# Department Seminars & Colloquia

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We use the geometry of symmetric products of curves to construct rank one symmetric Ulrich sheaves on the (higher) secant varieties. Time permitting, we will also give an application towards an algebraic theory of knots. This is joint work with M. Kummer and J. Park.

In this talk, we report some results on equations and the ideal of $\sigma_k(v_d(\mathbb{P}^n))$, the $k$-th secant variety of $d$-uple Veronese embedding of a projective space, in case of the $k$-th secant having a relatively small degree. Knowledge on defining equations of higher secant varieties is fundamental in the study of algebraic geometry and in recent years it also has drawn a strong attention in relation to tensor rank problems.
We first recall known results on the equation of a $k$-th secant variety and introduce key notions for this work, which are '$k$-secant variety of minimal degree' and 'del Pezzo $k$-secant variety', due to Ciliberto-Russo and Choe-Kwak, respectively. Next, we focus on the case of $\sigma_4(v_3(\mathbb{P}^3))$ in $\mathbb{P}^{19}$ as explaining our method and considering its consequences. We present more results which can be obtained by the same method. This is a joint work with K. Furukawa (Josai Univ.).

It is believed that one can attach a smooth mod-p representation of a general linear group to a mod-p local Galois representation in a natural way that is called mod-p Langlands program. This conjecture is quite far from being understood beyond GL₂(ℚₚ). However, for a given mod-p local Galois representation one can construct a candidate on the automorphic side corresponding to the Galois representation for mod-p Langlands correspondence via global Langlands. In this talk, we introduce automorphic invariants on the candidate that determine the given Galois representation for a certain family of mod-p Galois representations.

I will explain the notion of projection of syzygies, which was originally given by Ehbauer and later was much used by many mathematicians and then give two applications of it. We will firstly explain how it can be used to study the syzygies of canonical curves and in particular explain its application to a conjecture by Schreyer on the ranks of generating linear syzygies for general canonical curves. We will then explain an application of it to the study of linear syzygies of Veronese varieties.

In this talk, I will take you on a journey from mathematical concepts to their applications in brain and cognitive sciences. We will explore how differential equations and nonlinear dynamical systems can be employed to model complex biological systems, including the brain. I will also discuss how these models help in understanding higher-level processes such as sleep and circadian rhythms, offering a deeper glimpse into how the brain operates as a complex, dynamic system. Additionally, I will share my personal journey from my student days to my current role as a professor in brain and cognitive sciences, illustrating how my research path has evolved over time.

In this talk, I will first review the story about single/multi-parameter persistent homology and its algebraic abstraction, persistence modules, from the perspective of representation theory. Then, I will define the so-called interval rank invariant of persistence modules. This invariant can be computed easily by utilizing our proposed formula though its definition is purely algebraic, which will become the main part of this talk. One direct application of the formula is to show the relation between our invariant and the generalized rank invariant proposed by Kim-Memoli. If time permits, I will introduce some other applications and related content.

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.

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.

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.

In the 19th century, Kummer extensively studied quartic surfaces in the complex projective 3-space containing 16 nodes(=ordinary double points). One of his notable results states that a quartic surface cannot contain more than 16 nodes. This leads to a classic question: how many nodes may a surface of degree d contain? The answer to this question is known only for a very low degrees, namely, degrees 5 and 6. To find the optimal answer(31) for quintics, Beauville introduced the concept of "even sets of nodes," which turned out to be highly influential in the study of nodal surfaces. Based on the structure theorem of even sets by Casnati and Catanese, we will discuss some structure theorems of nodal quintics and sextics with maximal number of nodes. This talk is based on joint works with Fabrizio Catanese, Stephen Coughlan, Davide Frapporti, Michael Kiermaier, and Sascha Kurz.

Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on works with G. Farkas, Y. Kim, C. Raicu, A. Suciu, and J. Weyman I report on some recent results concerning the geometry of resonance schemes in the vector bundle setup.

https://kaist.zoom.us/j/81427312084?pwd=arF7jyUZ3aVbnQoKv74adW2Bx4Nh6g.1 Meeting ID: 814 2731 2084 Password: syzygies

https://kaist.zoom.us/j/81427312084?pwd=arF7jyUZ3aVbnQoKv74adW2Bx4Nh6g.1 Meeting ID: 814 2731 2084 Password: syzygies

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.

This is a reading seminar to be given Mr. Jaehong Kim (a graduate student in the department) on foundations of the intersection theory and the classification theory of complex algebraic surfaces. He will give three 2-hour long talks.

In this talk, I will discuss the expansion of the free energy of two-dimensional Coulomb gases as the size of the system increases. This expansion plays a central role in proving the law of large numbers and central limit theorems. In particular, I will explain how potential theoretic, topological, and conformal geometric information of the model arises in this expansion and present recent developments.

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자연과학동(E6) Room 1401 (최석정 강의실)
Algebraic Geometry
Doyoung Choi (KAIST)
Degree of the 3-secant variety

자연과학동(E6) Room 1401 (최석정 강의실)

Algebraic Geometry

We present a formula for the degree of the 3-secant variety of a nonsingular projective variety embedded by a 5-very ample line bundle. The formula is provided in terms of Segre classes of the tangent bundle of a given variety. We use the generalized version of double point formula to reduce the calculation into the case of the 2-secant variety. Due to the singularity of the 2-secant variety, we use secant bundle as a nonsingular birational model and compute multiplications of desired algebraic cycles. Additionally, we compute the Hilbert-Samuel multiplicity of 2-secant variety along given variety.

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.

The study of monomial ideals is central to many areas of commutative algebra and algebraic geometry, with Stanley-Reisner theory providing a crucial bridge between algebraic invariants and combinatorial structures. We explore how the syzygies and Betti diagrams of Stanley-Reisner ideals can be understood through combinatorial operations on simplicial complexes.
In this talk, we focus on the regularity of Stanley-Reisner ideals. We introduce a graph decomposition that bounds the regularity and a decomposition of simplicial complexes with respect to facets. In addition, we introduce secant complexes corresponding to the joins of varieties defined by Stanley-Reisner ideals and investigate the secant variety of minimal degree defined by the Stanley-Reisner ideals. This talk includes multiple collaborative works with G. Blekherman, J. Choe, J. Kim, M. Kim, and Y. Kim.