# Algebraic Structures in Geometry and Physics:

a conference on the occasion of Jae-Suk Park's 60th birthday

2024. 10. 31 ~ 11. 3

유니호텔 제주

시간표:

10월 31일 (목)

15:00 ~ 17:00 도착 및 체크인

17:00 ~ 18:00 Discussion

11월 1일 (금)

10:00 ~ 10:50 설석봉 (KIAS)

11:10 ~ 12:00 이석주 (IBS-CGP)

14:00 ~ 14:50 좌동욱 (IBS-CGP)

15:10 ~ 16:00 박지훈 (서울대학교)

16:20 ~ 17:10 박병도 (충북대학교)

11월 2일 (토)

09:30 ~ 10:20 이재혁 (POSTECH)

10:30 ~ 11:20 이경석 (POSTECH)

11:30 ~ 12:20 이상민 (서울대학교)

13:30 ~ 17:00 Excursion

17:00 ~ 17:50 박재석 (POSTECH)

18:40 ~ Banquet (뚱딴지 애월본점)

11월 03일 (일)

10:00 ~ 12:00 Discussion

프로그램

설석봉(Seokbong Seol): Kapranov’s L-infinity algebras are homotopy abelian

Given a Kähler manifold, Kapranov constructed an L-infinity algebra structure on its Dolbeault complex with coefficients in the tangent bundle, in his study of Rozanski—Witten invariants. Later, this construction has been generalised to differential graded manifolds and Lie algebroid pairs via formally defined exponential maps. In this talk, we show that both constructions generalise to the notion of differential graded Lie algebroid. Moreover, we show that such L-infinity algebras are homotopy abelian. In particular, we show that Kapranov’s L-infinity algebras are homotopy abelian. This is a joint work with Ruggero Bandiera, Mathieu Stiénon and Ping Xu.

이석주(Sukjoo Lee): The P=W phenomena

Cohomology of a geometric space has been one of the fundamental algebraic tools for studying geometric properties. For example, when X is a projective complex manifold or variety, the cohomology group H*(X) carries a Hodge structure, a decreasing filtration that satisfies certain properties. A natural and intriguing question is what additional structures arise if X becomes singular or non-compact. Two such structures are the perverse and weight filtrations. In algebraic geometry, a deep and mysterious relationship exists between these filtrations, known as the P=W phenomena, particularly in the contexts of non-abelian Hodge theory and mirror symmetry. In this talk, I will provide an overview of these connections and introduce several open problems in the subject.

좌동욱(Dongwook Choa): Spin Structures on 3-term Obstruction Theories

In this talk, I will discuss spin structures and their applications in Donaldson-Thomas theory. I will begin by briefly explaining obstruction theories, highlighting (-2)-shifted symplectic and (-1)-shifted Lagrangian structures. These structures arise naturally as moduli spaces of sheaves on CY/Fano 4 folds. The concept of spin structure will be used to construct a canonical two-periodic complex or matrix factorization in each case. This is joint work with Jeongseok Oh.

박지훈(Jeehoon Park): BF path integrals for elliptic curves and p-adic L-functions

In this talk, I will explain an arithmetic path integral formula for the inverse p-adic absolute values of the p-adic L-functions of elliptic curves over the rational numbers with good ordinary reduction at an odd prime p based on the Iwasawa main conjecture and Mazur’s control theorem. The talk is based on a joint work with Junyeong Park.

박병도(Byungdo Park): Aspects of differential K-theory

Differential K-theory is a refinement of topological K-theory that incorporates connections. It has interesting applications in Type IIA and IIB superstring theory, index theory, and the theory of infinity-categories. We will introduce differential K-theory as a differential cohomology theory and present several well-known constructions. After that, we shall explain cycle maps in differential K-theory to add some flavors from category theory and algebraic K-theory.

이재혁(Jaehyeok Lee): Grothendieck-Galois theory over a symmetric monoidal category

A Galois category is a category which is equivalent to the category of finite left G-sets for some profinite group G. The concept of Galois

category was established by Alexander Grothendieck, which was a consequence of his attempt to unify the Galois theory of field extensions and that of covering spaces. The development of such categorical language led him towards the invention of \'{e}tale fundamental groups of schemes.

A Tannakian category over a field k is a k-linear tensor category which is equivalent to the k-linear tensor category of finitely generated projective representations of some affine gerbe over the fpqc site of affine k-schemes. The concept of Tannakian category was suggested by Alexander Grothendieck, in his hope to understand various cohomology groups of a scheme X as different realizations of its motive m(X). The theory of Tannakian category was first developed by his student Neantro Saavedra Rivano, and it was later further developed by Pierre Deligne.

In this talk, I will present the current progress of a project whose aim is to simultaneously generalize and unify the two categorical concepts, Galois category and Tannakian category.

By generalizing them I mean that given an abstract symmetric monoidal category K, we define what is a group (groupoid, affine group scheme,

affine groupoid scheme) G in K and characterize which category is equivalent to the category of representations of G in K for some G. Some examples of K are the symmetric monoidal categories of (small) sets, simplicial sets, modules over a commutative ring, (co)chain complexes over a commutative ring, Banach spaces with linear contractions, compactly generated weakly Hausdorff spaces, symmetric spectra. Every elementary topos, hence every Grothendieck topos, is also an example of K.

By unifying them I mean that the project consists of two contexts, each of which corresponds to the generalization of Galois category and that of Tannakian category, and these two contexts are categorically dual to each other.

This is a joint work with Jae-Suk Park.

이경석(Kyoungseog Lee): Moduli spaces of vector bundles on Riemann surfaces

Moduli space of vector bundles on a Riemann surface is one of the most fundamental objects in modern mathematics. It plays important roles in many areas of mathematics, e.g., algebraic geometry, differential geometry, mathematical physics, number theory, symplectic geometry, topology to name a few. In the first part of this talk, I will briefly review the history of the subject. Then I will discuss recent developments about the derived categories of coherent sheaves and the motives of certain moduli spaces of vector bundles on Riemann surfaces.

이상민(Sangmin Lee): Classical eikonal from Magnus expansion

In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum mechanics, the classical eikonal admits an expansion in oriented tree graphs, where oriented edges denote retarded/advanced worldline propagators. The Magnus expansion, which takes the log of a time-ordered exponential integral, offers an efficient method to compute the coefficients of the tree graphs to all orders. We exploit a Hopf algebra structure behind the Magnus expansion to develop a fast algorithm which can compute the tree coefficients up to the 12th order (over half a million trees) in less than an hour. In a relativistic setting, our methods can be applied to the post-Minkowskian (PM) expansion for gravitational binaries. We demonstrate the methods by computing the 3PM eikonal and find agreement with previous results based on amplitude methods.

박재석(Jae-Suk Park): TBA

참가자 목록 (가나다 순):

김도형, 김범석, 김승원, 김영락, 김유식, 박병도, 박재석, 박종일, 박지훈, 박형주, 배한울, 설석봉, 안병희, 유필상, 유화종, 이경석, 이상민, 이상욱, 이석주, 이재혁, 조성윤, 조윤형, 조철현, 좌동욱, 현승준

지원: 삼성미래기술재단, QSMS

조직위원회: 조철현 (서울대학교 수리과학부), 유화종 (서울대학교 자유전공학부)