Seminar on geometry, symmetry, and physics, Fall 2018

(aka mirror symmetry/related topics)

Thursdays (usually) 1:30-2:30 pm in Serin Lab E372

Organized by Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward

September 11 Laura Starkston, UCSD. Symplectic Isotopy Problems.

Abstract: We will discuss some problems and results about symplectic surfaces in 4-manifolds, particularly in the complex projective plane. The main question is to classify symplectic surfaces up to symplectic isotopy. If the surface has singularities, we restrict the isotopies to the class of surfaces with the same model singularities.

September 20 No talk

September 27 Marco Castronovo, Rutgers New Brunswick

Homological mirror symmetry for Grassmannians: rectangles

Marsh-Rietsch proposed Landau-Ginzburg mirrors for the complex Grassmannians Gr(k,n), building on Peterson's work on the quantum cohomology of flag varieties. We confirm that they satisfy homological mirror symmetry when n=p prime. The proof describes an explicit correspondence between Lagrangian branes generating the Fukaya category of Gr(k,p) and sheaves generating the category of singularities of the mirror potential. The assumption n=p forces the singularities to lie in a special cluster chart of the mirror, that we call rectangular, by an argument that combines arithmetic properties of sums of roots of unity and Stanley's hook-content formula for the number of semi-standard tableaux on a Young diagram.

October 4 Double header

1:30 Yoel Groman, Columbia

Yoel Homological mirror symmetry for semi toric SYZ fibrations

I will discuss homological mirror symmetry for a class of examples introduced by Gross. Namely, special Lagrangian torus fibrations on the complement of an anti-canonical divisor in a toric Calabi-Yau 3-fold. This includes the local model for the positive singularity in SYZ mirror symmetry. Moreover it contains non-exact examples which demonstrate phenomena arising from the combining of multiple (positive) singularities, without introducing the full blown complexities associated with scattering phenomena. The discussion will be close in spirit to but ultimately independent of the family Floer approach.

Thursday October 4, 2:30-3:30pm, NHETC common room

Speaker: Mauricio Romo

Title: Hemisphere partition function, LG models and FJRW invariant Abstract: 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.

October 11 No talk

October 18 (ctw away) No talk (but Arkani-Hamed 4pm Friday)

October 25 No talk (but 2:30 Jan Manschot, Instantons and Mock Modular Forms, physics common room)

November 1 Timothy Magee (UNAM Oaxaca)

Title: Toric degenerations of cluster varieties

Abstract:Cluster varieties are a particularly nice class of log Calabi-Yau varieties the non-compact analogue of usual Calabi-Yaus. They come in pairs (A,X), with A and X built from dual tori. The punchline of this talk will be that compactified cluster varieties are a natural progression from toric varieties. Essentially all features of toric geometry generalize to this setting in some form, and the objects studied remain simple enough to get a hold of and do calculations. Compactifications of A and their toric degenerations were studied extensively by Gross, Hacking, Keel, and Kontsevich. These compactifications generalize the polytope construction of toric varieties a construction which is recovered in the central fiber of the degeneration. Compactifications of X were introduced by Fock and Goncharov and generalize the fan construction of toric varieties. Recently, Lara Bossinger, Juan Bosco Frías Medina, Alfredo Nájera Chávez, and I introduced the notion of an X-variety with coefficients, expanded upon the notion of compactified X-varieties, and for each torus in the atlas gave a toric degeneration where each fiber is a compactified X-variety with coefficients. We showed that these fibers are stratified, and each stratum is a union of compactified X-varieties with coefficients. In the central fiber, we recover the toric variety associated to the fan in question, and we show that strata of the fibers degenerate to toric strata. This talk is based on arXiv:1809.08369 [math.AG].

November 8

November 14 (SPECIAL DAY: Wednesday) Dmitry Tonkonog, Berkeley.

Location: Serin E372
Date & time: Wednesday, 14 November 2018 at 4:30PM - 5:30PM
The standard approach to mirror symmetry asserts that the mirror of a Fano manifold is a Landau-Ginzburg model, which can be constructed by patching together the holomorphic disk potentials of suitable Lagrangian tori. I will explain how certain classical mirror symmetry predictions can be proved purely by looking at Lagrangian tori. I will focus on two examples: the expression of quantum periods in terms of period integrals, and the quantum Lefschetz formula.

November 29

December 13 Marco Suen