# 학과 세미나 및 콜로퀴엄

The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.

In this series of lectures, we will discuss fundamental concepts of Bayesian inference and its applications to dynamical systems in the context of data assimilation. The focus is on the assimilation step, which combines observation data with a prediction model. The lectures will discuss Gaussian and non-Gaussian approaches, including Kalman and particle filters. We will also discuss implementations of various algorithms and consider their computational benefits and efficiency. It is recommended to know basic numerical analysis for predictions, but it is optional to understand the main ideas of the lectures.

There have been at least two surprising events to geometers in 80-90s that they had to admit physics really helps to solve classical problems in geometry. Donaldson proved the existence of exotic 4-dimensional Euclidean space using gauge theory and Givental counts rational curves in quintic threefolds using Feynman diagram in string theory. These events hugely popularised new mathematical topics such as mirror symmetry and enumerative geometry via moduli spaces. In this talk, we present what happened in this field in a past few decades and something happening right now.

I will describe recent joint work with Keller VandeBogert on constructing pure free resolutions over quadric hypersurface rings. Along the way I will describe some connections between total positivity and Koszul algebras and some conjectures regarding the homotopy Lie algebra and its "fattened" versions.

Zoom: https://kaist.zoom.us/j/81046219243?pwd=Py3uTm7NnFeUnshoaoctuZpsKptH5i.1 Meeting ID: 810 4621 9243 Password: syzygies

Zoom: https://kaist.zoom.us/j/81046219243?pwd=Py3uTm7NnFeUnshoaoctuZpsKptH5i.1 Meeting ID: 810 4621 9243 Password: syzygies

We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to other PDEs. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the two- and three-dimensional settings.

An Ulrich bundle E on an n-dimensional projective variety (X, O(1)) is a vector bundle whose module of twisted global sections is a maximal Cohen-Macaulay module having the maximal number of generators in degree 0. It was once studied by commutative algebraists, but after Eisenbud and Schreyer introduced its geometric viewpoint, many people discovered several important applications in wide areas of mathematics. In this motivating paper, Eisenbud-Schreyer asked a question whether a given projective variety has an Ulrich bundle, and what is the minimal possible rank of an Ulrich bundle if exists. The answer is still widely open for algebraic surfaces and higher dimensional varieties.
Thanks to a number of studies, the answer for the above question is now well-understood for del Pezzo threefolds. In particular, a del Pezzo threefold V_d of (degree d≥3) has an Ulrich bundle of rank r for every r at least 2. The Hartshorne-Serre correspondence translates the existence of rank-3 Ulrich bundle into the existence of an ACM curve C in V_d of genus g=2d+4 and degree 3d+3. In this talk, we first recall a construction of rank-3 Ulrich bundle on a cubic threefold by Geiss and Schreyer, by showing that a "random" curve of given genus and degree lies in a cubic threefold and satisfies the whole conditions we needed. We also discuss how this problem is related to the unirationality of the Hurwitz space H(k, 2g+2k-2) and the moduli of curves M_g. An analogous construction works for d=4, however, for d=5 a general curve of genus 14 and degree 18 does not belong to V_5. We characterize geometric conditions when does such a curve can be embedded into V_5 using the vanishing resonance. This is a joint work with Marian Aprodu.

We give L^2-signature obstructions to embedding closed 3-manifolds with infinite cyclic first homology in closed 4-manifolds with infinite cyclic fundamental group preserving first homology. From the obstructions, we obtain lower bounds on the double slice genus of a knot, and give examples of algebraically doubly slice knots with vanishing Casson-Gordon invariants whose double slice genera are arbitrarily large. This is a joint work with Taehee Kim.

We consider the Euler-Poisson system, which describes the ion dynamics in electrostatic plasmas. In plasma physics, the pressureless model is often employed to simplify analysis. However, the behavior of solutions to the pressureless model generally differs from that of the isothermal model, both qualitatively and quantitatively - for instance, in the case of blow-up solutions.
In previous work, we investigated a class of initial data leads to finite-time C^1 blow-up solutions. In order to understand more precise blow-up profiles, we construct blow-up solutions converging to the stable self-similar blow-up profile of the Burgers equation. For the isothermal model, the density and velocity exhibit C^{1/3} regularity at the blow-up time. For the pressureless model, we provide the exact blow-up profile of the density function, showing that the density is not a Dirac measure at the moment of blow-up.
We also consider the peaked traveling solitary waves, which are not differentiable at a point. Our findings show that the singularities of these peaked solitary waves have nothing to do with the Burgers blow-up singularity. We study numerical solutions to the Euler-Poisson system to provide evidence of whether there are solutions whose blow-up nature is not shock-like.
This talk is based on collaborative work with Junho Choi (KAIST), Yunjoo Kim, Bongsuk Kwon, Sang-Hyuck Moon, and Kwan Woo (UNIST)

In this talk we present a construction of quadratic equations and their weight one syzygies of tangent varieties using 4-way tensors of linear forms. This is in line with the 2-minor technique for quadratic equations of projective varieties and with the Oeding-Raicu theorem on equations of tangent varieties to Segre-Veronese varieties. We also discuss generalizations of the method if time permits. This is an early stage research.

In this talk, we study initial value problem for the Einstein equation with null matter fields, motivated by null shell solutions of Einstein equation. In particular, we show that null shell solutions can be constructed as limits of spacetimes with null matter fields. We also study the stability of these solutions in Sobolev space: we prove that solutions with one family of null matter field are stable, while the interaction of two families of null matter fields can give rise to an instability.

2-linear varieties are a rich topic. Sijong Kwak initiated the study of 3-regular varieties. In this talk I report on joined work Haoang Le Truong on the classification of smooth 3-regular varieties of small codimension 3. Some of these varieties are analogously to the 2-regular case determinantal. This first non-determinantal cases occurs in codimension 3. In this talk I report on the classification of varieties with Betti table
$$
\begin{matrix}
& 0 & 1 & 2 & 3\\ \hline
0: & 1 & . & . & .\\
1: & . & . & . & .\\
2: & . & 10 & 15 & 6
\end{matrix}
$$
Our approach consist of studying extension starting from curves. Let $X \subset \mathbb P^n$ be a variety. An e-extension $Y \subset \mathbb P^{n+e}$ of $X$ is a variety, which is not a cone, such that there exists a regular sequence $y_1,\ldots,y_e$ of linear forms for the homogeneous coordinate ring $S_Y$ of $Y$ such that $S_Y/(y_1,\ldots,y_e) = S_X$ is the coordinate ring of $X$. Using a computationally easy deformation theoretic method to compute extensions, we classify the extensions of 3-regular curves in $\mathbb P^4$ to surfaces in $\mathhbb P^5$ completely.

Distances such as the Gromov-Hausdorff distance and its Optimal Transport variants are nowadays routinely invoked in applications related to data classification. Interestingly, the precise value of these distances on pairs of canonical shapes is known only in very limited cases. In this talk, I will describe lower bounds for the Gromov-Hausdorff distance between spheres (endowed with their geodesic distances) which we prove to be tight in some cases via the construction of optimal correspondences. These lower bounds arise from a certain version of the Borsuk-Ulam theorem for discontinuous functions.

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. However, a critical question remains: How do we validate that PINNs accurately solve these PDEs?
This talk explores the types of mathematical validation required to ensure that PINNs can reliably approximate solutions to PDEs. We will discuss the conditions under which PINNs can converge to the correct solution, the relationship between minimizing residuals and achieving accurate results, and the role of optimization algorithms in this process. Our goal is to provide a clear understanding of the theoretical foundations needed to trust PINNs in practical applications while addressing the challenges in this emerging field.