I'll continue a broad introduction to the subject of integrable systems of PDE.
I'll discuss the construction of an infinite sequence of commuting flows and local conservation laws for the KdV equation, and if time permits I will also Backlund transformations, which can be used to construct multi-soliton solutions.
Integrable PDE, part 2
Nov. 19, 2013 1pm (MATH 220)
Grad Algebra/Logic
Charlie Scherer (CU Boulder)
X
Whitehead's problem is a question about abelian groups that was motivated by complex analysis via homology and eventually resolved by set theory. I use the word "resolved" rather than "solved" because it was shown that Whitehead's problem is undecidable using ordinary mathematics (ZFC). I want to discuss Whitehead's problem as an example of how Jensen's diamond principle from set theory can be applied in algebra. This week I will state Whitehead's problem and give the solution for countable groups.
Diamonds are a Group's Best Friend, Part 1
Nov. 19, 2013 1pm (MATH 350)
Number Theory
Mehmet Kiral (Brown University)
X
Given a modular form , the corresponding -function satisfies a functional equation as . One can view bounds of the form as improvements towards the case, the Lindelof Hypothesis. The functional equation together with the Phragmen Lindelof theorem give the bound for . This is called the convexity bound, and any result with is called the subconvexity result. The Lindelof Hypothesis is a consequence of the Riemann Hypothesis for and thus are expected to be true for all L-functions with a functional equation and an Euler product. We prove a subconvexity bound for in the conductor aspect where f is a half integral weight modular form. In this case the L function does not posess an Euler product.
Subconvexity bounds for half Integral weight modular forms
Nov. 19, 2013 3pm (MATH 350)
Algebraic Geometry
Matthew Satriano (U. Michigan)
X
We will discuss a technique which allows one to approximate singular varieties by smooth spaces called stacks. As an application, we will address the following question, as well as some generalizations: given a linear action of a group G on complex n-space C^n, when is the quotient C^n/G a singular variety? We will also mention some applications to Hodge theory and to derived equivalences.
Stacky Resolutions of Singularities Sponsored by the Meyer Fund
Nov. 19, 2013 3pm (MATH 220)
PDE/Analysis
Ryan Rosenbaum
X
Weyl's law describes the asymptotic distribution of the eigenvalues of the Laplacian on a compact Riemann manifold M with Dirichlet boundary condition. In the case that M is a torus, this quickly reduces to the classical problem of counting lattice points in a ball. We will prove Weyl's law in this case using a trace formula argument which generalizes considerably. Time permitting, we will sketch the proof of Weyl's law in a more general setting.
Integrable PDE, part 2