The structure of instability in moduli theory

Daniel Halpern-Leistner
Member, School of Mathematics
October 21, 2014
In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of a point in the stack. To such a degeneration one can often associate a real number which measures "how destabilizing" it is, and in these situations one can ask the question of whether there is a "maximal destabilizing" or "canonically destabilizing" degeneration of a given unstable point.

Extending differential forms and the Lipman-Zariski conjecture

Sándor Kovács
University of Washington; Member, School of Mathematics
October 22, 2014
The Lipman-Zariski conjecture states that if the tangent sheaf of a complex variety is locally free then the variety is smooth. In joint work with Patrick Graf we prove that this holds whenever an extension theorem for differential 1-forms holds, in particular if the variety in question has log canonical singularities.

An algebro-geometric theory of vector-valued modular forms of half-integral weight

Luca Candelori
Lousiana State University
October 23, 2014
We give a geometric theory of vector-valued modular forms attached to Weil representations of rank 1 lattices. More specifically, we construct vector bundles over the moduli stack of elliptic curves, whose sections over the complex numbers correspond to vector-valued modular forms attached to rank 1 lattices. The key idea is to construct vector bundles of Schrodinger representations and line bundles of half-forms over appropriate `metaplectic stacks' and then show that their tensor products descend to the moduli stack of elliptic curves.

Symplectic embeddings from concave toric domains into convex ones

Dan Cristofaro-Gardiner
Harvard University
October 24, 2014
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. These obstructions are known to be sharp in several interesting cases, for example for symplectic embeddings of one ellipsoid into another. We explain why ECH capacities give a sharp obstruction to embedding any "concave toric domain" into a "convex" one. We also explain why the ECH capacities of any concave or convex toric domain are determined by the ECH capacities of a corresponding collection of balls.

Beyond ECH capacities

Michael Hutchings
University of California, Berkeley
October 24, 2014
ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain is a "concave toric domain" and the target is a "convex toric domain" (see previous talk). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk.

Apery, irrationality proofs and dinner parties

Francis Brown
Centre national de la recherche scientifique, Institut des Hautes Études Scientifiques
October 27, 2014
After introducing an elementary criterion for a real number to be irrational, I will discuss Apery's famous result proving the irrationality of \(\zeta(3)\). Then I will give an overview of subsequent results in this field, and finally propose a simple geometric interpretation based on a classical dinner party game. The talk is intended for a general mathematical audience.

Exponential separation of information and communication

Gillat Kol
Member, School of Mathematics
October 28, 2014
In profoundly influential works, Shannon and Huffman show that if Alice wants to send a message \(X\) to Bob, it's sufficient for her to send roughly \(H(X)\) bits (in expectation), where \(H\) denotes Shannon's entropy function. In other words, the message \(x\) can be compressed to roughly \(H(X)\) bits, the information content of the message. Can one prove similar results in the interactive setting, where Alice and Bob engage in an interactive communication protocol?