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Homological Mirror Symmetry Seminar
September 15, 2016 @ 2:00 pm  4:00 pm
The seminar series, Homological Mirror Symmetry, will be held on selected Thursdays from 2PM – 4pm in CMSA Building, 20 Garden Street, Room G10.
The list of speakers is below and will be updated as details are confirmed.
Date  Name  Title 
091516  
092216  Netanel Blaier, Brandeis  “Intro to HMS.”
Abstract: This is the first talk of the seminar series. We survey the statement of Homological Mirror Symmetry (introduced by Kontsevich in 1994) and some known results, as well as briefly discussing its importance, and the connection to other formulations of Mirror Symmetry and the SYZ conjecture. Following that, we will begin to review the definition of the Aside (namely, the Fukaya category) in some depth. No background is assumed! Also, in the last half hour, we will divide papers and topics among participants. 
092916  Netanel Blaier, Brandeis  “Intro to HMS 2.”
Abstract: In the second talk, we review (some) of the nittygritty details needed to construct a Fukaya categories. This include basic Floer theory, the analytic properties of Jholomorphic curves and cylinders, Gromov compactness and its relation to metric topology on the compactified moduli space, and Banach setup and perturbation schemes commonly used in geometric regularization. We then proceed to recall the notion of an operad, Fukaya’s differentiable correspondences, and how to perform the previous constructions coherently in order to obtain $A_\infty$structures. We will try to demonstrate all concepts in the Morse theory ‘toy model’. 
100616 
Hansol Hong, CMSA

Title: Homological mirror symmetry for elliptic curves
Abstract: Notes Notes (Baris) 
101316 
YuWei Fan, Harvard 
Title: Semiflat mirror symmetry and FourierMukai transform Abstract: We will review the semiflat mirror symmetry setting in StromingerYauZaslow, and discuss the correspondence between special Lagrangian sections on the Aside and deformed HermitianYangMills connections on the Bside using real FourierMukai transform, following LeungYauZaslow. 
102016 
Tim Large, MIT 
Title: “Symplectic cohomology and wrapped Fukaya categories”
Abstract: While mirror symmetry was originally conjectured for compact manifolds, the phenomenon applies to noncompact manifolds as well. In the setting of Liouville domains, a class of open symplectic manifolds including affine varieties, cotangent bundles and Stein manifolds, there is an Ainfinity category called the wrapped Fukaya category, which is easier to define and often more amenable to computation than the original Fukaya category. In this talk I will construct it, along with symplectic cohomology (its closedstring counterpart), and compute some examples. We will then discuss how compactifying a symplectic manifold corresponds, on the Bside of mirror symmetry, to turning on a LandauGinzburg potential. 
102716 
Philip Engel, Columbia 
Title: Mirror symmetry in the complement of an anticanonical divisor” According to the SYZ conjecture, the mirror of a CalabiYau variety can be constructed by dualizing the fibers of a special Lagrangian fibration. Following Auroux, we consider this rubric for an open CalabiYau variety XD given as the complement of a normal crossings anticanonical divisor D in X. In this talk, we first define the moduli space of special Lagrangian submanfiolds L with a flat U(1) connection in XD, and note that it locally has the structure of a CalabiYau variety. The Fukaya category of such Lagrangians is obstructed, and the degree 0 part of the obstruction on L defines a holomorphic function on the mirror. This “superpotential” depends on counts of holomorphic discs of Maslov index 2 bounded by L. We then restrict to the surface case, where there are codimension 1 “walls” consisting of Lagrangians which bound a disc of Maslov index 0. We examine how the superpotential changes when crossing a wall and discuss how one ought to “quantum correct” the complex structure on the moduli space to undo the discontinuity introduced by these discs. 
110316 
Yusuf Baris Kartal, MIT 
I will present AurouxKatzarkovOrlov’s proof of one side of the homological mirror symmetry for Del Pezzo surfaces. Namely I will prove their derived categories are equivalent to the categories of vanishing cycles for certain LGmodels together with Bfields. I plan to show how the general Bfield corresponds to noncommutative Del Pezzo surfaces and time allowing may mention HMS for simple degenerations of Del Pezzo surfaces. The tools include exceptional collections( and mutations for degenerate case), explicit description of NC deformations, etc. 
111016  No seminar this week  
120816 
Lino Amorim, Boston University 
Title: The Fukaya category of a compact toric manifold Abstract: In this talk I will discuss the Fukaya category of a toric manifold following the work of FukayaOhOhtaOno. I will start with an overview of the general structure of the Fukaya category of a compact symplectic manifold. Then I will consider toric manifolds in particular the Fano case and construct its mirror. 