Yunhyung Cho (Sungkyunkwan University)
Title: Mirror symmetry of Fano varieties and toric degenerations
Abstract: Mirror symmetry of Fano varieties is a correspondence between Fano varieties and Laurent polynomials (called Landau-Ginzburg models) such that one can recover some enumerative data of a Fano variety from a corresponding LG-model. There are many perspectives for explaining this correspondence. In symplectic geometry, one can obtain a LG-model of a smooth Fano variety X as a Fukaya-Oh-Ohta-Ono's potential function of a monotone Lagrangian torus living in X. In algebraic side, one can obtain a LG-model by studying the solution of the quantum differential equation for X. Unfortunately it is extremely hard to compute a Laurent polynomial from a given Fano variety. In good cases, Nishinou-Nohara-Ueda computed a mirror using toric degenerations. In this talk, I will explain this story in as much detail as possible.
This talk will serve a backgroud materials as well as a motivation for Eunjeongs talks. If time permits, I will also explain a relationship between LG-models. In general, the mirror correspondance is one-to-many and LG-models of a fixed Fano variety are expected to be related to each other by so-called mutations. Those relations are deeply connected to the theory of cluster algebras when X is a flag variety.
Reference:
[ACGK] M. AKHTAR, T. COATES, S. GALKIN, and A. M. KASPRZYK, Minkowski polynomials and mutations, SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012)
[CO] Cheol-Hyun Cho and Yong-Geun Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), no.4, 773--814.
[FOOO] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono, Lagrangian Floer theory on compact toric manifolds. I. Duke Math. J. 151 (2010), no. 1, 23--174.
[NNU]Takeo Nishinou, Yuichi Nohara, Kazushi Ueda, Toric degenerations of Gelfand-Cetlin systems and potential functions. Adv. Math. 224 (2) (2010), 648--706.
Eunjeong Lee (Chungbuk National University)
Title: Exploring the combinatorics of string polytopes
Abstract: In this talk, we will explore the fascinating connections between geometry, topology, representations, and combinatorics. Let $G$ be a semisimple algebraic group and let $B$ be a Borel subgroup. The flag variety $G/B$ is a smooth projective variety that has a fruitful connection with representations. Specifically, the global sections of a line bundle $\mathcal{L}$ over $G/B$ relate to an irreducible representation of $G$ when $\mathcal{L}$ is a very ample line bundle by the Borel--Weil--Bott theorem. On the other hand, the string polytopes are combinatorial objects which encode the characters of irreducible $G$-representations. The geometry of flag varieties and the combinatorics of string polytopes are connected via the theory of Newton--Okounkov bodies and toric degenerations as shown by Kaveh [Kav15].
Among the various string polytopes, one of the most well-known examples is the Gelfand--Cetlin polytope. However, there exist other string polytopes that have distinct combinatorial structures. In this series of talks, we will explore the combinatorics of string polytopes. Our exploration contains the explicit description of string polytopes given by Gleizer and Postnikov [GP00], the classification and enumeration of Gelfand--Cetlin-type string polytopes [CKLP21, CKL22], and the construction of small toric resolutions of toric varieties associated with certain string polytopes [CKLP23].
This talk is based on collaborations with Yunhyung Cho, Jang Soo Kim, Yoosik Kim, and Kyeong-Dong Park.
References:
[CKL22] Yunhyung Cho, Jang Soo Kim, and Eunjeong Lee, Enumeration of Gelfand--Cetlin type reduced words, Electron. J. Combin. 29 (2022), no. 1, Paper No. 1.27.
[CKLP21] Yunhyung Cho, Yoosik Kim, Eunjeong Lee, and Kyeong-Dong Park, On the combinatorics of string polytopes, J. Combin. Theory Ser. A 184 (2021), Paper No. 105508.
[CKLP23] Yunhyung Cho, Yoosik Kim, Eunjeong Lee, and Kyeong-Dong Park, Small toric resolutions of toric varieties of string polytopes with small indices, Commun. Contemp. Math. 25 (2023), no. 1, Paper No. 2150112.
[GP00] Oleg Gleizer and Alexander Postnikov, Littlewood--Richardson coefficients via Yang--Baxter equation, Internat. Math. Res. Notices (2000), no. 14, 741--774.
[Kav15] Kiumars Kaveh, Crystal bases and Newton--Okounkov bodies, Duke Math. J. 164 (2015), no. 13, 2461--2506.
Naoki Fujita (Kumamoto University)
Title: Introduction to canonical bases and crystal bases
Abstract: In representation theory, it is a fundamental problem to give a concrete basis of a representation. Using quantized enveloping algebras, Lusztig and Kashiwara constructed a specific basis of an irreducible highest weight representation of a semisimple Lie algebra. This is called Lusztig's canonical basis (= Kashiwara's global basis). In this lecture series, we survey Kashiwara's algebraic approach to global bases in the case of special linear Lie algebras, following his survey [Kas95, Kas02]. The surveys [Kas95, Kas02] contain many common parts, but some contents in [Kas02] are not included in [Kas95]. His approach gives a combinatorial skeleton of the global basis, called a crystal basis.
In the theory of crystal bases, it is important to give their concrete realizations. Until now, many geometric or combinatorial realizations have been discovered. In this lecture series, we discuss some combinatorial realizations using Young tableaux and convex polytopes (Nakashima-Zelevinsky polytopes). A Nakashima-Zelevinsky polytope [NZ97] is a different kind of string polytope; string polytopes will be explained in the lecture series by Eunjeong Lee. Based on a joint work [FN17] with Satoshi Naito, we will also see a geometric aspect of Nakashima-Zelevinsky polytopes. If time permits, a relation [Fuj22] between Nakashima-Zelevinsky polytopes and Schubert calculus will be explained.
This lecture series will assume basic knowledge of representation theory of semisimple Lie algebras, for instance, written in [Hum78]. However, you can ask anything even if it is assumed.
References:
[Kas95] Masaki Kashiwara, On crystal bases, in Representations of Groups (Banff, AB, 1994), CMS Conf. Proc. Vol. 16, Amer. Math. Soc., Providence, RI, 1995, 155--197.
[Kas02] Masaki Kashiwara, Bases cristallines des groupes quantiques, Edited by Charles Cochet, Cours Spéc. 9, Société Mathématique de France, Paris, 2002.
[NZ97] Toshiki Nakashima and Andrei Zelevinsky, Polyhedral realizations of crystal bases for quantized Kac-Moody algebras, Adv. Math. 131 (1997), no. 1, 253-278.
[FN17] Naoki Fujita and Satoshi Naito, Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases, Math. Z. 285 (2017), no. 1-2, 325-352.
[Fuj22] Naoki Fujita, Schubert calculus from polyhedral parametrizations of Demazure crystals, Adv. Math. 397 (2022), Paper No. 108201.
[Hum78] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics Vol. 9, Springer-Verlag, New York-Berlin, 1978.
Grigory Solomadin: On acyclicity for torus orbit spaces of homogeneous manifolds
Hidemasa Suzuki: T*R^n上の概正則円盤について
Yuto Yamamoto: Tropical geometry and period integrals
Nobukazu Kowaki: Toric degenerations of Grassmannians by SAGBI basis
Kentaro Yamaguchi: Toric submanifolds associated to affine subspaces
Azuna Nishida: 重み付き射影空間に対するSYZトーラスファイバー束を用いたホモロジー的ミラー対称性予想について