We briefly introduce the restriction theory in harmonic analysis and its connections with PDEs through Strichartz estimate. We then discuss the Kakeya and multilinear Kakeya estimates, which naturally arise from restriction theory. The main part of the talk will focus on Larry Guth’s proof of the multilinear Kakeya estimate via the induction on scales method.

The syzygy scheme is the scheme defined by the quadric forms associated to the linear syzygies of certain order of a given scheme. It is natural to ask whether the syzygy scheme is equal to the scheme itself. In this talk, I will discuss about the classification of the second syzygy schemes for 4-gonal canonical curves of genus at least 6. This talk is based on the work by Aprodu-Bruno-Sernesi.

TBA

Geometric evolution equations describe how geometric objects such as curves, surfaces, or metrics evolve toward more symmetric or optimal shapes. Among the most fundamental examples are the mean curvature flow and the Ricci flow, which have played central roles in modern differential geometry and topology. In this talk, I will give an introduction to these flows, explaining how curvature acts as a driving mechanism that smooths and reshapes geometry. I will also outline the key ideas behind Perelman’s proof of the Poincaré conjecture, focusing on the role of singularity formation and the classification of canonical neighborhoods. Finally, I will discuss the problem of classifying singularity models arising under geometric flows and present some recent progress on the classification of ancient oval solutions, together with possible further directions.

This talk explores the relationship between 3-dimensional lens spaces and smooth 4-manifolds that bound them under various topological constraints—topics that connect to several central conjectures in low-dimensional topology. After reviewing the classifications of Lisca, Greene, and Aceto–McCoy–JH Park, I will present recent joint work with Wookhyeok Jo and Jongil Park investigating which lens spaces can bound smooth 4-manifolds with second Betti number one. In particular, we exhibit infinite families of lens spaces that bound simply connected 4-manifolds with b₂ = 1, yet do not bound 4-manifolds consisting of a single 0-handle and 2-handle. Moreover, we construct infinite families of lens spaces that bound 4-manifolds with b₁ = 0 and b₂ = 1, but do not bound simply connected 4-manifolds with b₂ = 1. These constructions are motivated by the study of rational homology projective planes with cyclic quotient singularities.

Self-improving properties are a kind of fundamental regularity results in the theory of elliptic equations. However, for nonlocal elliptic equations, establishing this property is significantly more comlicated than in the local setting. In the first part of this talk, we will discuss several results and arguments that demonstrate this self-improving property. We will also present an ongoing project on self-improving properties for equations with fractional Orlicz growth, which is joint work with Kyeong Song (KIAS). In the second part, we will address the optimality of these self-improving properties. Classically, this optimality was established by Meyers in 1963 by constructing a counterexample. We will present a nonlocal analogue of Meyers' example. The construction the example is based on Fourier transform techniques for distributional convolutions. One of the key feature of our example is its robustness: It remains valid as the order of the nonlocal operator converges to 2, the order of a classical second-order elliptic operator. This is joint work with Anna Balci, Lars Diening, and Moritz Kassmann (Bielefeld University).

In this talk, we introduce a generalized Schauder theory for degenerate and singular parabolic equations. The key idea is an approximation scheme based on fractional-order polynomials—s-polynomials—which replace constant coefficients in the classical setting. This approach not only recovers the classical regularity results for uniformly parabolic equations but also extends to operators where traditional bootstrap arguments are difficult to apply.

The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations. In this talk, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski to solve the existence of the stationary Boltzmann equation in a slab domain. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished. This talk is based on a collaborative work with Daisuke Kawagoe and Chun-Hsiung Hsia.

Consider a general Turan-type problem on hypergraphs. Let $\mathcal{F}$ be a family of $k$-subsets of $[n]$ that does not contain sets $F_1, \ldots, F_s$ satisfying some property $P$. We show that if $P$ is low-dimensional in some sense (e.g., is defined by intersections of bounded size) then, under polynomial dependencies between $n, k$ and the parameters of $P$, one can reduce the problem of maximizing the size of the family $|\mathcal{F}|$ to a finite extremal set theory problem independent of $n$ and $k$. We show that our technique implies new bounds in a number of Turan-type problems including the Erdős-Sós forbidden intersection problem, the Duke-Erdős forbidden sunflower problem, forbidden $(t, d)$-simplex problem and the forbidden hypergraph problem. Furthermore, we also briefly discuss the connection between the aforementioned reduction and the measure boosting argument based on the action of a certain semigroup on the Boolean cube.  This connection turns out to be fruitful when extending extremal set theory problems to domains different from $\binom{[n]}{k}$. Joint work with Liza Iarovikova, Andrey Kupavskii, Georgy Sokolov and Nikolai Terekhov

In this talk, I will present the local existence theory for quasilinear symmetric hyperbolic systems, based on Sections 1.3 and 2.1 of [1]. I will begin by reviewing the framework of symmetric systems and then explain how it is applied to establish local-in-time existence of classical solutions. The main focus will be on the iteration scheme, energy estimates, and convergence arguments. We aim to understand how regularity and a priori bounds are used to construct solutions from smooth initial data.

This is a reading seminar of a graduate student, following the Fields medal work of Daniel Quillen on the foundation of the higher algebraic K-theory.

In this presentation, we discuss recent existence results for nonlinear diffusion equations with a divergence-type drift term, which are broadly applicable to various reaction-diffusion equations, including Keller-Segel models. We focus on identifying appropriate functional spaces for the drift, guided by the nonlinear diffusion and initial data. Using techniques from the theory of Wasserstein spaces, we construct weak solutions and establish their regularity properties.

Associated to a group action on a bifoliated plane, satisfying some reasonable conditions, one can associate a combinatorial object known as a veering triangulation. Since their introduction by Agol (in a very different setting), these triangulations have recently played an interesting role in studying pseudo-Anosov flows, the structure of fibered 3-manifolds, algorithmic properties of mapping class groups, and fixed points of surface homeomorphisms, to name just a few (from my own biased perspective). This talk will be an overview of these applications, starting with the most basic properties from the initial bifoliated plane.

In this talk, we will discuss Leray-Hopf solutions to the incompressible Navier-Stokes equations with vanishing viscosity. We explore important features of turbulence, focusing around the anomalous energy dissipation phenomenon. As a related result, I will present a recent result proving that for two-dimensional fluids, assuming that the initial vorticity is merely a Radon measure with nonnegative singular part, there is no anomalous energy dissipation. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. We will also discuss possible extensions to the viscous SQG equation in the context of Hamiltonian conservation and existence of weak solutions for a rough initial data.

Given a $k$-uniform hypergraph $F$, its Turán density $\pi(F)$ is the infimum over all $d\in [0,1]$ such that any $n$-vertex $k$-uniform hypergraph $H$ with at least $d\binom{n}{k}+o(n^k)$ edges contains a copy of $F$. While Turán densities are generally well understood for graphs ($k=2$), the problem becomes notoriously difficult for $k\geq 3$, even for small hypergraphs. We study two well-known variants of this Turán problem for hypergraphs: first, under minimum codegree conditions and, second, with a quasirandom edge distribution. Each variant defines a distinct extremal parameter, generalising the classical Turán density. Here we present recent results in both settings, with a particular emphasis on the case of hypergraphs where every link is itself quasirandom. Our results include exact solutions for key hypergraphs and general results about the behaviour of the Turán density functions.

For many variant kinetic equations, we choose an appropriate approx- imatation equations. Also, this approximation equations are solvable more easier than the original equations and it retains the expected a priori bounds. Then, we use the variant compactness theorem to pass to the limit in the sense of distributions in the approximation equations. In the kinetic theory, this compactness theorem is called the averaing lemma, that is, the averaging in velocity improves regularity in the space and time variables. For this PDE seminar, we study the basic averaging lemma. In other words, we investigate the basic properties of the free transport operator ∂t + v · ∇x.

In this talk, I will try to explain how the essence of the Weierstrass representation formula and the Bjorling representation formula for minimal surfaces in $E^3$ can be suitably applied to zero/constant mean curvature surfaces in the three-dimensional spaceforms in the Lorentz-Minkowski four-space.

TBA

In this talk, we introduce the concept of t-core partitions. We discuss the generating function and modularity, along with some results and applications of t-core partitions. Recent results on simultaneous core partitions will also be presented. Toward the end of the talk, we introduce numerical semigroups and explore connections between numerical semigroups (or numerical sets) and partitions. Additionally, we present some open problems related to these topics.

We study the partial dimensional semi-classical Weyl’s laws, describing the quantum subband structures for two-dimensional electron gases (2DEGs). As a simple application, we derive lowest free energy states for the subband models describing non-interacting 2DEGs.

We study the partial dimensional semi-classical Weyl’s laws, describing the quantum subband structures for two-dimensional electron gases (2DEGs). As a simple application, we derive lowest free energy states for the subband models describing non-interacting 2DEGs.

We consider a continuous model of graphs, introduced by Dearing and Francis in 1974, where each edge of G to be a unit interval, giving rise to an infinite metric space that contains not only the vertices of G but all points on all edges of G. Several standard graph problems can be defined and studied on continuous graphs, yielding many surprising algorithmic results and combinatorial connections. The motivation can be exemplified by the well-known Independent Set problem on graphs. Given a graph G, we want to place k facilities that are pairwise at least a distance 2 edge lengths apart. In some applications, such as when the underlying graph represents a street network, it is reasonable to allow placing a facility not only at a crossroad but also somewhere within a street, that is, not only at a vertex but also at any point on an edge between two vertices. This motivates the study of the Independent Set problem on continuous graphs. In such a setting, for example, the problem corresponds to requiring pairwise distance r=2 for the placed facilities. However, we may also study the problem where we fix r to any positive integer, rational, or even irrational number. Other problems studied in the continuous model include Dominating Set, TSP, and Coloring. In the first part of this talk, we will give a general overview of research on continuous graphs and computational problems in this setting. In the second part, we explore, as part of recent work, a coloring problem on continuous graphs akin to the well-known Hadwiger-Nelson Problem. Based on joint work with Fabian Frei, Florian Hörsch, Tom Janßen, Stefan Lendl, Dániel Marx, Prahlad Narasimhan, and Gerhard Woeginger.

The Lyapunov-Schmidt reduction is a powerful tool to solve PDEs. This method reduces the equations, which are essentially infinite-dimensional, to finite-dimensional ones. In this talk, we illustrate the reduction by showing the existence of a positive solution to the singularly perturbed problem in for positive smooth and appropriate . To show the existence, we first construct an -dimensional surface of approximate solutions. Then, we reduce the problem onto that surface by the Lyapunov-Schmidt reduction. The key to the reduction is proving the invertibility of a certain operator, which in turn, is proved by a certain uniqueness result. After the reduction, we end the proof by solving the equation on the -dimensional surface.

This is a reading seminar of a graduate student, following the Fields medal work of Daniel Quillen on the foundation of the higher algebraic K-theory.

In recent years, syzygies of projections of algebraic varieties have drawn a lot of attentions. It turns out that their Betti diagrams carry geometric information like the codimension of the projection and the position of the projection center, by the investigations of E. Park, S. Kwak and so on. In this talk, I will show that for a generic canonical curve $C$ in $\mathbb{P}^{g−1}$, its projection $C'$ away from a generic point into $\mathbb{P}^{g−2}$ is cut out by quadrics for $g \geq 9$. I will also give the predictions of the Betti diagrams with the help of Macaulay2.

In this talk, we will discuss the discrete argmin inference problem in high-dimensional settings. Given n observations from a d dimensional vector, the goal is to test whether the rth component of the mean vector is the smallest among all components. We propose dimension-agnostic tests that maintain validity regardless of how d scales with n, and regardless of arbitrary ties in the mean vector. Notably, our validity holds under mild moment conditions, requiring little more than finiteness of a second moment, and permitting possibly strong dependence between coordinates. In addition, we establish the local minimax separation rate for this problem, which adapts to the cardinality of a confusion set, and show that the proposed tests attain this rate. Empirical results illustrate the strong performance of our approach in terms of type I error control and power compared to existing methods.

Vertex Cover is perhaps the most-studied problem in parameterized complexity that frequently serves as a testing ground for new concepts and techniques. In this talk, I will focus on a generalization of Vertex Cover called Component Order Connectivity (COC). Given a graph G, an integer k and a positive integer d, the task is to decide whether there is a vertex set S of size at most k such that each connected component of G – S has size at most d. If d = 1, then COC is the same as Vertex Cover. While almost all techniques to obtain polynomial kernels for Vertex Cover extend well to COC parameterized by k + d, the same cannot be said for structural parameters. Vertex Cover admits a polynomial kernel parameterized by the vertex deletion distance to treewidth 1 graphs, but not when parameterized by the deletion distance to treewidth 2 graphs. The picture changes when considering COC: It was recently shown that COC does not admit a polynomial kernel parameterized by the vertex deletion distance to treewidth 1 graphs with pathwidth 2, even if d ≥ 2 is a fixed constant. Complementing this, we show that COC does admit a polynomial kernel parameterized by the distance to graphs with pathwidth at most 1 (plus d). Hence, the deletion distance to pathwidth 1 vs. pathwidth 2 forms a similar line of tractability for COC as the distance to treewidth 1 vs. treewidth 2 does for Vertex Cover. In this talk, I will highlight the ideas and techniques that make this kernelization result possible.

In this talk, I will survey Liouville quantum gravity (LQG) surfaces in the supercritical regime, where the matter central charge exceeds 1. Unlike the subcritical case---where many models from mathematical physics (such as percolation and Ising) are known to converge to LQG surfaces---no canonical model is currently available in this regime.I will discuss how the six-vertex model may serve as a candidate whose scaling limit is a supercritical LQG surface, and I will present some preliminary progress and many conjectures in this direction, based on joint work with Ewain Gwynne and Andrew Elvey Price.

Robertson and Seymour introduced branch-width as a connectivity invariant of graphs in their proof of the Wagner conjecture. Decompositions based on this invariant provide a natural framework for implementing dynamic-programming algorithms to solve graph optimization problems. We will discuss the computational issues involved in using branch-width as as a general tool in discrete optimization.

We study support properties of solutions to stochastic heat equations $\partial_t u = \Delta u + \sigma(u) \xi$ where $\xi$ is Gaussian noise. For $\sigma(u) = u^\lambda$ with colored noise, we show the compact support property holds if and only if $\lambda \in (0, 1)$. Here, the compact support property refers to the property that if the initial function has compact support, then so does the solution for all time. For space-time white noise with general $\sigma$, we characterize when solutions maintain compact support versus become strictly positive. We also discuss how the initial function influences these support properties. This is based on joint work with Beom-Seok Han and Jaeyun Yi.

This is a reading seminar of a graduate student, following the Fields medal work of Daniel Quillen on the foundation of the higher algebraic K-theory.

We investigate compact minimal surfaces in the Einstein-Maxwell theory with both electric and magnetic charges and a negative cosmological constant. A two-sided, embedded and strictly stable minimal surface that maximizes the magnetically charged Hawking mass naturally corresponds to the event horizon of a black hole. Our main theorem shows that the geometry near such a surface is rigid: a neighborhood is isometric to the dyonic Reissner-Nordstrom-Anti-de Sitter space, the canonical model of a charged black hole in Anti-de Sitter spacetime. In addition, we provide an area estimate for the surface that depends only on its topology and the relevant physical parameters.

A graph $G$ is (list, DP) $k$-critical if the (list, DP) chromatic number is $k$ but for every proper subgraph $G’$ of $G$, the (list, DP) chromatic number of $G’$ is less than $k$. For $k\geq 4$, we show a bound on the minimum number of edges in a DP $k$-critical graph, and our bound is the first bound that is asymptotically better than the corresponding bound for proper $k$-critical graphs by Gallai from 1963. Our result also improves the best bound on the list chromatic number. This is joint work with Bradshaw, Kostochka, and Xu.

In this talk, we prove that the inviscid surface quasi-geostrophic (SQG) equation is strongly ill-posed in critical Sobolev spaces: there exists an initial data $H^2(\mathbb{R}^2)$ without any solutions in $L^{\infty}_tH^2$. Then, we introduce similar ill-posedness results for $\alpha$-SQG and two-dimensional incompressible Euler equations. This talk is based on joint works with In-Jee Jeong(SNU), Young-Pil Choi(Yonsei Univ.), Jinwook Jung(Hanyang Univ.), and Min Jun Jo(Duke Univ.).

In this talk, we extend the classical theory of minimal surfaces studied in Euclidean and Riemannian geometry to a more general framework in Weitzenböck and Riemann-Cartan geometry, which incorporates torsion. After providing a gentle introduction to minimal surface theory, we present theorems that generalize classical results concerning the holomorphic nature of the Hopf differential, the conformality of the Gauss map, and the minimality of surfaces.

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation given noisy observation data on its solution at a set of spatial points. We consider the Gaussian prior probability for the log-slowness, which is expressed as a countable linear expansion of mutually independent normal random variables. The well-posedness of the inverse problem is established, using the variational formulation of the eikonal equation. We approximate the posterior by finitely truncating the expansion of the log-slowness, with an explicit error estimate in the Hellinger metric with respect to the truncation level. Solving the truncated eikonal equation by the Fast Matching Method, we obtain an approximation for the posterior in terms of the truncation level and the discrete grid size in the Fast Matching Method resolution. Using this result, we develop and justify the convergence of a Multilevel Markov Chain Monte Carlo (MLMCMC) method. In comparison to the case of a forward log-normal elliptic equation, proving error estimate for the MLMCMC method is technically more complicated, as the available result on the error of the Fast Matching Method only holds when the grid size is not more than a threshold, which is not uniform for all the realizations of the log-normal slowness. Using the heap sort procedure for the Fast Marching Method, our MLMCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity, which is equivalent to that of the forward solver. This reduces the computation complexity drastically, in comparison to the plain Monte Carlo method where a large number of realizations of the forward equation are solved with equal high accuracy. Numerical examples confirm the theoretical results on the convergence rate of the method and the optimal complexity. This is a joint work with Zhan Fei Yeo.