학과 세미나 및 콜로퀴엄
I will provide a brief introduction to the canonical metric problem in Kähler geometry and related objects. Then I'll explain how generalizations of these objects naturally appear in the context of partition functions of determinantal point processes on polarized Kähler manifolds. The talk will be aimed at beginning geometry students and I will be rather pedagogical. Especially, I will focus on geometric aspects of the topic, so probabilistic or physical discussion will be postponed or omitted. This is based on my recent preprint.
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.
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).
A normal projective surface with the same Betti numbers of the
projective plane CP2 is called a rational homology projective plane (briefly Q-homology CP2 or QHCP2). People working in algebraic geometry and topology have long studied a Q-homology CP2 with possibly quotient singularities. It has been known that it has at most five such singular points, but it is still mysterious so that there are many unsolved problems left.
In this talk, I’ll review some known results and open problems in this field which might be solved and might not be solved in near future. In particular, I’d like to review the following two topics and
to report some recent progress:
1. Algebraic Montgomery-Yang problem.
2. Classification of Q-homology CP2 with quotient singularities.
This is a joint work with Woohyeok Jo and Kyungbae Park..
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 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 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.
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.
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.
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
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).
The singular limit problem is an important issue in various forms of ODEs and PDEs, and it is particularly known as a fundamental problem in equations derived from fluid dynamics. In this presentation, I will introduce some general phenomena of the singular limit problem through several examples. Subsequently, I will examine how the solution of the Euler-Maxwell equations converges to the MHD equations under the assumption that the speed of light approaches infinity, and how the Boussinesq equations converge to the QG equations in certain regimes.
Wavelets provide a versatile framework for signal representation and analysis, integrating ideas from harmonic analysis, approximation theory, and practical algorithm design. In this talk, we introduce foundational concepts in wavelet theory, focusing on classical results regarding wavelet expansions and approximations. Building on these basics, we explore modern developments and discuss how these approaches can balance theoretical rigor with practical convenience. The presentation aims to offer both a solid introduction to classical wavelet theory and a glimpse into current and future research directions. Part of the talk is based on joint work with Hyojae Lim.
A surface can be decomposed into a union of pairs of pants, a construction known as a pants decomposition. This fundamental observation reveals many important properties of surfaces. By forming a simplicial graph whose vertices represent pants decompositions, connecting two vertices with an edge whenever the corresponding decompositions differ by a simple move, we obtain a graph that is quasi-isometric to the Weil–Petersson metric on Teichmüller space. Meanwhile, topologists often study a structure called a rose, formed by attaching multiple circles at a single point. A rose is homotopy equivalent to a compact surface with boundary. Consequently, we can define a pants decomposition of a rose as the pants decomposition of a surface homotopy equivalent to it. In this talk, we will explore the concept of pants decompositions specifically in the context of roses.
In this note, we investigate threshold conditions for global well-posedness and finite-time blow-up of solutions to the focusing cubic nonlinear Klein–Gordon equation (NLKG) on $\bbR^{1+3}$ and the focusing cubic nonlinear Schrödinger equation (NLS) on $\bbR$. Our approach is based on the Payne–Sattinger theory, which identifies invariant sets through energy functionals and conserved quantities. For NLKG, we review the Payne–Sattinger theory to establish a sharp dichotomy between global existence and blow-up. For NLS, we apply this theory with a scaling argument to construct scale-invariant thresholds, replacing the standard mass-energy conditions with a $\dot{H}^{\frac12}$-critical functional. This unified framework provides a natural derivation of global behavior thresholds for both equations.
In this talk, we will discuss about smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role to understand group actions. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures.
In this talk, we will study non-linear actions on surfaces using random dynamics. We will discuss about absolutely continuity and exact dimensionality of stationary measures as well as classification of orbit closures. This talk will be mostly about the ongoing joint work with Aaron Brown, Davi Obata, and Yuping Ruan.
We establish the generic local Langlands correspondence by showing the equality of Langlands-Shahidi L-functions and Artin L-functions in the case of even unitary similitude groups. As an application, with one assumption on L-function, we prove both weak and strong versions of the generic Arthur packet conjectures in the cases of even unitary similitude groups and even unitary groups. Furthermore, we describe and define generic L-packets and therefore we were able to remove the above assumption. With our definition of L-packets, we recently prove its expected properties such as Shahidi's conjecture and finiteness of L-packets. This is in preparation and joint work with Muthu Krishnamurthy and Freydoon Shahidi.
The Langlands program, introduced by Robert Langlands, is a set of conjectures that attempt to build bridges between two different areas: Number Theory and Representation Theory (Automorphic forms). The program is also known as a generalization of a well-known theorem called Fermat’s Last Theorem. More precisely, when Andrew Wiles proved Fermat’s Last Theorem, he proved a special case of so-called Taniyama-Shimura-Weil Conjecture, which states that every elliptic curve is modular. And as a corollary, he was able to prove Fermat’s Last Theorem since Taniyama-Shimura-Weil Conjecture implies that certain elliptic curves associated with Fermat-type equations must be modular, leading to a contradiction. Note that the Langlands program is a generalization of the Taniyama-Shimura-Weil conjecture. In the first part of the colloquium, we briefly go over the following subjects:
(1) Fermat’s Last Theorem
(2) Taniyama-Shimura-Weil conuecture
And then, in the remaining of the talk, we start to explain a bit of the Langlands program
(3) Langlands program and L-functions
(4) (If time permits) Recent progress
This colloquium will be accessible to graduate students in other fields of mathematics (and undergraduate students who are interested in Number theory) at least in the first part.
In this talk, we will introduce vector field method for the wave equation. The key step is to establish the Klainerman-Sobolev inequality developed in [1]. Using this inequality, we will provide dispersive estimates of the linear wave equation, and prove small-data global existence for some nonlinear wave equations. The main reference will be Chapter II in [2].
참고문헌:
[1]. Sergiu Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. MR 784477
[2]. Christopher D. Sogge, Lectures on Nonlinear Wave Equations, Second Edition
Abstract:
We consider the initial-boundary value problem (IBVP) for the 1D isentropic Navier-Stokes equation (NS) in the half space. Unlike the whole space problem, a boundary layer may appear due to the influence of viscosity.
In this talk, we first briefly study the asymptotic behavior for the initial value problem of NS in the whole space. Afterwards, we will present the characterization of the expected asymptotics for the IBVP of NS in the half space. Here, we focus only on the inflow problem, where the fluid velocity is positive on the boundary.
Reference:
Matsumura, Akitaka. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl. Anal. 8 (2001), no. 4, 645–666.
We study stochastic motion of objects in micrometer-scale living systems: tracer particles in living cells, pathogens in mucus, and single cells foraging for food. We use stochastic models and state space models to track objects through time and infer properties of objects and their surroundings. For example, we can calculate the distribution of first passage times for a pathogen to cross a mucus barrier, or we can spatially resolve the fluid properties of the cytoplasm in a living cell. Recently developed computational tools, particularly in the area of Markov Chain Monte Carlo, are creating new opportunities to improve multiple object tracking. The primary remaining challenge, called the data association problem, involves mapping measurement data (e.g., positions of objects in a video) to objects through time. I will discuss new developments in the field and ongoing efforts in my lab to implement them. I will motivate these techniques with specific examples that include tracking salmonella in GI mucus, genetically expressed proteins in the cell cytoplasm, active transport of nuclei in multinucleate fungal cells, and raphid diatoms in seawater surface interfaces.