Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In recent joint work with Purnaprajna Bangere we give a positive answer to this question.

Let L be a ample line bundle on a projective scheme X. We say that (X,L) satisfies property QR(k) if the homogeneous ideal can be generated by quadrics of rank less than or equal to k. In the previous paper, we show that the Veronese embedding satisfies property QR(3). Let (X,L) be a Segre-Veronese embedding where X is a product of P^{a_i} with i=1,...,l and L is a very ample lines bundle O_X(d_1,d_2,...,d_l). In the paper [Linear determinantal equations for all projective schemes, SS2011], they prove that (X,L) satisfies QR(4) and it is determinantally presented if at least l-2 entries of d_1,...,d_l are at least 2. in this talk, we prove that (X,L) satisfies Qr(3) if and only if all the entries of d_1,...,d_l are at least 2. For one direction, we compute the radical ideal of 4 by 4 minors of a big matrix with linear forms, and for the other direction, we use the inducution on the sum of entries of (d_1,...,d_l).

A fundamental problem at the confluence of algebraic geometry, commutative algebra and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. By embedding smaller flag varieties as Schubert subvarieties in larger ones, one can compare cohomology groups on different spaces and study their eventual asymptotic behavior. In this context I will describe a sharp stabilization result, and discuss some consequences and illustrative examples. Joint work with Keller VandeBogert.

A vector bundle on projective space is called "Steiner" if it can be recognized simply as the cokernel of a map given by a matrix of linear forms. Such maps arise from various geometric setups and one can ask: from the Steiner bundle, can we recover the geometric data used to construct it? In this talk, we will mention an interesting Torelli-type result of Dolgachev and Kapranov from 1993 that serves as an origin of this story, as well as other work that this inspired. We'll then indicate our contribution which amounts to analogous Torelli-type statements for certain tautological bundles on the very ample linear series of a polarized smooth projective variety. This is joint work with R. Lazarsfeld.

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.

We examine the dynamics of short-range interacting Bose gases with varying diluteness and interaction strength. Using a combination of mean-field and semiclassical methods, we show that, for large numbers of particles, the system’s local mass, momentum, and energy densities can be approximated by solutions to the compressible Euler system (with pressure P = gρ2 ) up to a blow-up time. In the hard-core limit, two key results are presented: the internal energy is derived solely from the many-body kinetic energy, and the coupling constant g = 4πc0 where c0 the electrostatic capacity of the interaction potential. The talk is based on our recent work arXiv:2409.14812v1. This is joint work with Shunlin Shen and Zhifei Zhang. The talk will be delivered in English and is meant for the general audience.

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.

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.

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

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.

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.

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 by 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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심사위원장: 황강욱, 심사위원: 강완모, 정연승, 하우석, 안정연(산업및시스템공학과)

This is a reading seminar to be given by 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.

This is a reading seminar to be given by 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.

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심사위원장: 안드레아스 홈슨, 심사위원: 박진형, 김은정(전산학부), 엄상일(ibs), 권오정(한양대학교)

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 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.

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.

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.

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.

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Graph Neural Networks 에 관심이 있는 누구든 환영합니다. 총 4개의 강의 중 마지막 강의입니다.

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총 4개의 강좌 중 세 번째 강좌입니다. 10월 4일에 계속됩니다.

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.

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.