YONSEI ALGEBRAIC GEOMETRY SEMINARS

• 2020.10.14 1:30 Science hall
Speaker: 윤영호 (성균관대)
Title: Cohomologies in Topology, differential and algebraic geometry
Abstract: There are several cohomology theories in various areas of mathematics. Also, there are well-known relations between these theories. In this talk we review the relations between singular cohomology, de Ram cohomology and sheaf cohomology on a complex variety.

• 2018.9.19 WED. 4:30 SCIENCE HALL, Room TBA
SPEAKER: Igor Krylov (KIAS)
TITLE: Semistability of del Pezzo surfaces over rings: good models and reductions.
ABSTRACT: Kollar has introduced a notion of stability of a hypersurface in projective space over a ring. This notion can be used to find good reductions of hypersurfaces to finite characteristic. It also has application to birational geometry. If a cubic fibration over an affine curve is a semistable cubic surface over the coordinate ring of the base, then the fibration is a Mori fiber space and the total space has only Gorenstein singularities. This is an improvment over an existing result of Corti, whose version gave models with also Gorenstein but worse singularities. On the other hand, Corti's method allowed to prove similar result for del Pezzo fibrations of degree 2. I will talk about extending the results of Kollar to del Pezzo surfaces of degree 2 and 1. As a corollary we improve Corti's result and prove an analogue in degree 1.

• 2018.3.30 FRI. 4:00 Science bldg 254
SPEAKER: Kangjin Han (DGIST)
TITLE: Singular loci of secant varieties and its application to tensors
ABSTRACT: Let $X$ be an irreducible closed subvariety in a projective space. There is so-called a ‘secant construction’ which allows a natural way to construct a new variety from a given algebraic variety. In general, on singular loci of secants, it is classically known that the singular locus of $r$-th secant of X contains the previous one. Thus, a natural question is to ask its properness in the containment. For matrices,the variety of matrices with rank bounded by $r$ is singular exactly in the variety of matrices with rank bounded by $r-1$. But, for tensors, there are some known examples where the inclusion is strict.

In this talk, we introduce basic notions, review some known results briefly, and explain a recent result on set of rank 3 symmetric tensors by the author. We also consider some application of the result to a problem of tensor decomposition.

• 2018.3.8 Thu (1:30-3:30) & 9 Fri (2:00-5:00) Lecture Series (5 hours) Science bldg 254
SPEAKER: Kyoung-Seog Lee (IBS-CGP)
TITLE: Derived Categories and Birational Geometry
ABSTRACT: Recently, there are lots of progress in birational geometry and theory of derived categories. Moreover it turns out that derived categories of coherent sheaves are closely related to birational geometry. In these talks, I will discuss these relations and survey recent developments.

2018.3.8 Thur. 5pm Science Bldg 213 - DEPARTMENT COLLOQUIUM
SPEAKER: Kyoung-Seog Lee (IBS-CGP)
TITLE: Derived categories and their applications
ABSTRACT: The notion of derived category is a powerful tool in homological algebra invented by Grothendieck and Verdier. In particular, derived categories of coherent sheaves play important roles in many areas of algebraic geometry in these days. In this talk I will discuss basic definitions and properties about derived categories. Then I will discuss several applications of these categories.

• 2018.1.29 Mon
3:00pm, Science bldg 254
SPEAKER: Inkyun Kim (Seoul National University)
TITLE: WEAKLY EXCEPTIONAL SINGULARITIES OF COMPLETE INTERSECTION LOG DEL PEZZO SURFACES
ABSTRACT: We calculate the global log canonical thresholds of log del Pezzo surfaces embedded in weighted projective spaces as codimension two. As important applications, we show that most of them are weakly exceptional and admit K ?ahler-Einstein metrics. This is a joint work with Joonyeong Won.

4:30pm, Science bldg 254
SPEAKER:
Joonyeong Won (Institute for Basic Science)
TITLE: Survey of winter school of K-stability
ABSTRACT: I will survey the story of the talks by Chenyang Xu in the winter school of K-stability.

• 2017.9.14 Thur. 5pm Science Bldg 213 - DEPARTMENT COLLOQUIUM
SPEAKER:
MILES REID (University of Warwick)
TITLE: Classification of varieties and classification of Godeaux surfaces
ABSTRACT: The talk consist of two parts. The first gives an overview of the modern point of view of the classification of projective varieties via Mori theory. The aim is to use the intuitive idea of curvature familiar from plane geometry, where zero curvature (or flat geometry) corresponds to Euclidean geometry, positive curvature to spherical geometry, and hyperbolic non-Euclidean geometry corresponds to negative curvature, like the surface of a saddle or a Pringle's potato chip. The second part focuses on a particular case of surfaces of general type, that is, complex projective surfaces with negative Ricci curvature. Godeaux surfaces are the surfaces of general type with the smallest possible invariants p_g = 0, K^2 = 1. Despite being the "first case" in terms of invariants, these surfaces have a rich culture, and I describe two different programs of work in progress that aspire to give a complete treatment of their moduli. The more recent of these is by Isabel Stenger, a PhD student of Frank-Olaf Schreyer and Wolfram Decker at Kaiserslautern.

• 2017.7.27. Thur. 4pm. Science bldg 254
SPEAKER:
Wanmin Liu (IBS, Center for Geometry and Physics)
TITLE: How to use Bridgeland stability wall-crosssings to study birational geometry of moduli spaces?
ABSTRACT : This is an introductory talk on Bridgeland's idea of stability conditions. I will give examples on Bridgeland stability conditions on smooth projective surfaces, and give you an basic example of using wall-crosssings of stability conditions to explain birational flips inside Hilbert scheme of points over Hirzebruch surface or elliptic fibered surface.

• 2017.6.1. Thur. 1pm. Science Bldg 254
SPEAKER:
Jaiung Jun (State University of New York at Binghamton)
TITLE: Geometry of hyperfields
ABSTRACT: Hyperrings and hyperfields are algebraic structures which generalize commutative rings and fields. In this talk, we aim to introduce these exotic structures and illustrate how hyperrings and hyper- fields show up and fit into the classical theory, in particular, algebraic geometry and combinatorics.

• 2017.5.18 Thur. 5pm Science Bldg 213 - DEPARTMENT COLLOQUIUM
SPEAKER: 박의성 (고려대)
TITLE: Guido Castelnuovo의 대수곡선론에 관한 업적들
ABSTRACT : Guido Castelnuovo(1865-1952)는 이탈리아 수학자였습니다. 그는 19세기 중반에서 20세기 초중반에 크게 번영을 누렸던 Italian Algebraic Geometry School의 리더 중 한명이었고 대수 곡선 이론과 대수 곡면 이론의 발전에 큰 공헌을 하였습니다. 이 발표에서는 그가 대수 곡선 이론에 어떠한 기여를 하였는지 살펴보고자 합니다. 그리고 그의 대수 곡선 이론에 대한 업적들과 직접적으로 관련이 되는 현대 대수기하학의 몇 가지 세부 분야들 - Castelnuovo-Mumford regularity, Projecitve normality and syzygies, Castelnuovo Theory - 에 대해서도 간략하게 설명하고자 합니다.

• 2017.5.18. Thur. 11am. Science Bldg 254
SPEAKER: Mehdi Tavakol (Institute for Basic Science, Pohang)
TITLE: Tautological classes with twisted coefficients.
ABSTRACT: Let $M_g$ be the moduli space of smooth genus $g$ curves. We define a notion of Chow groups of $M_g$ with coefficients in a representation of $Sp(2g)$, and we define a subgroup of tautological classes in these Chow groups with twisted coefficients. Studying the tautological groups of $M_g$ with twisted coefficients is equivalent to studying the tautological rings of all fibered powers of the universal curve over $M_g$ simultaneously. By taking the direct sum over all irreducible representations of the symplectic group in fixed genus, one obtains the structure of a twisted commutative algebra on the tautological classes. We obtain some structural results for this twisted commutative algebra, and we are able to calculate it explicitly when $g \leq 4$. Thus we determine $R^\bullet(C_g^n)$ completely, for all $n$, in these genera. We also give some applications to the Faber conjecture. Joint with Dan Petersen and Qizheng Yin.

• 2017.4.26. Wed. 2pm. Science Bldg 254
SPEAKER:
Nero Budur (Katholieke Universiteit Leuven, Belgium)
TITLE: Absolute sets and the Decomposition Theorem
ABSTRACT: The celebrated Monodromy Theorem states that the eigenvalues of the monodromy of a polynomial are roots of unity. In this talk we give on overview of recent results on local systems achieving a vast generalization of the Monodromy Theorem. We end up with a conjecture of Andre-Oort type for special loci of local systems. The conjecture is true in rank one, and if true in general, it would provide a simple proof in all generality of the DecompositionTheorem of Beilinson-Bernstein-Deligne-Gabber. Joint work with Botong Wang.

• 2016.12.15. 4pm. Science Bldg 254
SPEAKER:
Constantin Shramov (Steklov Mathematical Institute of Russian Academy of Sciences)
TITLE: Rational nodal quartic threefolds
ABSTRACT: I will speak about rationality (and non-rationality) constructions for a special family of quartic threefolds with large automorphism groups. Among them are classical Burkhardt and Igusa quartic, and some explicit constructions go back to Todd. The talk is based on joint works (partially in progress) with Ivan Cheltsov.

• 2016.11.28. 3pm. Science Bldg 262
SPEAKER:
Jeehoon Park (POSTECH)
TITLE: Arithmetic Chern-Simons theory
ABSTRACT: The Chern-Simons theory is a gauge theory which is a version of (2+1)-dimensional TQFT (topological quantum field theory). It provided a useful framework and tools to understand the topology of knots in a 3-manifold, for example, the Jones polynomial of knots. The arithmetic Chern-Simons theory, initiated by Minhyong Kim, is an arithmetic analogue of the Chern-Simons theory, which aims to attack the number theory problem (Galois theory problem, L-functions, Iwasawa theory, and etc) guided by physics (quantum field theory) and topology principles and techniques appearing in the Chern-Simons theory. In this talk, we will briefly explain the analogy between primes in a number field and knots in a 3-manifold, and define the arithmetic Chern-Simons action. Then we will provide its arithmetic application to a certain Galois embedding problem based on an explicit computation of the arithmetic Chern-Simons action. This is a joint work with Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, and Hwajong Yoo.

• 2016.11.17 Thur. 5pm Science Bldg 213, DEPARTMENT COLLOQUIUM
SPEAKER: DongSeon Hwang (Ajou University)
TITLE: Cascades of singular del Pezzo surfaces
ABSTRACT: After a brief review on what it means by classification in topology and in algebraic geometry, I will talk about cascades of del Pezzo surfaces.

• 2016.11.24. 2pm. Science Bldg 254
SPEAKER: Donghoon Jang (KIAS)
TITLE: Equilibrium points of symplectic periodic maps
ABSTRACT: During this talk, we discuss equilibrium points of flows in the case that a manifold admits a symplectic structure and a flow on the manifold preserves the symplectic structure. We discuss main theorems on equilibrium points of symplectic periodic maps, and their relation to the question of when a symplectic flow is Hamiltonian. Next, we discuss the classification of a symplectic periodic flow, when the dimension of a manifold or the number of equilibrium points is small.

• 2016. 8.25
SPEAKER:
Joonyeong Won (Institute for Basic Science)
TITLE: K-stability of smooth del Pezzo surfaces
ABSTRACT: We introduce new invariant, delta-invariant for K-stability of Fano variety. By using the invariant, in algebro-geometric way, we completely determine whether smooth del Pezzo surfaces are K-stable.

• 2016. 6.24
SPEAKER:
Jaiung Jun (State University of New York at Binghamton)
TITLE: Introduction to hyperrings and hyperfields.
ABSTRACT:We introduce hyperrings and hyperfields. These are algebraic structures which generalize the classical commutative rings and fields. In this talk, we aim to introduce these rather exotic structures and illustrate several examples. We also discuss how hyperrings and hyperfields show up and fit into the classical theory, in particular, algebraic geometry and combinatorics.

• 2016. 6.22
SPEAKER:
Sungmun Cho (University of Toronto)
TITLE: An inductive formula of the Gross-Keating invariant of a quadratic form
ABSTRACT: The Gross-Keating invariant of a half-integral symmetric matrix B over the ring of integers of a non-archimedean local field plays an important role to make a direct connection between arithmetic intersection numbers for some moduli stack and the Fourier coefficients of Siegel Eisenstein series of level one and weight 2 for Sp_{2g}, g<4. Recently Ikeda and Katsurada described the Fourier coefficients in terms of Gross-Keating invariants. In this talk, I will explain an explicit inductive formula to compute the Gross-Keating invariant for any given such B. In conjunction with Ikeda-Katsurada's works, this would be used to prove an equality between some arithmetic intersection numbers and the Fourier coefficients of Siegel Eisenstein series of level one and weight 2 for Sp_{2g} for g=4. This is a joint work with Takuya Yamauchi.