School of Mathematics

The four-color theorem and an instanton invariant for spatial graphs II

Tomasz Mrowka
Massachusetts Institute of Technology
October 13, 2015
Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional $\mathbf{Z}/2$ vector space. The main result about the instanton homology is a non-vanishing theorem, proved using techniques from 3-dimensional topology: if the graph is bridgeless, its instanton homology is non-zero.

Bernoulli convolutions for algebraic parameters

Péter Varjú
University of Cambridge
May 8, 2015
The Bernoulli convolution with parameter $\lambda$ is the law of the random variable: $\sum X_i \lambda^i$, where $X_i$ are independent unbiased $+1/-1$ valued random variables. If $\lambda \lambda > 1/2$, the question whether the Bernoulli convolution is singular or a.c. is a very interesting open problem. Recent papers of Hochman and Shmerkin prove that the set of $\lambda$'s such that the measure is singular is of Hausdorff dimension 0. I will discuss the problem for parameters $\lambda$ that are algebraic. Work in progress, joint with Emmanuel Breuillard.

Reductions of Galois representations of small slopes

Eknath Ghate
Tata Institute of Fundamental Research
May 7, 2015
We investigate the shape of the reduction of certain crystalline Galois representations of integral slope 1 and of fractional slopes in (1,2). The proof uses the compatibility between the p-adic and mod p Local Langlands Correspondences with respect to the process of reduction. We give an essentially complete description of the reduction, and show that in the former case it is generically reducible, whereas in the latter case it is generically irreducible. This is joint work with Shalini Bhattacharya for slopes in (1,2), and also with Sandra Rozensztajn for slope 1.

Derived categories of cyclic covers and their branch divisors

Alexander Perry
Harvard University
April 29, 2015
Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, I will discuss an interesting relation between the derived categories of a cyclic cover of $Y$ and its branch divisor. As examples, I will describe the cases of cyclic cubic fourfolds and "special Gushel--Mukai varieties," where we get a description of the equivariant category of certain K3 categories. This is joint work with Alexander Kuznetsov.

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 $C$.