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## Department of Mathematics

### Seminars and Workshops

The M-Center organizes a regular seminar, as well as the M-Center Lecture Series. The Center also supports and co-organizes workshops at other venues.The Mathematics Department has other Seminars and hosts a number of Distinguished Lectures.

### M-Seminar & Related Events

#### Fall 2018

###### The seminar meets at 3:30pm in CW 120.
• September 6, Yan Soibelman, KSU:

• Title: Fukaya categories of complex symplectic surfaces and periodic monopoles.

Abstract: I will explain why a parabolic version of the Fukaya category of the partially compactified symplectic surfaces of certain type should be related to the 3-dimensional dimensional reductions of the 4-dimensional autoduality equations.

• September 13, Timothy Logvinenko, Cardiff University:

• Title: Perverse schobers and orbifolds.

Abstract: Perverse schobers are a recent notion introduced by Kapranov and Schechtman. It is a categorification of perverse sheaves on a stratified topological space. In this talk, I will discuss perverse sheaves and perverse schobers on an orbifold disc.

• September 20, Gabriel Kerr, KSU:

• Title: Spheres in complex hypersurfaces.

Abstract: Given a hypersurface X of the complex torus, mirror symmetry predicts a quasi-equivalence between the Fukaya category F(X) of X and a certain category of graded matrix factorizations MF (W) on a toric Calabi-Yau variety. In this talk, I will describe this correspondence, as well as how it fits in the larger picture of homological mirror symmetry. Exploring the algebraic side, one finds there are many spherical objects in MF (W) which have combinatorial descriptions. Using phase tropical varieties, I will provide a prediction of the topological mirrors to these objects in F(X) and discuss some generalizations. This is based on joint work with Ilia Zharkov.

• September 27, Gabriel Kerr, KSU:

• Title: Spheres in complex hypersurfaces II.

Abstract: Given a hypersurface X of the complex torus, mirror symmetry predicts a quasi-equivalence between the Fukaya category F(X) of X and a certain category of graded matrix factorizations MF (W) on a toric Calabi-Yau variety. In this talk, I will describe this correspondence, as well as how it fits in the larger picture of homological mirror symmetry. Exploring the algebraic side, one finds there are many spherical objects in MF (W) which have combinatorial descriptions. Using phase tropical varieties, I will provide a prediction of the topological mirrors to these objects in F(X) and discuss some generalizations. This is based on joint work with Ilia Zharkov.

• October 4, Peter Koroteev, Perimeter Institute:

• Title: DAHA and Brane Quantization.

Abstract: Double affine Hecke algebras (DAHA) arise in many places in mathematics and string theory. I. Cherdnik used DAHA to prove Macdonald conjectures while knot theorists constructed homological invariants of torus knots using this algebras. It was shown by Oblomkov that A-type DAHA arises as deformation quantization of the character variety of the once-punctured torus. I shall discuss geometric representation theory of DAHA using branes in 2d A model on the Hitchin moduli space M_H. We shall arrive to a conjecture that the category of A-branes on M_H is equivalent to a representation category of DAHA.

• October 11, Qiang Wang, KSU:

• Title: Wall Crossing Structures in Seiberg-Witten Integrable Systems.

Abstract: Wall crossing structure (WCS) is a formalism that could be used to encode the Donaldson-Thomas invariants in mathematics. In this talk, we will review the notion of WCS and see how it can be induced from the data coming from a complex integrable system. In particularly, we will apply this formalism to the  Seiberg-Witten integrable systems that are relevant for the N=2 super symmetric Yang-Mills theory for gauge groups SU(2) and SU(3), which would enable us to encode the known BPS spectrum of the theories.

#### Spring 2018

• January 25, Gabriel Kerr, KSU:
• Homological mirror symmetry of birational cobordisms I

Abstract: A birational cobordism is a basic variation of GIT from a stack $X_+$ to $X_-$. On the level of derived categories, this results in a semi-orthogonal decomposition $D(X_+) = \left< D(X_- ) , \mathcal{T} \right>$ where $\mathcal{T}$ has a full exceptional collection. There is a mirror to this basic operation which yields a homological mirror to $\mathcal{T}$. This is a Landau-Ginzburg model on a higher dimensional pair-of-pants. This talk will explain this result and sketch the proof of the one dimensional case. The general case follows from dimensional induction which will be explored if time permits.

• February 1, Gabriel Kerr, KSU:
• Homological mirror symmetry of birational cobordisms II

Abstract: A birational cobordism is a basic variation of GIT from a stack $X_+$ to $X_-$. On the level of derived categories, this results in a semi-orthogonal decomposition $D(X_+) = \left< D(X_- ) , \mathcal{T} \right>$ where $\mathcal{T}$ has a full exceptional collection. There is a mirror to this basic operation which yields a homological mirror to $\mathcal{T}$. This is a Landau-Ginzburg model on a higher dimensional pair-of-pants. This talk will explain this result and sketch the proof of the one dimensional case. The general case follows from dimensional induction which will be explored if time permits.
• February 8, Rina Anno, KSU:
• Geometric Invariant Theory I
• February 15, Rina Anno, KSU:
• Geometric Invariant Theory II
• February 20, Weiwei Wu , University of Georgia:
• Spherical twists and projective twists in Fukaya categories

Seidel's Lagrangian Dehn twist exact sequence has been a cornerstone of the theory of Fukaya categories. In the last decade, Huybrechts and Thomas discovered a new autoequivalence in the derived cateogry of coherent sheaves using the so-called "projective objects", which are presumably mirrors of Lagrangian projective spaces. On the other hand, Seidel's construction of Lagrangian Dehn twists as symplectomorphisms can be easily generalized to Lagrangian projective spaces. The induce auto-equivalence on Fukaya categories are conjectured to be the mirror of Huybrechts-Thomas's auto-equivalence on B-side. This remains open until recently, and I will explain my joint work with Cheuk-Yu Mak on the solution to this conjecture using the technique of Lagrangian cobordisms. Moreover, we will explain a recent progress, again joint with Cheuk-Yu Mak, on pushing this further to Lagrangian embeddings of finite quotients of rank-one symmetric spaces, leading to another new class of auto-equivalences, which are different from the classical spherical twists only in coefficients of finite characteristics.
• February 22, Rina Anno, KSU:
• Geometric Invariant Theory III
• March 1, Benjamin Gammage, UC-Berkeley:
• Two perspectives on mirror symmetry for hypersurfaces

Abstract: Recent developments in the theory of microlocal sheaves allow us to compute symplectic invariants from the skeleton of a symplectic manifold. We will explain this in the context of affine hypersurfaces, for which we will describe two types of skeleta and derive proofs of homological mirror symmetry
• March 8, Ludmil Katzarkov , University of Miami:
• Categrical Brill Noether invariants.

Abstract: In this talk we will introduce some new categorical invariants. We connect them with some classical questions in combinatorics and geometry
• March 29, Junwu Tu, University of Missouri:
• Gromov-Witten invariants of Calabi-Yau categories
Classical mirror symmetry relates Gromov-Witten invariants in symplectic geometry to Yukawa coupling invariants in algebraic geometry. Through non-commutative Hodge theory, one can define categorical Gromov-Witten invariants associated to (Calabi-Yau A-infinity) categories. Conjecturally, this construction should reproduce the Gromov-Witten invariants and Yukawa coupling invariants, when applied Fukaya categories and Derived categories, respectively. In this talk, I describe a first computation of categorical Gromov-Witten invariants at positive genus. This is a joint work with Andrei Caldararu.
• April 12, Yu-Wei Fan, Harvard University:
• Systoles, Special Lagrangians, and Bridgeland stability conditions

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.
• April 19, Nikon Kurnosov, University of Georgia:
• Generalized Kuga-Satake construction for hyperkahler manifolds

Abstract: The classic Kuga-Satake construction gives us a Hodge structure of weight one by K3-type Hodge structure. I will talk about joint work with A.Soldatenkov and M.Verbitsky, in which we generalize Kuga-Satake construction and embed cohomology of hyperk\"ahler manifolds to cohomology of some complex torus which is morphism of Hodge structures. I will discuss also some applications of this result.
• April 26, Jamie Peabody, KSU:
• GIT fan for Mori Dream Space

Abstract: In this talk I will introduce the secondary fan of a marked polytope and the secondary fan of a closed subgroup of an algebraic torus acting on affine space. I will then use these constructions to describe how the unstable locus changes as we move between chambers in the secondary fan

### M-Center Lecture Series

• Yan Soibelman, Kansas State.Moduli Spaces of Higgs Bundles in Mathematics and Physics(slides)
• Ilia Itenberg, Strasbourg.Enumeration of real rational curves on surfaces
•  Abstract: Welschinger invariants can be seen as real analogs of genus zero Gromov-Witten invariants and are designed to bound from below the number of real rati
onal curves passing through a given generic real collection of points on a real rational surface. In some cases, these invariants can be calculated using G. Mikhalkin's approach which deals with a corresponding count of tropical curves. The general theory of Welschinger invariants is still under construction, but they gave already a number of non-trivial applications in real enumerative geometry.In these lectures, based on joint works with V. Kharlamov and E. Shustin, we discuss some basic properties of Welschinger invariants (positivity, monotonicity, asymptotic growth) obtained in the case of Del Pezzo surfaces (with $K^2\ge 3$ for the moment) using the tropical approach and real analogues of Caporaso-Harris type recursive formulas.
• Slava Kharlamov, Strasbourg.First Steps In Real Enumerative Geometry
• Abstract: Surprisingly, in several real enumerative problems the number of real solutions happens to be comparable (for example, in logarithmic scale) with the number of complex ones. Currently, such a phenomenon is best studied in the case of interpolation of real points on a real rational surface by real rational curves. The key tool here is given by the Welschinger invariants (which can be seen as a real analogue of genus zero Gromov-Witten invariants).

In this talk, based on joint works with I. Itenberg and E. Shustin, I will explain what Welschinger invariants look like in the case of surfaces, point certain modifications, and present some recursive formulas that allow to control Welschinger invariants in the case of Del Pezzo surfaces. Using this, I'll explain how to establish some basic properties of the invariants that imply, in particular, the abundance of real solutions.
• Eric Zaslow, Northwestern.Constructible Sheaves and Mirror Symmetry
• I will review some results relating the theory of constructible sheaves to the Fukaya category, and the relation to mirror symmetry for toric varieties. With this in hand, I will then discuss conjectural constructible models of the categories involved in mirror symmetry of Calabi-Yau manifolds at the large complex / large radius limit points.