• Abstract: Quantum integrable systems provide a rich meeting point between physics, algebra, and geometry. In this talk, we begin with the spin chain as a simple and concrete example, and explain how it gives rise to quantum groups. We then describe how the same structures can be understood geometrically, through equivariant constructions on holomorphic symplectic manifolds. The goal is to give an accessible overview of this circle of ideas from complementary algebraic and geometric viewpoints.

• Speaker: 유필상(서울대학교)

• Time & Location: 3월 27일 금요일 14-16시, 서울대학교 129동 104호

• Abstract: The quantum connection is a flat connection arising from genus 0 Gromov--Witten theory. They can be defined integrally for sufficiently positive symplectic manifolds, allowing one to consider their characteristic p or p-adic versions which bear similarity to Gauss--Manin connections in arithmetic geometry. I would like to survey aspects of this theory, focusing on the case of Calabi--Yau threefolds and the role of quantum power operations. This talk is mostly based on joint work with Shaoyun Bai and Daniel Pomerleano.

• Speaker: 이재희 (Stanford)

• Time & Location: 3월 27일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호 .

• Abstract: Mirror symmetry predicts a deep relation between moduli of Lagrangian branes and coherent sheaves. On the algebraic geometric side, locally, moduli stack of coherent sheaves can be described by quiver varieties via deformation theory. From the symplectic perspective, Floer theory plays the role of Ext groups, and it is natural to expect a quiver description of moduli of Lagrangian branes.

In this talk, I will recall recent results realizing Nakajima quiver varieties as deformation spaces of framed Lagrangian branes. I will then focus on bulk deformations and explain how they correspond to noncommutative deformations of the mirror quiver algebra. This yields a symplectic framework for noncommutative ADHM-type constructions introduced by Baranovsky–Ginzburg–Kuznetsov, and the noncommutative models introduced by Kawamata. Finally, I will explain how these results fit into local homological mirror symmetry and their relation to Calabi-Yau algebras. This talk is based on an ongoing project with Hansol Hong and Siu-Cheong Lau. 

• Speaker: Ju Tan (IBS CGP)

• Time & Location: 3월 20일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호 .

• Abstract: Nakajima quiver varieties appear in various fields, including symplectic resolutions, instanton moduli spaces, and representation theory. In this talk, I will introduce the basic notions of quiver representations and outline the construction of quiver varieties using Geometric Invariant Theory (GIT). Following this, I will explain how to construct highest-weight representations of Kac-Moody algebras via the convolution product on the Borel-Moore homology of Lagrangian subvarieties, illustrated with a tractable example.  

• Speaker: 정정우 (서울대학교)

• Time & Location: 3월 13일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호 .

• Abstract: Wall-crossing refers to the phenomenon that invariants defined using parameters can change discontinuously as those parameters vary. The Kontsevich–Soibelman formula provides a precise framework for organizing such changes in certain settings. I will explain the basic setup and work through an elementary example to illustrate how the formula captures wall-crossing behavior.  

• Speaker: 유필상 (서울대학교)

• Time & Location: 2월 13일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호 .

• Abstract: Donaldson–Thomas (DT) invariants are numerical invariants that count stable coherent sheaves on Calabi–Yau threefolds. However, DT invariants are well-defined only when the moduli space of stable coherent sheaves is proper. Kontsevich–Soibelman and Joyce–Song later independently developed invariants for 3-Calabi–Yau categories which make it possible to study generalized DT invariants. In this seminar, we introduce generalized DT invariants for categories associated with quivers with superpotentials.  

• Speaker: 어재혁

• Time & Location: 1월 30일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호 .

• Abstract:  We know that every Weinstein manifold admits a Lefschetz fibration, but it is not generally known how to construct a Lefschetz fibration of a given Weinstein manifold. Recently, Breen-Roy-Wang uploaded a preprint on arXiv, which claims that if one has a "regular" Lagrangian, it helps us to construct a Lefschetz fibration. In this talk, we will discuss the preprint, after introducing the notion of regular Lagrangians.   

• Speaker: 이상진 (KIAS)

• Time & Location: 12월 19일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 406호 .

• Abstract:  In the 1970s, Meyer and many other mathematicians studied the bifurcation of Hamiltonian orbits, obtaining a number of fundamental results. One of the most important is that, in the generic case, such bifurcations fall into eight well-known types. This classification includes the familiar Lyapunov, birth–death, and period-doubling bifurcations, among others. In this talk, I will review this classical classification of bifurcations, together with several classical theorems used in their analysis, and illustrate them with a few examples. I will also discuss how these classical results relate to the modern theory of symplectic homology. This talk is based primarily on Meyer’s 1970 paper and the book by Abraham and Marsden. 

• Speaker: 이동호 (QSMS)

• Time & Location: 12월 12일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호 .

• Abstract:  A spectral network is a graph on a Riemann surface subordinate to a branched covering, with many applications in geometry and mathematical physics. In this talk I will introduce the notions of a WKB spectral network and its topological generalization. I will briefly indicate the physical ideas behind WKB spectral networks and explain the geometric motivation for topological spectral networks via non-abelianization of flat connections. I will also discuss how spectral networks give natural coordinate systems on moduli spaces of flat connections on the surface. 

• Speaker: 하준수 (서울대학교)

• Time & Location: 12월 5일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract:  In this talk, I will review the symplectic homology associated with Lefschetz fibrations introduced by McLean. I will explain that the symplectic homology of a Lefschetz fibration is isomorphic to that of its total space viewed as a Liouville manifold. Furthermore, I will discuss a spectral sequence converging to the symplectic homology of a Lefchetz fibration, whose first page is described in terms of the fixed Floer homology of the monodromy map.

The main references for this talk are: 1. Mark McLean, Lefscehtz fibrations and symplectic homology. 2. Mark McLean, Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map.

• Speaker: 배한울 (QSMS)

• Time & Location: 11월 13일 목요일, 오전 10-12시, 서울대학교 상산수리과학관 309호 & 11월 21일 금요일, 오후 3-5시, 서울대학교 상산수리과학관 406호

• Abstract:  We will have a series of seminars that discuss (symplectic) Lefschetz fibration and related topics. In the first talk of this series, I will introduce basic definitions and results. (The targeting audiences are students, so we will start at a very basic level, such as definitions of Lefschetz fibration, vanishing cycle, etc.)  

• Speaker: 이상진 (고등과학원)

• Time & Location: 11월 7일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Notes

• Abstract:  I will discuss a relationship between moduli spaces of quadratic differentials on compact Riemann surfaces and spaces of Bridgeland stability conditions on a certain class of 3-Calabi–Yau categories, following the work of Bridgeland and Smith. While I won’t discuss this aspect in detail, these 3-CY categories are expected to be (and in some cases are) Fukaya categories of Calabi–Yau threefolds, making the connection particularly relevant from the perspective of symplectic geometry. 

• Speaker: 유필상 (서울대)

• Time & Location: 9월 26일 금요일, 오전 10-12시, 서울대학교 27동 220호 (평소와 다른 장소에서 진행됩니다.)

• Abstract:  A triangulation of a marked surface gives a quiver with potential, from which a 3-Calabi–Yau triangulated category can be constructed. In this talk, we will introduce the connections between surface triangulations, quivers with potentials, and their associated derived categories. We will explain how flips of a triangulation correspond to mutations of the associated quiver with potential, and how these mutations induce derived equivalences between the corresponding categories.  

• Speaker: 송석원 (서울대)

• Time & Location: 9월 19일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract:  In this talk, I will discuss a noncommutative analogue of a (smooth, compact) Calabi–Yau manifold. As a concrete local model for this analogue, I construct a quiver with superpotential. The definitions and the construction are explained through detailed B-side examples. If time permits, I will also explain the general pattern underlying this construction, known as Koszul duality. 

• Speaker: 정명진 (서울대)

• Time & Location: 9월 12일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract:  A fundamental problem in quantitative symplectic geometry is to understand in which ways a Hamtilonian flow can "squeeze" phase space. The special case of ellipsoids has been a great source of motivation for the last several decades, in many ways mirroring various important developments in the field (e.g. Gromov-Witten theory, Floer homology, symplectic field theory, embedded contact homology, and more). In this talk, I will survey some new developments in the study of high dimensional symplectic embeddings, and in particular the recent resolution of the so-called stabilized ellipsoid conjecture. Our framework sets up a bridge between quantitative symplectic geometry and the classical study of singular algebraic curves, studying the latter using tools from log Calabi-Yau mirror symmetry. I will not assume familiarity with any of this background. 

• Speaker: Kyler Siegel (University of Southern California & 서울대)

• Time & Location: 9월 5일 금요일, 오전 11-13시, 서울대학교 상산수리과학관 309호

• Abstract: We introduce and develop the theory of spectral networks in real contact and symplectic topology. First, we establish the existence and pseudoholomorphic characterization of spectral networks for Lagrangian fillings in the cotangent bundle of a smooth surface. These are proven via analytic results on the adiabatic degeneration of Floer trajectories and the explicit computation of continuation strips. Second, we construct a Family Floer functor for Lagrangian fillings endowed with a spectral network and prove its equivalence to the non-abelianization functor. In particular, this implies that both the framed 2d-4d BPS states and the Gaiotto-Moore-Neitzke non-abelianized parallel transport are realized as part of the A∞-operations of the associated 4d partially wrapped Fukaya categories. To conclude, we present a new construction relating spectral networks and Lagrangian fillings using Demazure weaves, and show the precise relation between spectral networks and augmentations of the Legendrian contact dg-algebra. 

• Speaker: 노윤재

• Time & Location: 8월 8일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract: 지난 두 번의 세미나(김종명 박사님, 유필상 교수님 발표)에서 stability condition에 관해 논의하였습니다. 이번 세미나에서는 symplectic geometry와 stablity condition이 어떻게 연결되는지 살펴보려 합니다. 

구체적으로는 Joyce의 conjecture를 소개하고, Joyce가 왜/어떻게 holomorphic volume form, special Lagrangian과 같은 기하학의 object들을 stability condition이라는 대수적인 object와 연결시켰는지 살펴보려 합니다. 

• Speaker: 이상진 (KIAS)

• Time & Location: 8월 1일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Notes

• Abstract: Abstract: In their 2023 survey, Lekili and Segal proposed a conjecture relating equivariant Lagrangian Floer theory to the Floer theory of a singular symplectic reduction. In this talk, I will present their conjecture, discuss a broad class of supporting evidence they provided, and explain why and how it is expected to hold. 

• Speaker: 좌동욱 (충북대학교)

• Time & Location: 7월 25일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract: Various types of duality results in both symplectic and algebraic geometry are well-known to be related to Calabi-Yau structures on the relevant categories. In this talk, I'll discuss an abstract approach, as a categorical analogue to Beilinson-Tate residue theory, to provide ingredients for constructing Calabi-Yau structures on dg/A-infinity categories that are not necessarily proper. Applications include duality type results for Rabinowitz Fukaya categories, singularity categories, and so on.

• Speaker: Yuan Gao (Nanjing University)

• Time & Location: 7월 9일 수요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract: In this talk, I will introduce the recent result of Lekili-Ueda. They conjectured HMS for Rabinowotz Fukaya category of Milnor fibers of invertible polynomials and proved it for Brieskorn-Pham singularities which are not of Calabi-Yau type. If time permits, I will explain its application for a calculation of the Rabinowitz Floer homology of the Milner fiber. 

• Speaker: 정원보 (서강대학교)

• Time & Location: 7월 4일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract: In this expository talk, we will review the notions of t-structures and torsion theories, with an emphasis on an explicit example. We will then introduce the notion of a 3-Calabi–Yau category via a quiver with potential, and revisit torsion theories in terms of the quiver data. Time permitting, we will discuss the correspondence between quiver mutations and surface triangulations. 

• Speaker: 유필상(서울대학교)

• Time & Location: 6월 27일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 301호

• Abstract:  I will talk about the basics of Bridgeland stability conditions and a conjectural description of the spaces of stability conditions as (almost) Frobenius manifolds. 

• Speaker: 김종명(QSMS)

• Time & Location: 6월 13일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract:  We introduce a categorical framework for simple homotopy theory in Fukaya categories, based on the fundamental group of the ambient symplectic manifold. As a consequence, we prove that any two closed exact Lagrangians that are isomorphic in the Fukaya category of a Weinstein manifold are simple homotopy equivalent, under the assumption that one of them is homotopy equivalent to the ambient symplectic manifold. This generalizes Abouzaid-Kragh's result for cotangent bundles to general Weinstein manifolds. 

• Speaker: 김용환(MIT)

• Time & Location: 6월 10일 화요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Abstract: Fukaya categories of smooth surfaces can be defined topologically, which makes them easier to compute with. Through their formal generators, topological Fukaya categories (TFC) can be related to a certain class of graded algebras, called (graded) gentle algebras. Conversely, derived categories of gentle algebras are realized as TFCs of some smooth surfaces. In this talk, I will generalize TFC construction to Z/2Z-orbifold surfaces, which introduces a new class of algebras, semi-gentle algebras. Also, we will see how to use this result to show the class of semi-gentle algebras is closed under derived equivalence. This talk is based on the joint work with Cheol-Hyun Cho and ongoing project with Severin Barmeier, Cheol-Hyun Cho, Kyungmin Rho, Sibylle Schroll, and Zhengfang Wang. 

• Speaker: 김경모 (퀼른대학교)

• Time & Location: 6월 2일 월요일, 오후 3-5시, 서울대학교 상산수리과학관 309호

• Abstract: Rabinowitz Floer homology is a Floer homology theory for contact hypersurfaces in symplectic manifolds. Recently, Ganatra-Gao-Venkatesh introduced its categorification, called the Rabinowitz Fukaya category. In this talk, I will discuss its construction and some examples. If time permits, I will explain that for any Weinstein manifold, its Rabinowitz Fukaya category is quasi-equivalent to the categorical formal punctured neighborhood of infinity of the wrapped Fukaya category.

• Speaker: 배한울 (QSMS)

• Time & Location: 5월 9일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Notes

• Abstract:  이번 발표에서는 지난 발표에서 정의한 topological Fukaya category의 hall algebra에 대해 공부하고 그 Hall algebra가 derived Skein relation을 만족한다는 Cooper-Samuelson의 결과(https://arxiv.org/abs/1708.00889)를 소개하려 합니다. Cooper-Samuelson은 local-to-global approach를 사용하여 Hall algebra를 연구하였고 이와 관련된 conjecture를 제시하였습니다. 이번 발표는 local-to-global approach를 소개하는 것으로 시작하여 이를 이용하여 어떻게 derived Skein relation을 증명할 수 있는지 간략하게 살펴보려 합니다. 

• Speaker: 이상진(고등과학원)

• Time & Location: 4월 18일 금요일, 오전 10-12시, 서울대학교 상산수리과학관 309호

• Notes

• Abstract:  Skein algebra is an algebra assigned to a surface that carries interesting information, such as quantum group actions or cluster algebra structures. It is conjectured and partially proven by Cooper-Samuelson and others that the skein algebra of a surface has a deep relation with the hall algebra of the Fukaya category of the surface. The expository talk will start with motivational questions about the conjectured relation and will discuss the result of Cooper-Samuelson with examples.  

• Speaker: 이상진(고등과학원)

• Time & Location: 4월 11일 금요일, 오후 1-3시, 서울대학교 상산수리과학관 309호

• Notes

• Abstract: A Hall algebra is an invariant of a category that encodes extension data between objects. In this expository talk, we will introduce the Hall algebra associated to a "finite" category and present Ringel's theorem, which reveals a striking relationship between Hall algebras and quantum groups. If time permits, we will also discuss how this framework extends beyond the classical setting. 

• Speaker: 유필상(서울대학교)

• Time & Location: 4월 4일 금요일, 오후 1-3시, 서울대학교 상산수리과학관 309호