RTG Seminar

from Thursday
June 01, 2017 to Sunday
December 31, 2017
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Instructions for subscribing to Stony Brook Math Department Calendars

Monday
September 11, 2017

1:00 PM
5-127
Charlie Cifarelli, Stony Brook University
Introduction to Calabi-Yau manifolds II

This is intended to be a second introductory talk to the general theory of Calabi-Yau manifolds. We will begin by giving an introduction to the Calabi conjecture and Yau's solution thereof, focusing on the existence of Ricci-flat metrics for manifolds with vanishing first Chern class. We will then discuss some interesting properties of Calabi-Yau manifolds, and time permitting we will introduce some more examples. As with the previous talk, only a familiarity with Chapter 0 of Griffiths and Harris will be assumed at this stage.


Monday
September 18, 2017

1:00 PM
5-127
Erik Gallegos Baņos, Stony Brook University
Introduction to deformations of complex manifolds

Today, we will introduce the idea of deforming an almost complex structure on a smooth manifold. We will then give a condition for when the deformed almost complex structure is integrable and discuss a guiding example. Depending on time, we will discuss the theorem of Kodaira, Nirenberg and Spencer.


Monday
September 25, 2017

1:00 PM
5-127
Jean-Francois Arbour, Stony Brook University
The Bogomolov-Tian-Todorov theorem

After quickly reviewing the basics of deformation theory, I will discuss the possible obstruction to extending a first order deformation. I will then give Tian's proof of the fact that there is no such obstruction in the case of a Calabi-Yau manifold.


Monday
October 02, 2017

1:00 PM - 2:00 PM
5-127
Hang Yuan, Stony Brook University
Introduction to Gromov-Witten invariants

We will begin by introducing the ideas of Gromov-Witten invariants and related moduli spaces heuristically; and then assuming good properties of moduli, we give their preliminary definitions (via both algebraic geometry and symplectic geometry). Next we provide further details of the construction of moduli spaces (of pseudo-holomorphic curves) and thereby the rigorous definition of GW invariants in symplectic side. If time permits, we will list a number of axioms which are satisfied by both AG and SG GW invariants, and use them, for example, to obtain a recursion formula for N_d, the number of rational curves of degree d in CP^2 passing through 3d-1 generic points.


Monday
October 09, 2017

1:00 PM - 2:00 PM
5-127
Yuhan Sun, Stony Brook University
Gromov-Witten invariants and the (1,1) Yukawa coupling

I will calculate basic Gromov-Witten invariants and explain why they are related to curve-counting. Also I will introduce the (1,1)-Yukawa coupling.


Monday
October 16, 2017

1:00 PM - 2:00 PM
5-127
Michael Albanese, Stony Brook University
Variations of Hodge Structures

We begin by introducing Hodge structures and Hodge filtrations. In order to motivate the definition of a variation of Hodge structures, we then consider a local system associated to a family of complex manifolds (given by a higher direct image sheaf). There is a corresponding holomorphic vector bundle equipped with a connection known as the Gauss-Manin connection. The compatibility of the connection with the Hodge filtration, known as Griffiths transversality, is the final ingredient needed for the abstract definition of a variation of Hodge structures.

Given a family of Calabi-Yau threefolds, we will define the associated (1, 2)-Yukawa coupling and period map. The latter will lead to a local Torelli theorem for Calabi-Yau threefolds.


Monday
October 30, 2017

1:00 PM
5-127
Yoonjoo Kim, Stony Brook University
Degenerations of Hodge Structures

TBA


Monday
November 06, 2017

1:00 PM
5-127
John Sheridan, Stony Brook University
The Yukawa couplings, canonical coordinates, and the mirror map

In this talk, we will aim to introduce the remaining structure needed to state the mirror symmetry conjecture (with at least enough precision for the upcoming example of the quintic threefold). We will first recall the definition of the (1,2)-Yukawa coupling as well as the nilpotent orbit theorem. The latter allows for good approximation of the period map, and in particular of the canonical coordinates near a large complex structure limit point (both to be defined), of a family of Calabi-Yau 3-folds.


Monday
November 13, 2017

1:00 PM
5-127
Marlon de Oliveira Gomes, Stony Brook University
A mirror to the quintic threefold

I'll begin by reviewing the topology of a smooth quintic threefold, and discussing the (1,1) Yukawa coupling. Next I'll introduce a 1-parameter family of singular Calabi-Yau threefolds obtained from a family of quintics, and study the resolution of singularities. This will lead to our candidates for mirror symmetry: a family of smooth Calabi-Yau threefolds whose Hodge diamond mirrors that of the quintic.


Monday
November 27, 2017

1:00 PM
5-127
Francois Greer, Stony Brook University
Yukawa (1,2)-coupling on the quintic mirror

In this final talk of the semester, we will finish our study of the quintic 3-fold and its Calabi-Yau mirror by computing the Yukawa (1,2)-coupling of the latter and using this to predict (via the mirror conjecture) some counts of rational curves on the generic quintic 3-fold.


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars