School of Mathematics

Extensions of the Gross-Zagier formula

Kartik Prasanna
University of Michigan
April 23, 2015
I will first discuss the general conjectural picture relating algebraic cycles to L-functions and some extensions of the Gross-Zagier formula involving $p$-adic L-functions. This leads naturally to the question of constructing algebraic cycles corresponding to the vanishing of certain Rankin-Selberg L-functions at the center of symmetry. Finally, I will outline some new constructions of such cycles, based on work in progress with A. Ichino.

A topological approach for investigating the intrinsic structure of neural activity

Vladimir Itskov
Pennsylvania State University
April 18, 2015
Experimental neuroscience is achieving rapid progress in the ability to collect neural activity and connectivity data. Detecting meaningful structure in this data is challenging because the measured quantities are related to more "fundamental" variables by an unknown nonlinear transformation. We find that combinatorial topology can be used to obtain meaningful answers to questions about the structure of neural activity and introduce an approach that extracts features of the data invariant under arbitrary nonlinear monotone transformations.

Entanglement of embedded graphs

Toen Castle
University of Pennsylvania
April 18, 2015
Even simple graphs can be embedded in space ($\mathbb E^3$ or $\mathbb S^3$) in a topologically complex way. If there is a cycle in the graph then there can be knots in the embedding, if there are disjoint cycles then there can be links. However there are also other entanglement modes known as 'ravels', which contain neither knots nor links. Potentially familiar examples of ravels include Thurston's 'tripus' and Kinoshita's embedded theta graph.

Characterizing force-chain network architecture in granular materials

Danielle Bassett
University of Pennsylvania
April 18, 2015
Force chains form heterogeneous physical structures that can constrain the mechanical stability and acoustic transmission of granular media. However, despite their relevance for predicting bulk properties of materials, there is no agreement on a quantitative description of force chains. Consequently, it is difficult to compare the force-chain structures in different materials or experimental conditions and to quantify their impact on materials properties.

Unlinked fixed points of Hamiltonian diffeomorphisms and a dynamical construction of spectral invariants

Sobhan Seyfaddini
Massachusetts Institute of Technology
April 17, 2015
Hamiltonian spectral invariants have had many interesting and important applications in symplectic geometry. Inspired by Le Calvez's theory of transverse foliations for dynamical systems of surfaces, we introduce a new dynamical invariant, denoted by $N$, for Hamiltonians on surfaces (except the sphere). We prove that, on the set of autonomous Hamiltonians, this invariant coincides with the classical spectral invariant. This is joint work with Vincent Humilière and Frédéric Le Roux.

Syzygies, gonality and symmetric products of curves

Robert Lazarsfeld
Stony Brook University
April 14, 2015
In the mid 1980s, Mark Green and I conjectured that one could read off the gonality of an algebraic curve $C$ from the syzygies among the equations defining any one sufficiently positive embedding of $C$. Ein and I recently noticed that a small variant of the ideas used by Voisin in her work on canonical curves leads to a quick proof of this gonality conjecture. The proof involves the geometry of certain vector bundles on the symmetric product of $C$.

Factorization of birational maps on steroids

Dan Abramovich
Brown University
April 14, 2015
Searching literature you will find the following statement (I'm paraphrasing): "If $X_1,X_2$ are nonsingular schemes proper over a complete DVR $R$ with residue characteristic 0, and $\phi: X_1 \to X_2$ is birational, then $\phi$ can be factored as a sequence of blowups and blowdown between nonsingular schemes proper over $R$, with nonsingular blowup centers." along with a demonstration: "The method of [Włodarczyk] or [AKMW] works word-for-word." In revenge you will find elsewhere (I'm paraphrasing): "Since a proof of weak factorization of birational maps over a complete D

Embedding the derived category of a curve into a Fano variety

Alexander Kuznetsov
Steklov Mathematical Institute, Moscow
April 14, 2015
According to the conjecture of Bondal, the derived category of coherent sheaves on any smooth projective variety can be embedded as a semiorthogonal component into the derived category of a Fano variety of higher dimension. I will explain how this embedding can be constructed for general curves of arbitrary genus. This is a joint work with Anton Fonarev.

Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

János Kollár
Princeton University
April 13, 2015
We consider elliptic curves whose coefficients are degree 2 polynomials in a variable $t$. We prove that for infinitely many values of $t$ the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. The main step of the proof is to show that such conic bundles are unirational. (joint work with M. Mella)

A new approach to the sensitivity conjecture

Michael Saks
Rutgers University
April 13, 2015
The sensitivity conjecture is a major outstanding foundational problems about boolean functions is the sensitivity conjecture. In one of its many forms, it asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that interpolates the function) is bounded above by some fixed power of its sensitivity (which is the maximum vertex degree of the graph defined on the inputs where two inputs are adjacent if they differ in exactly one coordinate and their function values are different).