Monday September 11, 2017 1:00 PM 5127
 Charlie Cifarelli, Stony Brook University
Introduction to CalabiYau manifolds IIThis is intended to be a second introductory talk to the general theory of CalabiYau manifolds. We will begin by giving an introduction to the Calabi conjecture and Yau's solution thereof, focusing on the existence of Ricciflat metrics for manifolds with vanishing first Chern class. We will then discuss some interesting properties of CalabiYau 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 5127
 Erik Gallegos Baņos, Stony Brook University
Introduction to deformations of complex manifoldsToday, 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 5127
 JeanFrancois Arbour, Stony Brook University
The BogomolovTianTodorov theoremAfter 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 CalabiYau manifold.

Monday October 02, 2017 1:00 PM  2:00 PM 5127
 Hang Yuan, Stony Brook University
Introduction to GromovWitten invariantsWe will begin by introducing the ideas of GromovWitten 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 pseudoholomorphic 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 3d1 generic points.

Monday October 09, 2017 1:00 PM  2:00 PM 5127
 Yuhan Sun, Stony Brook University
GromovWitten invariants and the (1,1) Yukawa couplingI will calculate basic GromovWitten invariants and explain why they are related to curvecounting. Also I will introduce the (1,1)Yukawa coupling.

Monday October 16, 2017 1:00 PM  2:00 PM 5127
 Michael Albanese, Stony Brook University
Variations of Hodge StructuresWe 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 GaussManin 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 CalabiYau threefolds, we will define the associated (1, 2)Yukawa coupling and period map. The latter will lead to a local Torelli theorem for CalabiYau threefolds.

Monday October 30, 2017 1:00 PM 5127
 Yoonjoo Kim, Stony Brook University
Degenerations of Hodge StructuresTBA

Monday November 06, 2017 1:00 PM 5127
 John Sheridan, Stony Brook University
The Yukawa couplings, canonical coordinates, and the mirror mapIn 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 CalabiYau 3folds.

Monday November 13, 2017 1:00 PM 5127
 Marlon de Oliveira Gomes, Stony Brook University
A mirror to the quintic threefoldI'll begin by reviewing the topology of a smooth quintic threefold, and discussing the (1,1) Yukawa coupling. Next I'll introduce a 1parameter family of singular CalabiYau 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 CalabiYau threefolds whose Hodge diamond mirrors that of the quintic.

Monday November 27, 2017 1:00 PM 5127
 Francois Greer, Stony Brook University
Yukawa (1,2)coupling on the quintic mirrorIn this final talk of the semester, we will finish our study of the quintic 3fold and its CalabiYau 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 3fold.

