Geometry and Physics Seminar
Student No.:40
Time:Tue 13:30-15:00, Oct.8-Jan.15
Instructor:Li Si, Zong Zhengyu, Zhou Jie, Tang Xinxing  
Place:Jing Zhai 105
Starting Date:2018-10-8
Ending Date:2019-1-15

2019-1-8 Tuesday

Speaker: Chenglong Yu (University of Pennsylvania)

Title: Period integrals and tautological systems

Tautological systems are Picard-Fuchs differential systems introduced by Lian-Song-Yau that governs the period integrals of Calabi-Yau complete intersections in ambient spaces with large symmetry. I will describe a generalization to zero loci of vector bundle sections and discuss the Riemann-Hilbert type problems, including the solution rank formulas and geometric realizations of solutions. This is a joint work with Huang, Lian and Yau.

2018-12-25 Tuesday

Speaker: Meng Guo (PI)

Title: Cobordism and topological quantum field theory

We will show how to use cobordism to classify positivity reflection invertible TQFTs with symmetry groups. We use this to give a general formula for SPT phases by cobordism. We will also show some examples of certain symmetry groups and their computations of cobordism.

2018-12-18 Tuesday

Speaker: Emanuel Scheidegger (Peking University)

Title: Periods and quasiperiods of modular forms and the mirror quintic at the conifold

We review the theory of periods of modular forms and extend it to quasiperiods. General motivic conjectures predict a relation between periods and quasiperiods of certain weight 4 Hecke eigenforms associated to hypergeometric one-parameter families of Calabi-Yau threefolds. We show how to verify this prediction. Time permitting, we discuss some of its implications on D-brane masses at the conifold on the corresponding mirrors.

2018-12-11 Tuesday

Speaker: Jie Gu (University of Geneva)

Title: Solutions to Minimal 6d SCFTs by Modularity and Blowup Equations

The elliptic genera of a 6d SCFT contain the BPS invariants of an associated noncompact elliptic Calabi-Yau threefold. We propose two methods to compute the elliptic genera of 6d SCFTs, the first exploiting the modular properties of the elliptic genera, the second using a generalized version of the G\"ottsche-Nakajima-Yoshioka's blowup equations. We demonstrate these two methods with the examples of some minimal 6d SCFTs. We also emphasize the universality of the generalized blowup equations, which seem to be valid for any noncompact Calabi-Yau threefold.

2018-11-27 Tuesday

Speaker: Zhiwei Zheng(YMSC)

Title: Geometry of cubic fourfolds and hyper-Kahler manifolds

I will start with an introduction to K3 surfaces, cubic fourfolds and hyper-Kahler manifolds, and the motivation to study them. Then I will talk about some recent work (with Chenglong Yu) on moduli of K3 or cubic fourfolds with symmetry or singular points, and (with Radu Laza) on a classification of symplectic automorphism groups of cubic fourfolds.

2018-11-20 Tuesday

Speaker: Zhengfang Wang (BICMR)

Title: Tate-Hochschild Cohomology

The Tate-Hochschild cohomology of a singular space X is defined as the graded endomorphism ring of the diagonal inside the singularity category of X x X. Singularity categories are introduced by Buchweitz in representation theory and then rediscovered by Orlov in algebraic geometry and homological mirror symmetry. By Keller's very recent result, the Tate-Hochschild cohomology of a hypersurface is isomorphic to the Hochschild cohomology of the dg category of matrix factorization associated to the hypersurface.
In this talk, an explicit complex will be constructed to compute the Tate-Hochschild cohomology. We prove that there is a natural action of the little 2-discs operad on this complex. We will also talk about a conjecture recently proposed by Keller.

2018-11-13 Tuesday

Speaker: Mauricio Romo (YMSC)

Title: Hemisphere partition function, LG models and FJRW invariants

We consider LG orbifolds and the central charges of their B-branes (equivariant matrix factorizations) in the context of Gauged Linear Sigma Models (GLSM). We will focus on the hemisphere partition function on the GLSM extension of certain LG orbifolds, and how this provides information about their Gamma class, I/J-function and some predictions about FJRW invariants. This is joint work with J. Knapp and E. Scheidegger.

2018-10-30 Tuesday

Speaker: Gaetan Borot (MPI)

Title: Mapping class group invariants via geometric recursion

I will explain the formalism of the geometric recursion (GR) recently developed with Andersen and Orantin. Its aim is to produce, from a small amount of initial data, invariant elements in a given tower of vector spaces equipped with mapping class group representations of arbitrary surfaces, by induction on the Euler characteristic of the surfaces. I will mainly focus on GR taking values on the space of functions on the Teichmuller space. GR can be seen as a lift of the topological recursion (TR, which is related to a B-model quantization) to hyperbolic geometry. Besides, integrating over the moduli space of bordered surfaces the functions obtained by GR give quantities computed by the topological recursion.
I will give a few examples, on the TR side corresponding to Weil-Petersson volume, Gromov-Witten theory of a point, twisted volumes, and whose GR counterparts are the constant function 1, a yet not independently identified function on the moduli space, and statistics of length of multicurves. The first example was provided by the work of Mirzakhani. The last one can be seen as a new generalization of Mirzakhani-McShane identity, which arises by considering the lift of certain elements of the Givental group to the hyperbolic world.

2018-10-23 Tuesday

Speaker: Yu Qiu (YMSC)

Title: q-Stability Conditions via q-Quadratic Differentials

We first review the works of Bridgeland-Smith and Haiden-Katzarkov-Kontsevich, where they realize spaces of stability conditions on two types of Fukaya categories as moduli spaces of certain type of quadratic differentials. Then we introduce the q-deformation of stability conditions on Calabi-Yau-S categories. Finally, we realize a class of such spaces as moduli spaces of multi-valued quadratic differentials, that relates the works of BS and HKK. This is a joint work with Akishi Ikeda.

2018-10-16 Tuesday

Speaker: Shuai Guo(Peking University)

Title: BCOV's Feynman graph sum formula via NMSP

I will talk about the physics and mathematics approaches to the Gromov-Witten invariants of quintic 3-fold, and how they are related via the language of R-matrix action on CohFT. This is a joint work with H-L Chang and Jun Li.

2018-10-08 Monday

Speaker: Alfredo Najera

Title: Toric degenerations of cluster varieties, cluster duality and mirror symmetry

Cluster varieties are a special kind of lof-Calabi-Yau varieties. They come in pairs (A,X), with A and X built gluing dual tori via cluster transformations. Partial compactifications of A-varities and their toric degenerations have been studied extensively by Gross, Hacking, Keel, and Kontsevich (GHKK). These partial compactifications generalize the polytope construction of toric varieties, a construction which is recovered in the toric central fiber of the degeneration.

In this talk we introduce the notion of X-cluster varieties with coefficients. We use this notion to construct partial compactifications of an arbitrary X-cluster variety. Moreover, we use it to construct a flat degenerations of any specially completed X-cluster variety (in the sense of Fock-Goncharov) to the toric variety associated to its cluster complex. Our construction generalizes the fan construction of toric varieties. We further show that our degeneration is cluster dual to GHKK's toric degeneration of A-cluster varieties. If time permits we will outline two applications:

1) We can show that the toric degeneration of the Grassmannian Gr_k(C^n) constructed by Rietsch-Williams in 2017 coincides with GHKK's toric degeneration.

2) We can use our approach to give a precise relation of cluster duality and Batyrev-Borisov duality of Gorensteintoric Fanos in the context of mirror symmetry.

This is based on joint work with Lara Bossinger, Juan Bosco Frias Medina, and Timothy Magee, and if we have time to address the Batyrev-Borisov connection, Man-Wai Cheung as well.

2018-8-28 Tuesday

Speaker: Gao Honghao

Title: Augmentations and Sheaves for Knot Conormals

Abstract: Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the Chekanov-Eliashberg differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. Nadler-Zaslow's theorem suggests a connection between the two types of invariants. In this talk, I will manifest the correspondence explicitly.

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