Thermodynamical approach to the Markoff-Hurwitz equation

Michael Magee
Yale University
April 19, 2017
I'll first introduce the Markoff-Hurwitz equation and explain how it plays a fundamental role in different areas of mathematics. The main result I'll discuss is a true asymptotic formula for the number of real points in a fixed orbit of the automorphism group of the Markoff-Hurwitz variety with bounded maximal entry. In particular this establishes an asymptotic count for the number of integer solutions to the Markoff-Hurwitz equation of bounded height.

Bounds on roots of polynomials (and applications)

Adam Marcus
Princeton University; von Neumann Fellow, School of Mathematics
April 18, 2017
I will discuss methods for deriving bounds on the roots of polynomials, and how one can use such bounds to assert the existence of combinatorial structures with certain spectral properties. This will include introducing the "method of interlacing polynomials" and showing how one can use it prove the existence of Ramanujan graphs. Lastly, I will show how one can interpret these methods as a finite version of the previous week's results.

Efficient empirical revenue maximization in single-parameter auction environments

Yannai Gonczarowski
Hebrew University of Jerusalem and Microsoft Research
April 17, 2017
We present a polynomial-time algorithm that, given samples from the unknown valuation distribution of each bidder, learns an auction that approximately maximizes the auctioneer's revenue in a variety of single-parameter auction environments including matroid environments, position environments, and the public project environment. The valuation distributions may be arbitrary bounded distributions (in particular, they may be irregular, and may differ for the various bidders), thus resolving a problem left open by previous papers.

Mirror symmetry for moduli of flat bundles and non-abelian Hodge theory

Tony Pantev
University of Pennsylvania
April 14, 2017
I will discuss a construction of the homological mirror correspondence on algebraic integrable systems arising as moduli of flat bundles on curves. The focus will be on non-abelian Hodge theory as a tool for implementing hyper Kaehler rotations of objects in the Fukaya category. I will discuss in detail a specific example of the construction building automorphic sheaves on the moduli space of rank two bundles on the projective line with parabolic structure at five points. This is a joint work with Ron Donagi.

Congruences between motives and congruences between values of $L$-functions

Olivier Fouquet
Université Paris-Sud
April 13, 2017
If two motives are congruent, is it the case that the special values of their respective $L$-functions are congruent? More precisely, can the formula predicting special values of motivic $L$-functions be interpolated in $p$-adic families of motives? I will explain how the formalism of the Weight-Monodromy filtration for $p$-adic families of Galois representations sheds light on this question (and suggests a perhaps surprising answer).

Algebraic proofs of degenerations of Hodge-de Rham complexes

Andrei Căldăraru
University of Wisconsin, Madison
April 12, 2017

In the first half of the talk I shall present a new algebraic proof of a result of Deligne-Illusie about the degeneration of the Hodge-de Rham spectral sequence. The idea is to reduce the main technical point of their proof to a question about the formality of a derived intersection in an Azumaya space.

Soliton resolution for energy critical wave and wave map equations

Hao Jia
Member, School of Mathematics
April 12, 2017
It is widely believed that the generic dynamics of nonlinear dispersive equations in the whole space is described by solitary waves and linear dispersions. More precisely, over large times, solutions tend to de-couple into solitary waves plus radiation. It remains an open problem to rigorously establish such a description for most dispersive equations. For energy critical wave equations in the radial case, we have better understanding, using tools such as ``channel of energy inequalities" firstly introduced by Duyckaerts-Kenig-Merle, and monotonicity formulae.

Mirror symmetry for moduli of flat bundles and non-abelian Hodge theory

Tony Pantev
University of Pennsylvania
April 12, 2017
I will discuss a construction of the homological mirror correspondence on algebraic integrable systems arising as moduli of flat bundles on curves. The focus will be on non-abelian Hodge theory as a tool for implementing hyper Kaehler rotations of objects in the Fukaya category. I will discuss in detail a specific example of the construction building automorphic sheaves on the moduli space of rank two bundles on the projective line with parabolic structure at five points. This is a joint work with Ron Donagi.