Mathematics Colloquium Winter 2008

January 10, 2008

The tautological classes of the moduli spaces of stable maps

Dr. Dragos Oprea
Stanford University
Special Colloquium Speaker

We will define a system of tautological classes on the moduli spaces of stable maps to smooth projective varieties. The intersection theory of these classes give the Gromov-Witten invariants. I will explain that the cohomology of the moduli spaces of genus 0 stable maps to type A flag varieties is generated by the tautological classes.

January 15, 2008

Complex projective structures and applications

Dr. David Dumas
Brown University
Special Colloquium Speaker

We will discuss the theory of complex projective structures on surfaces (which are also known as Mobius structures) and applications of this theory to related areas of low-dimensional geometry and complex analysis. In particular we will explain connections with classical Teichmuller theory, 3-dimensional hyperbolic geometry and Kleinian groups, and how these connections enable new approaches to some analytic and geometric problems.

January 17, 2008

Heegaard Splittings and Hyperbolic Geometry

Dr. Hossein Namazi
Princeton University
Special Colloquium Speaker

The relationship between the geometry and topology 3-manifolds has been the focus of lots of research in the area of low dimensional topology. In principle, this is supported by Mostow rigidity theorem and other rigidity theorems that followed. More importantly this became a central theme after Thurston's phenomenal work which showed how topological and geometric properties interact and one can be used in the understanding of the other one. We introduce a new approach where combinatorics of a Heegaard splitting can be used to construct a geometric structure and then the geometric structure can be used to answer many topological questions about the 3-manifold. In particular in joint work with Brock, Minsky and Souto we use this approach in many cases to construct combinatorial models for the geometry of hyperbolic closed 3-manifolds.

January 24, 2008

"Say X x A^1 = A^n. Please solve for X" ... and related questions

Dr. Brent Doran
Institute for Advanced Study - Princeton University
Special Colloquium Speaker

Starting with the problem of the title -- in algebraic geometry, this is known as the Zariski Cancellation problem -- we will show how it touches upon a number of deep problems in mathematics. Classification of contractible manifolds (and an historic mis-proof of the Poincare conjecture), non-reductive group actions and quotients, classical invariant theory, Hilbert's 14th problem, algebraic vector bundles, counting rational curves, the extent to which detailed algebraic geometry can be captured by new techniques from algebraic topology ... all of these emerge naturally, and at heart are rather simple to explain, at least conceptually. Indeed, the basic object of study is quite friendly: free additive group actions on affine space. Pretty examples abound.

January 25, 2008

A p-adic approach to Hilbert's 12th problem

Dr. Samit Dasgupta
Harvard University
Special Colloquium Speaker

It is well known that the square root of any integer can be written as a linear combination of roots of unity. A generalization of this fact is the "Kronecker-Weber Theorem", which states that in fact any element which generates an abelian Galois extension of the field of rational numbers Q can also be written as such a linear combination. The roots of unity may by viewed as the special values of the analytic function e(x) = exp(2*pi*i*x) where x is taken to be a rational number. Broadly speaking, Hilbert's 12th problem is to find an analogous result when Q is replaced by a general algebraic number field F, and in particular to find the analytic functions which play the role of e(x) in this general setting. Hilbert's 12th problem has been solved in the case where F is an imaginary quadratic field, with the role of e(x) being played by certain modular forms. All other cases are, generally speaking, unresolved. In this talk I will discuss the case where F is a real quadratic field, and more generally, a totally real field. I will describe relevant conjectures of Stark and Gross, as well as current work using a p-adic approach and methods of Shintani. A proof of these conjectures would arguably provide a positive resolution of Hilbert's 12th problem in these cases.

January 29, 2008

Exotic 4-manifolds with small Euler characteristics

Dr. Anar Akhmedov
Georgia Institute of Technology
Special Colloquium Speaker

It is known that many simply connected, smooth topological 4-manifolds admitinfinitely many smooth structures. The smaller the Euler characteristic, the harder it is to construct exotic smooth structure. In this talk we present examples of symplectic 4-manifolds with same integral cohomology as S^2 x S^2. We also discuss the generalization of these examples to #{2n-1}(S^2 x S^2) for n > 1. As an application of these symplectic building blocks, we construct exotic smooth structure on small 4- manifolds such as CP^2#k(-CP^2) for k = 2, 3, 4, 5 and 3CP^2#l(-CP^2) for l = 4, 5, 6, 7. We will also discuss an interesting applications to the geography of minimal symplectic 4-manifolds.

January 31, 2008

Curves, abelian varieties, and the moduli of cubic threefolds

Dr. Sebastian Casalaina-Martin
Harvard University
Special Colloquium Speaker

A result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this talk I will discuss the possible degenerations of these abelian varieties, and give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra will also be considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves. This is joint work with Radu Laza.

February 12, 2008

From combinatorics to geometry for knots and 3-manifolds

Dr. David Futer
Michigan State University
Special Colloquium Speaker

Powerful theorems of Thurston, Perelman, and Mostow tell us that almost every 3-manifold admits a hyperbolic metric, and that this metric is unique. Thus, in principle, there is a 1-to-1 correspondence between a combinatorial description of a 3-manifold and its geometry. On the other hand, a concrete dictionary between combinatorial features and geometric measurements has been much harder to obtain. I will survey some recent results that explicitly relate the combinatorics of a knot diagram to geometric features of the 3-manifolds obtained by surgery on the knot. There are also interesting connections to the behavior of surfaces and the Jones polynomial of the knot.

February 19, 2008

On the McKay Conjecture

Dr. Gabriel Navarro
Mathematical Sciences Research Institute

The McKay conjecture is one of the main problems in the representation theory of finite groups. We will introduce the audience to the conjecture, to several of its recent refinements, and we will explain how we think it is eventually going to be proved.

March 4, 2008

Fusion Systems

Dr. Marcus Linckelmann
Mathematical Sciences Research Institute

The p-local structure of a finite group G consists of information including the structure of a Sylow-p-subgroup P of G and the way P is embedded in G. This was formalised in the early 1990's by L. Puig,leading to the notion of fusion systems. As it turned out, not all fusion systems need to arise from finite groups. Nonetheless, many algebraic and topological invariants of finite groups can be associated more generally with fusion systems: we have classifying spaces and numbers of irreducible characters of finite groups which may not exist...