero. In a previous joint work with S. Mukai and its sequels, we developed techniques to compute obstructions to deforming curves on a threefold, under the assumption that the curves lie on a "good" surface (e.g., del Pezzo, K3, Enriques, etc.) contained in the threefold. In this talk, I will review some known results in the case where the intermediate surface is a K3 surface and the ambient threefold is Fano. Finally, I will discuss the deformations of certain space curves lying on a complete intersection K3 surface, and the construction of a generically non-reduced component of the Hilbert scheme of P^5.

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심사위원장: 이지운, 심사위원: 남경식, 황강욱, 서성미(충남대학교), 변성수(서울대학교)

We consider a nonlocal semilinear elliptic equation in a bounded smooth domain with the inhomogeneous Dirichlet boundary condition, which arises as the stationary problem of the Keller-Segel system with physical boundary conditions describing the boundary-layer formation driven by chemotaxis. This problem has a unique steady-state solution which possesses a boundary-layer profile as the nutrient diffucion coefficient tends to zero. Using the Fermi coordinates and delicate analysis with subtle estimates, we also rigorously derive the asymptotic expansion of the boundary-layer profile and thickness in terms of the small diffusion rate with coefficients explicitly expressed by the domain geometric properties including mean curvature, volume and surface area. By these expansions, one can explicitly find the joint impact of the mean curvature, surface area and volume of the spatial domain on the boundary-layer steepness and thickness.

A seminal result of Tutte asserts that every 4-connected planar graph is hamiltonian. By Wagner’s theorem, Tutte’s result can be restated as: every 4-connected graph with no $K_{3,3}$ minor is hamiltonian. In 2018, Ding and Marshall posed the problem of characterizing the minor-minimal 3-connected non-hamiltonian graphs. They conjectured that every 3-connected non-hamiltonian graph contains a minor of $K_{3,4}$, $\mathfrak{Q}^+$, or the Herschel graph, where $\mathfrak{Q}^+$ is obtained from the cube by adding a new vertex and connecting it to three vertices that share a common neighbor in the cube. We recently resolved this conjecture along with some related problems. In this talk, we review the background and discuss the proof.

Lecture 1: Artem Pulemotov (University of Queensland), 4:15-5:15PM
Title: The prescribed Ricci curvature problem on homogeneous spaces
Abstract: We will discuss the problem of recovering the ``shape" of a Riemannian manifold $M$ from its Ricci curvature. After reviewing the relevant background material and the history of the subject, we will focus on the case where $M$ is a homogeneous space for a compact Lie group. Based on joint work with Wolfgang Ziller (The University of Pennsylvania).

Lecture 2: Mikhail Feldman (University of Wisconsin-Madison), 5:30-6:30PM
Title: Self-similar solutions to two-dimensional Riemann problems with transonic shocks
Abstract: Multidimensional conservation laws is an active research area with open questions about existence, uniqueness, and stability of properly defined weak solutions, even for fundamental models such as the compressible Euler system. Understanding particular classes of weak solutions, such as Riemann problems, is crucial in this context. This talk focuses on self-similar solutions to two-dimensional Riemann problems involving transonic shocks for compressible Euler systems. Examples include regular shock reflection, Prandtl reflection, and four-shocks Riemann problem. We first review the results on existence, regularity, geometric properties and uniqueness of global self-similar solutions of regular reflection structure in the framework of potential flow equation. A significant open problem is to extend these results to compressible Euler system, i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions of regular reflection structure have low regularity. We further discuss existence, uniqueness and stability of renormalized solutions to the transport equation for vorticity in this low regularity setting.

***Tea Time 3:45PM-4:15PM in Room 1410***

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***Tea Time 3:45PM-4:15PM in Room 1410***

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심사위원장: 임미경, 심사위원: 김용정, 전현호, Elena Beretta(NYU Abu Dhabi), 이재용(중앙대학교)

Confidence sequence provides ways to characterize uncertainty in stochastic environments, which is a widely-used tool for interactive machine learning algorithms and statistical problems including A/B testing, Bayesian optimization, reinforcement learning, and offline evaluation/learning.In these problems, constructing confidence sequences that are tight and correct is crucial since it has a significant impact on the performance of downstream tasks. In this talk, I will first show how to derive one of the tightest empirical Bernstein-style confidence bounds, both theoretically and numerically. This derivation is done via the existence of regret bounds in online learning, inspired by the seminal work of Raklin& Sridharan (2017). Then, I will discuss how our confidence bound extends to unbounded nonnegative random variables with provable tightness. In offline contextual bandits, this leads to the best-known second-order bound in the literature with promising preliminary empirical results. Finally, I will turn to the $[0,1]$-valued regression problem and show how the intuition from our confidence bounds extends to a novel betting-based loss function that exhibits variance-adaptivity. I will conclude with future work including some recent LLM-related topics.

Given a distribution, say, of data or mass, over a space, it is natural to consider a lower dimensional structure that is most “similar” or “close” to it. For example, consider a planning problem for an irrigation system (1-dimensional structure) over an agricultural region (2-dimensional distribution) where one wants to optimize the coverage and effectiveness of the water supply. This type of problem is related to “principal curves” in statistics and “manifold learning” in AI research. We will discuss some recent results in this direction that employ optimal transport approaches. This talk will be based on joint projects with Anton Afanassiev, Jonathan Hayase, Forest Kobayashi, Lucas O’Brien, Geoffrey Schiebinger, and Andrew Warren.

The investigation of $G_2$-structures and exceptional holonomy on 7-dimensional manifolds involves the analysis of a nonlinear Laplace-type operator on 3-forms. We will discuss the existence of solutions to the Poisson equation for this operator. Based on joint work with Timothy Buttsworth (The University of New South Wales).

The now classical theorem of Erdős, Ko and Rado establishes the size of a maximal uniform family of pairwise-intersecting sets as well as a characterization of the families attaining such upper bound. One natural extension of this theorem is that of restricting the possiblechoices for the sets. That is, given a simplicial complex, what is the size of a maximal uniform family of pairwise-intersecting faces. Holroyd and Talbot, and Borg conjectured that the same phenomena as in the classical case (i.e., the simplex) occurs: there is a maximal size pairwise-intersecting family with all its faces having some common vertex. Under stronger hypothesis, they also conjectured that if a family attains such bound then its members must have a common vertex. In this talk I will present some progress towards the characterization of the maximal families. Concretely I will show that the conjecture is true for near-cones of sufficiently high depth. In particular, this implies that the characterization of maximal families holds for, for example, the independence complex of a chordal graph with an isolated vertex as well as the independence complex of a (large enough) disjoint union of graphs with at least one isolated vertex. Under stronger hypothesis, i.e., more isolated vertices, we also recover a stability theorem. This talk is based on a joint work with Russ Woodroofe.

We study the gradient theory of phase transitions through the asymptotic analysis of variational problems introduced by Modica (1987). As the perturbation parameter tends to zero, minimizers converge to two-phase functions whose interfaces minimize area. The proof uses techniques from the theory of functions of bounded variation and Γ-convergence. This framework has applications in materials science and the study of minimal surfaces. 4참고자료: L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), 123–142.

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.

(This is part of the reading seminar given by the undergrad student Mr. Naing Zaw Lu for his Individual Study project.) This is an introductory talk on homotopy theory in model categories. Over the course of three lectures, we will familiarize ourselves with model categories, see how powerful cofibrant/fibrant objects can be, and build up the tools necessary to define the (Quillen) homotopy category of a model category.

In this talk, we discuss the paper “Quantifying and correcting bias in transcriptional parameter inference from single-cell data” by Ramon Grima and Pierre-Marie Esmenjaud, Biophysical journal, 2024.

In this talk, we discuss the paper “Direct Estimation of Parameters in ODE Models Using WENDy: Weak-Form Estimation of Nonlinear Dynamics” by David M. Bortz, Daniel A. Messenger, and Vanja Dukic, Bulletin of Mathematical Biology, 2023.

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심사위원장: 김완수, 심사위원: 박진현, 박진형, 유필상(서울대학교), 조성문(포항공과대학교)

We present an alternative proof of the uniform and Hausdorff exponential convergence results for Michael Gage's area-preserving curve shortening flow (APCSF) using Leon Simon's framework based on Łojasiewicz-Simon inequalities. We introduce a functional that combines length and area on $L^2(\mathbb S^1)$ and establish the optimal Łojasiewicz-Simon inequality for it to achieve the desired convergence.

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(석사논문심사)

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심사위원장: 백형렬, 심사위원: 김우진, 박정환, 김상현(KIAS), 니콜라스 블라미스(CUNY)

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.

In this talk I shall describe how the components of J_b-orbits of an affine Deligne-Lusztig varieties contribute to the components of the basic locus of a Shimura variety. Then we shall present results for the cases of PEL-type A and C and describe the mass formula part for type C. This is based on joint works with Jiangwei Xue and Yasuhiro Terakado.

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This is also a KAI-X invited lecture. There will be coffee right before the seminar.

Motivated by colouring minimal Cayley graphs, in 1978 Babai conjectured that no-lonely-colour graphs have bounded chromatic number. We disprove this in a strong sense by constructing graphs of arbitrarily large girth and chromatic number that have a proper edge colouring in which each cycle contains no colour exactly once. The result presented is the joint work with James Davies and Liana Yepremyan.

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 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.

One of the main topics in Random Matrix Theory(RMT) is universality. In this talk, we focus on edge universality in Wigner matrices. With overall description of significant findings including spiked Wigner matrix and BBP transition, we introduce our recent topic, fluctuations of the largest eigenvalues of transformed spiked Wigner matrices. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes. This is a joint work with Prof. Ji Oon Lee (KAIST).

TBD

If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

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.

If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

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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.

If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

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: 강문진, 심사위원: 권순식, 변재형, 권봉석(UNIST), 김정호(경희대학교)

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심사위원장: 강문진, 심사위원: 권순식, 변재형, 권봉석(UNIST), 김정호(경희대학교)

(This is part of the reading seminar given by the undergrad student Mr. Naing Zaw Lu for his Individual Study project.) This is an introductory talk on homotopy theory in model categories. Over the course of three lectures, we will familiarize ourselves with model categories, see how powerful cofibrant/fibrant objects can be, and build up the tools necessary to define the (Quillen) homotopy category of a model category.

I will discuss how to study the mod p non-vanishing problem for Dirichlet L-values with characters whose kernels are unbounded as the conductors grow. Main ingredients are a dynamical reformulation of the problem using a variant of trace defined by Poisson kernel and a study on the spectral properties of transfer operators related to p-Bernoulli map on the interval. This is ongoing research joint with A. Burungale.

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.

Arithmetic of Elliptic curves, one of fundamental research themes in modern number theory, is encoded in the special values of elliptic L-functions. For example, Mazur-Rubin heuristics on the distribution of the special L-values predict the behavior of ranks of elliptic curves. The average version of the heuristics was proved by several researchers including myself several years ago. In the talk, I will present how to use the dynamics of continued fractions to study the problem and introduce an approach to study the original problem, i.e., non-average version, after introducing a classical result, so-called Lochs' theorem, which compares entropies of two distinct dynamical systems.

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PPT Slide: English

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.

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.

In this talk, we discuss the paper “Network inference from short, noisy, low time-resolution, partial measurements: Application to C. elegans neuronal calcium dynamics” by Amitava Banerjee, Sarthak Chandra, and Edward Ott, PNAS, 2023.

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.

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 part of the reading seminar given by the undergrad student Mr. Naing Zaw Lu for his Individual Study project.) This is an introductory talk on homotopy theory in model categories. Over the course of three lectures, we will familiarize ourselves with model categories, see how powerful cofibrant/fibrant objects can be, and build up the tools necessary to define the (Quillen) homotopy category of a model category.

In undergraduate PDE course, one may have learned that the (classical) diffusion equation can be expressed as $u_t=D \Delta u$, where $D$ is a constant diffusivity. This is true for homogeneous environment. However, for (spatially) heterogeneous environment, $D$ is no longer a constant, and diffusion phenomena in those environments such as fractionation, or Soret effect, cannot be explained with the classical diffusion equation. In this talk, I will first discuss how to model and derive some of the diffusion equations in heterogeneous environment by using basic random walk theory. We will see that the heterogeneity of components, such as speed, walk length, sojourn time, etc, can explain the diffusion phenomena. Then, I will give some specific examples how such models can be applied in science, based on my recent works.