Mirror Symmetry Seminar

The Mirror Symmetry seminar series will be held Fridays from 11:00am-12:00pm in Science Center 530.

The schedule will be updated below as details are confirmed.

Date……… Speaker…………….. Title/Abstract
1-26-2018 Matthew Young

Chinese University of Hong Kong

Algebra and geometry of orientifold Donaldson-Thomas theory

Abstract: This talk will be an overview of the orientifold Donaldson-Thomas theory of quivers. Roughly speaking, orientifold Donaldson-Thomas theory is a counting theory for principal orthogonal or symplectic bundles in three dimensional Calabi-Yau categories. I will explain an approach via geometrically defined representations of cohomological Hall algebras. In this way we are able to categorify the orientifold wall-crossing formula appearing in the string theory literature, formulate and prove an orientifold variant of the integrality conjecture of Kontsevich-Soibelman and give a geometric interpretation of orientifold Donaldson-Thomas invariants

2-2-2018 Minxian Zhu

Yau Mathematical Sciences Center, Tsinghua University

On the hyperplane conjecture for periods of Calabi-Yau hypersurfaces in P^n

Abstract: Hosono, Lian, and Yau made a conjecture in the 90s describing the solutions to the Gelfand-Kapranov-Zelevinsky hypergeometric equations which arise as periods of CY hypersurfaces in a Gorenstein Fano toric variety. We will prove this conjecture for projective spaces. This is joint work with Bong Lian.

2-9-2018 Dan Xie

CMSA/Harvard

Three dimensional mirror symmetry

Abstract: I will discuss the basic features of three dimensional mirror symmetry. Several interesting class of examples will be discussed.

2-16-2018 Man-Wai Cheung

Harvard

Quiver representations and theta functions

Abstract: Scattering diagrams theta functions and broken lines were developed in order to describe toric degenerations of Calabi-Yau varieties and construct mirror pairs. Later, Gross-Hacking-Keel-Kontsevich unravel the relation of those objects with cluster algebras. In the talk, we will discuss how we can combine the representation theory with these objects. We will also see how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting.

2-23-2018 Jingyu Zhao

Brandeis/Harvard

Connection on S^1-equivariant Floer theory

Abstract: We will discuss Seidel’s construction of Getzler’s connection on S^1-equivariant Floer cohomology on the closed string A-model.

3-2-2018 Chuck Doran

University of Alberta and ICERM

Mirror Symmetry for Lattice Polarized del Pezzo Surfaces

Abstract: We describe a notion of lattice polarization for rational elliptic surfaces and weak del Pezzo surfaces, and describe the complex moduli of the former and the Kähler cone of the latter. We then propose a version of mirror symmetry relating these two objects, which should be thought of as a form of Fano-LG correspondence. Finally, we relate this notion to other forms of mirror symmetry, including Dolgachev-Nikulin-Pinkham mirror symmetry for lattice polarized K3 surfaces and the Gross-Siebert program. This is joint work with Alan Thompson based on arXiv:1709.00856.

3-9-2018 Shinobu Hosono

Gakushuin University

Gluing monodromy nilpotent cones of a family of K3 surfaces

Abstract: I consider a K3 surface which is known as Cayley model of Reye congruences, and construct its mirror family. This mirror family turns out to have three degeneration points (LCSLs), for each of which we can define monodromy nilpotent cones. I will show that these nilpotent cones are naturally glued together to make a larger cone which can be identified with the ample cone of the Cayley model.

3-9-2018

*9:30-11am*

Guangbo Xu

Princeton University

Open quantum Kirwan map

Abstract: (Joint work with Chris Woodward) Consider a Lagrangian submanifold $\bar L$ in a GIT quotient $\bar X = X//G$. Besides the usual Fukaya $A_\infty$ algebra $Fuk(\bar L)$ defined by counting holomorphic disks, another version, called the quasimap Fukaya algebra $Fuk^K(L)$, is defined by counting holomorphic disks in $X$ modulo group action. Motivated from the closed string quantum Kirwan map studied by Ziltener and Woodward, as well as the work of Fukaya–Oh–Ohta–Ono, Chan–Lau–Leung–Tseng, we construct an open string version of the quantum Kirwan map. This is an $A_\infty$ morphism from $Fuk^K(L)$ to a bulk deformation of $Fuk(\bar L)$. The deformation term is defined by counting affine vortices (point-like instantons) in the gauged sigma model, while the $A_\infty$ morphism is defined by counting point-like instantons with Lagrangian boundary condition. It has several useful consequences. For example, the weakly unobstructedness of $Fuk^K(L)$ (which is easy to check) implies the weak unobstructedness of $Fuk(\bar L)$ after bulk deformation (which is hard to check). It recovers the “open mirror theorem” of Chan–Lau–Leung–Tseng for semi-Fano toric manifolds which says the Lagrangian Floer potential coincides with the Hori–Vafa potential after a coordinate change, and extends to general toric manifolds.

3-30-2018 Yu-Wei Fan

(Harvard)

Systoles, Special Lagrangians, and Bridgeland stability conditions

Abstract: Motivated by Loewner’s torus inequality which relates the least length of a non-contractible loop (systole) on a torus to its volume, one can ask whether there is a higher-dimensional analogue, with torus replaced by Calabi-Yau manifolds and loops replaced by special Lagrangian submanifolds.

We attempt to answer this question in the case of mirror quartic K3 surfaces via mirror symmetry and Bridgeland stability conditions. We define the notion of categorical systole and systolic ratio of a Bridgeland stability condition, and show that the aforementioned question is related to a purely lattice-theoretic problem. We prove that the answer to the lattice-theoretic problem is finite and give an explicit upper bound.

4-13-2018 Fabian Haiden

(Harvard)

Title: Geometric flows, iterated logarithms, and balanced weight filtrations

Abstract: The long term behavior of certain gradient flows, both finite and infinite dimensional, is governed by an algebraically defined balanced weight-type filtration which provides a canonical refinement of the Harder-Narasimhan filtration. In recent joint work with Katzarkov, Kontsevich, and Pandit ( arXiv:1802.04123 ) we consider in detail the case of the Yang-Mills flow on the space of metrics on a holomorphic bundle over a Riemann surface, as well as a modified curve shortening flow on the cylinder. I will discuss these results and conjectural generalizations.

4-20-2018

9:30-10:45am

Xiaomeng Xu

(MIT)

Stokes phenomenon, quantum groups and 2d topological field theory

Abstract: This talk will include a general introduction to differential equations with singularities, and its relation with symplectic geometry and representation theory. In particular, we will focus on the Stokes phenomenon of linear systems of ordinary differential equations, and construct the braiding of Drinfeld-Jimbo quantum groups as quantum Stokes matrices. We then introduce an isomonodromic deformation of Knizhnik–Zamolodchikov type equations, and relate its classical limit to the theory of Frobenius manifolds.

4-20-2018

10:45-12:00pm

Alan Thompson

(Cambridge)

Threefolds fibred by K3 surfaces and mirror symmetry
Abstract: I will present a number of results about the classification of threefolds fibred by K3 surfaces. These will start out very general but rapidly specialize, first to the case where the K3 surface fibres admit a certain class of lattice polarization, then further to the case where the threefold total space is Calabi-Yau. In the process, we will begin to notice links between this classification and the classification of Fano threefolds. This turns out not to be a coincidence, but a manifestation of mirror symmetry, relating Calabi-Yau threefolds fibred by K3 surfaces to Calabi-Yau threefolds constructed by gluing together Fanos and smoothing the result. This is joint work with C. Doran, A. Harder, R. Kooistra, and A. Novoseltsev.

 

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