09-12-2016 Mathematical Physics Seminar
2016-09-12 17:10 (America/New_York) | MA

Homological Mirror Symmetry Seminar
2016-09-15 14:00 - 16:00
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

09-15-16

09-22-16
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 A-side (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.
Lecture Slides

09-29-16
Netanel Blaier, Brandeis
“Intro to HMS 2.”
Abstract: In the second talk, we review (some) of the nitty-gritty details needed to construct a Fukaya categories. This include basic Floer theory, the analytic properties of J-holomorphic 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’.
Lecture Slides

10-06-16

Hansol Hong, CMSA

Title: Homological mirror symmetry for elliptic curves
Abstract:
We survey the proof of homological mirror symmetry by Polishchuk and Zaslow. Some of more recent methods to prove HMS for elliptic curves will be discussed also,
which use homological algebra techniques and formal deformation theory of Lagrangians etc.
Notes
Notes (Baris)

10-13-16

Yu-Wei Fan, Harvard

Title: Semi-flat mirror symmetry and Fourier-Mukai transform
Abstract: We will review the semi-flat mirror symmetry setting in Strominger-Yau-Zaslow, and discuss the correspondence between special Lagrangian sections on the A-side and deformed Hermitian-Yang-Mills connections on the B-side using real Fourier-Mukai transform, following Leung-Yau-Zaslow.

10-20-16

Tim Large, MIT

Title: “Symplectic cohomology and wrapped Fukaya categories”
Abstract: While mirror symmetry was originally conjectured for compact manifolds, the phenomenon applies to non-compact 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 A-infinity 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 closed-string counterpart), and compute some examples. We will then discuss how compactifying a symplectic manifold corresponds, on the B-side of mirror symmetry, to turning on a Landau-Ginzburg potential.
Notes

10-27-16

Philip Engel, Columbia

Title: Mirror symmetry in the complement of an anticanonical divisor”
According to the SYZ conjecture, the mirror of a Calabi-Yau variety can be constructed by dualizing the fibers of a special Lagrangian fibration. Following Auroux, we consider this rubric for an open Calabi-Yau variety X-D 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 X-D, and note that it locally has the structure of a Calabi-Yau 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.
Notes

11-03-16

Yusuf Baris Kartal, MIT

HMS for Del Pezzo surfaces
I will present Auroux-Katzarkov-Orlov’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 LG-models together with B-fields. I plan to show how the general B-field corresponds to non-commutative 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.

11-10-16
No seminar this week

12-08-16

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 Fukaya-Oh-Ohta-Ono. 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.
Video

09-19-2016 Mathematical Physics Seminar
2016-09-19 17:11

09-21-2016 Random Matrix & Probability Theory Seminar
2016-09-21 17:13

09-22-2016 Homological Mirror Symmetry Seminar
2016-09-22 14:00 - 16:00
References:

D. Auroux, A beginner’s introduction to Fukaya categories. arXiv:1301.7056
I. Smith, A symplectic prolegomenon. arXiv:1401.0269
D. Auroux, “Topics in geometry: mirror symmetry”, Fall 2009 (MIT Math 18.969)
Nick Sheridan’s IAS and Jussieu lectures.
Sheel Gantara “Topics in symplectic topology”, Spring 2016 (Stanford Math 257B)

09-26-16 Mathematical Physics Seminar
2016-09-26 17:32

09-28-2016 Random Matrix & Probability Theory Seminar
2016-09-28 17:35

09-29-2016 Homological Mirror Symmetry Seminar
2016-09-29 17:48

10-03-16 Mathematical Physics Seminar
2016-10-03 18:13

10-05-2016 Random Matrix & Probability Theory Seminar
2016-10-05 17:41

10-05-2016 Homological Mirror Symmetry Seminar
2016-10-05 18:17

Duality String Seminar, Thursdays
2016-10-11 16:15 - 2016-10-12 18:00
The Duality String Seminar is held every Thursday at 4:15pm in Jefferson Lab, 453.
For details, please visit the website.
* The Duality String Seminar is sponsored by the Center of Mathematical Sciences and Applications’ Cheng Yu-Tong Fund, for Research at the Interface of Mathematics and Physics.

10-12-2016 Random Matrix & Probability Theory Seminar
2016-10-12 18:16

10-13-2016 Homological Mirror Symmetry Seminar
2016-10-13 18:19

10-14-16 CMSA Members’ Seminar
2016-10-14 18:24

10-17-16 Mathematical Physics Seminar
2016-10-17 18:26

10-18, 10-25, 11-01-16 CMSA Special Seminar Series, Tuesdays
2016-10-18 18:11 - 2016-10-25 18:12

10-19-2016 Random Matrix & Probability Theory Seminar
2016-10-19 18:22

10-20-2016 Homological Mirror Symmetry Seminar
2016-10-20 14:00 - 16:00
References:

P. Seidel, “Biased view of symplectic cohomology” https://arxiv.org/abs/0704.2055
D. Auroux, “Beginners introduction to Fukaya categories” https://arxiv.org/pdf/1301.7056v1.pdf
D. Auroux, “Mirror symmetry in complement of anti-canonical divisor” https://arxiv.org/abs/0706.3207
Auroux’s talks at 2016 IMJ-PRG summer school, notes taken by Kevin Sackel, available http://school2016.imj-prg.fr/abstract.html

10-24-2016 Random Matrix & Probability Theory Seminar
2016-10-24 18:18

10-24-16 Mathematical Physics Seminar
2016-10-24 18:30

10-26-2016 Random Matrix & Probability Theory Seminar
2016-10-26 18:27

10-28-16 CMSA Special Seminar
2016-10-28 18:34

10-31-16 Mathematical Physics Seminar
2016-10-31 19:05

11-01-2016 Social Sciences Applications Forum
2016-11-01 18:36

11-02-2016 Random Matrix & Probability Theory Seminar
2016-11-02 10:32

11-02-16 Special Seminar (Science Center)
2016-11-02 10:36

11-03-2016 Homological Mirror Symmetry Seminar
2016-11-03 10:43

11-04-16 Special Seminar
2016-11-04 10:42

11-07-16 Mathematical Physics Seminar
2016-11-07 10:38
