Stony Brook Colloquium

Schedule of Talks

September 20: Pawel Nurowsky (Stony Brook)
Differential Equations and Conformal Geometry
We discuss a few examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry. The third example exhibits the one-to-one correspondence between point equivalent classes of 2nd order ODEs and 4-dimensional conformal Fefferman-like metrics of neutral signature. The fourth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature (3,2). The Cartan normal conformal connection for these geometries is reduced to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group G2. All the examples are deeply rooted in Elie Cartan's works on exterior differential systems.

September 27: Oleg Viro (Stony Brook)
Classical and Tropical Constructions of Algebraic Curves
In topology of real algebraic varieties one needs to construct real algebraic varieties keeping track of their topological properties. I will tell about the problems, results and methods, following the evolution of the subject starting from the XIXth century. This development motivated, in particular, genesis of tropical geometry.

October 4: Micheal Thaddeus (Columbia)
Mirror Symmetry for Finite Quotients of Tori
The Strominger-Yau-Zaslow version of mirror symmetry calls for a Calabi-Yau and its mirror to be fibered by dual families of special Lagrangian tori. We will exhibit a large class of examples where such dual families exist for straightforward reasons. They are quotients of an abelian variety by a finite group acting by automorphisms and translations. The orbifold Hodge numbers can be calculated and seen to satisfy the mirror relationship. However, a complicating feature is the "turning on of the B-field": we must work with a flat U(1)-gerbe and take cohomology with the corresponding local coefficients.

October 11: Dan Silver (USA/Stony Brook)
Scottish Physics and the Origins of Modern Knot Theory
Knot theory is an active area of mathematics today. A mildly eccentric trefoil of 19th century Scottish scientists, William Thomson (a.k.a. Lord Kelvin), Peter Guthrie Tait and James Clerk Maxwell, are to blame. Using pictures, puns and poetry, we answer the question: What could they have possibly been thinking?

November 1: Zhiqin Lu (UCI)
Chern-Weil Forms on CY Moduli
We will first give a brief review of the geometry of CY moduli, including the result on the proof of the B-side of the BCOV conjecture (joint with Fang and Yoshikawa). Then we will prove that the integrals of the Chern-Weil forms on CY moduli are always rational numbers. This result follows from a more general one: the integrals of the Chern-Weil forms of the Hodge bundles on any coarse moduli spaces are rational numbers. We will also discuss the applications in string theory. The second part of the result is joint with M. Douglas.

November 29: Samuel Grushevsky (Princeton)
Integrable Equations, Secants, and Prym Varieties
Prym varieties are a special class of abelian varieties, naturally embedded in Jacobians of curves with involutions, and are perhaps the best understood abelian varieties beyond Jacobians. Many of the geometric properties of Jacobians can be generalized to Prym varieties. In particular the theta functions associated to Prym varieties give solutions to (and are characterized by) certain integrable hierarchies. In this talk we will discuss a geometric characterization of Prym varieties by their (Kummer) image having a pair of quadrisecant planes. This is the Prym version of the trisecant conjecture for Jacobians. This is joint work with Igor Krichever.

January 29: Alexei Oblomkov (Princeton)
Counting Curves in Space
A curve in space can be characterized either by its parametrization or by the set of equations defining it. The number of the parametrized curves of fixed degree and genus passing through some collection of lines and points can be computed as an integral over the completed space of the parametrized curves. This number is called a Gromov-Witten (GW) invariant of the space. Analogously, if we use the completed space of the equations to count curves we obtain a Donaldson-Thomas (DT) invariant. These two invariants are very far from being equal but the recent conjecture by Maulik, Nekrasov, Okounkov and Pandharipande gives a way to obtain GW invariants from DT invariants and vice versa. In my talk I will discuss the motivations for definitions of the invariants and describe recent progress in the proof of the conjecture which is due to Maulik, Okounkov, Pandharipande and the speaker.

January 31: Larry Guth (Stanford)
Ways to Describe the Size of a Riemannian Manifold
The two most common invariants that describe how big a Riemannian manifold is are the volume and the diameter. In this talk, I will introduce some less well-known invariants which also describe the size of a manifold. In particular, I will introduce the Uryson width. For each new size invariant, I'll discuss the following questions.
A. What are the sizes of simple examples?
B. How does this size invariant relate to other invariants? For example, how are the Uryson width and the volume related?
C. Local-to-global problems. Suppose each unit ball in a Riemannian manifold has a very small Uryson width. What does that imply about the large-scale geometry of the manifold? What if each unit ball has a very small volume? etc.
D. Isoperimetric problems. Let U be a bounded open set in Euclidean n-space. Suppose that the boundary of U has small size in terms of one of our new invariants. What does that tell us about U?

February 7: Pan Peng (Harvard)
String Duality and the Integrality Structure in Calabi-Yau Geometry
There have been a lot of marvelous results revealed by string theory, which deeply relate different aspects of mathematics. All these mysterious relations are connected by a core idea in string theory called "duality". In this talk, I will focus on a duality between Chern-Simons gauge theory and topological string theory. Based on large N Chern-Simons/topological string duality, in a series of papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants in the topological string theory. I will discuss a proof of this conjecture and its application and relation to other problems, for example, the famous volume conjecture.

February 14: Ravi Vakil (Stanford)
Murphy's Law in algebraic geometry: Badly-behaved moduli spaces
   We consider the question: ``How bad can the deformation space of an object be?'' (Alternatively: ``What singularities can appear on a moduli space?'') The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space can be arbitrarily bad.'' We show this for a number of important moduli spaces.
   More precisely, up to smooth parameters, every singularity that can be described by equations with integer coefficients appears on moduli spaces parameterizing: smooth projective surfaces (or higher-dimensional manifolds); smooth curves in projective space (the space of stable maps, or the Hilbert scheme); plane curves with nodes and cusps; stable sheaves; isolated threefold singularities; and more. The objects themselves are not pathological, and are in fact as nice as can be. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise.
   I will begin by telling you what ``moduli spaces'' and ``deformation spaces'' are. The complex-minded listener can work in the holomorphic category; the arithmetic listener can think in mixed or positive characteristic. This talk is intended to be (mostly) comprehensible to a broad audience.

February 21: B.C. Ngo (IAS)
Geometry of Orbital Integrals
According to Langlands' conjectures, automorphic forms should explain many mysterious but basic properties of the Galois representations attached to algebraic varieties. The Arthur-Selberg trace formula, which is one of the most powerful tools to study automorphic representations, displays a kind of duality between automorphic representations and orbital integrals. The conjectural basic structure of the collection of automorphic representations should derive from the trace formula and a certain elementary-looking identity involving orbital integrals called the "fundamental lemma". Orbital integrals for p-adic groups are much more concrete objects than automorphic representations, but are otherwise extremely difficult to compute. Via Grothendieck's dictionary "faisceaux-fonctions", orbital integrals can be interpreted geometrically as certain algebraic varieties called affine Springer fibers and Hitchin fibers. I will explain how these objects interplay and how their geometry controls the asymptotic behaviour of orbital integrals and ultimately yields a proof of the "fundamental lemma".

Macrh 6: David Nadler (Northwestern/IAS)
Applications of the Geometric Langlands Program
The Geometric Langlands program can be understood from the viewpoint of topological field theory. This perspective leads to many beautiful predictions about the representation theory of Lie groups. In this talk, I would like to explain the basic structures in this story assuming no specific background. Time permitting, I will describe results in derived algebraic geometry involved in establishing these predictions. This is joint work with David Ben-Zvi (Texas-Austin).

Macrh 13: Jake Rasmussen (Princeton)
Knot Homologies
Polynomial invariants of knots have been around since the work of Alexander in the 20's. Later examples include the Jones and HOMFLY polynomials. More recently, work of Khovanov-Rozansky and Ozsvath-Szabo has led to homological generalizations of these invariants. I'll discuss the relations between these "knot homologies" and their applications to topology in three and four dimensions.

April 10: Qing Han (Notre Dame)
Isometric Embeddings of Compact Surfaces in R³
In early 1950s, Nirenberg and Pogorelov independently gave an affirmative answer to the Weyl problem: Any smooth metric on S² with positive Gauss curvature admits a smooth isometric embedding in R³. This is the only result concerning isometric embeddings of compact surfaces in R³. In this talk, we will discuss isometric embeddings of other compact surfaces in R³. When Gauss curvature changes sign, fundamental equations for isometric embedding are of mixed type, elliptic when Gauss curvature is positive and hyperbolic when Gauss curvature is negative. Very little is known for global solutions of differential equations of mixed type. Our strategy to construct isometric embeddings is to solve these equations first where Gauss curvature is positive and then extend solutions to regions where Gauss curvature is negative. An important step is to ensure that a closed surface is formed. Rigidity of compact surfaces will also be discussed.

April 24: Sasha Killilov (Stony Brook)
TBA
  

May 8: Bernard Maskit (Stony Brook)
TBA
  

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