Title: Ramanujan Conjectures, Density Hypotheses and Applications for Arithmetic Groups.
Abstract: The Generalized Ramanujan Conjecture (GRC) for GL(n) is a central open problem in modern number theory. Its resolution is known to yield applications in many fields, such as: Diophantine approximation and arithmetic groups. For instance, Deligne's proof of the Ramanujan-Petersson conjecture for GL(2) was a key ingredient in the work of Lubotzky, Phillips and Sarnak on Ramanujan graphs.
Title: The Grothendieck--Serre conjecture for classical groups in low dimensions
A famous conjecture of Grothendieck and Serre predicts that if G is a reductive group scheme over a semilocal regular domain R and X is a G-torsor, then X has a point over the fraction field of R if and only if it has an R-point. Many instances of the conjecture have been established over the years. Most notably, Panin and Fedorov--Panin proved the conjecture when R contains a field.
I will explain how Feynman diagrams arise in pure algebra: how the computation of compositions of maps of a certain natural class, from one polynomial ring into another, naturally leads to a certain composition operation of quadratics and to Feynman diagrams.
I will also explain, with very little detail, how this is used in the construction of some very well-behaved poly-time computable knot polynomials, and then with better detail, why I care about having such invariants.
Mirror symmetry relates the algebraic and symplectic geometry of spaces which are related by dualizing a Lagrangian torus fibration. From the perspective of representation theory this is particularly interesting, with ties to geometric Langlands duality, in cases where the spaces are hyperkähler, and the Lagrangian tori are actually holomorphic Lagrangian. Such spaces, which arise as moduli spaces of four-dimensional field theories, include character varieties, multiplicative quiver varieties, and the "K-theoretic Coulomb branches" of Braverman-Finkelberg-Nakajima.
In joint work with K. Zhang we construct some explicit canonical geometries on various classes of complex manifolds, following a general symmetry principle pioneered by Calabi in the 70's. Our focus is to allow edge type singularities (that are the natural higher-dimensional analogues of conical Riemann surfaces studied by Picard and others since the 19th century) and study Gromov-Hausdorff limits as the angle in the cone tends to zero.
Title: Cup products oncurves over finite fields
Abstract: This is joint work with Ted Chinburg.
Let C be a smooth projective curve over a finite field k, and
let l be a prime number different from the characteristic of k.
In this talk I will discuss triple cup products on the first etale
cohomology group of C with coefficients in the constant
sheaf of l-th roots of unity. These cup products are important
for finding explicit descriptions of the l-adic completion of the
etale fundamental group of C and also for cryptographic
Physicists have observed in the '80s that Calabi-Yau manifolds come in pairs so that quantum cohomology on the one is related to period integrals on the other. This phenomenon, known as mirror symmetry, has since evolved into a deeper understanding that symplectic geometry on a manifold is typically encoded in the complex geometry of another, its mirror. I will discuss in some simple examples of how the relation arises naturally from the study of Hamiltonian Floer cohomology associated with invariant sets of an integrable system.
Physicists have observed in the '80s that Calabi-Yau manifolds come in pairs so that quantum cohomology on the one is related to period integrals on the other. This phenomenon, known as mirror symmetry, has since evolved into a deeper understanding that symplectic geometry on a manifold is typically encoded in the complex geometry of another, its mirror. I will discuss in some simple examples how the relation arises naturally from the study of Hamiltonian Floer cohomology associated to invariant sets of an integrable system.