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.

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.

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.

A new potential theory for the Maxwell equations

Leslie Greengard
New York University
April 18, 2015
Existing formulations of Maxwell's equations encounter numerical difficulties in geometries with sub-wavelength features and/or non-trivial genus. We will describe a new system of boundary value problems for the electromagnetic vector and scalar potentials that permits stable and robust solutions of scattering problems for all frequencies. This is joint work with Felipe Vico, Miguel Ferrando and Zydrunas Gimbutas.

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.

Sensors, sampling, and scale selection: a homological approach

Don Sheehy
University of Connecticut
April 18, 2015
In their seminal work on homological sensor networks, de Silva and Ghrist showed the surprising fact that its possible to certify the coverage of a coordinate free sensor network even with very minimal knowledge of the space to be covered. We give a new, simpler proof of the de Silva-Ghrist Topological Coverage Criterion that eliminates any assumptions about the smoothness of the boundary of the underlying space, allowing the results to be applied to much more general problems. The new proof factors the geometric, topological, and combinatorial aspects of this approach.

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.

Heavy Element Synthesis in the Universe

Enrico Ramirez-Ruiz
University of California, Santa Cruz
April 28, 2015

The source of about half of the heaviest elements in the Universe has been a mystery for a long time. Although the general picture of element formation is well understood, many questions about the nuclear physics processes and particularly the astrophysical details remain to be answered. Here I focus on advances in our understanding of the origin of the heaviest and rarest elements in the Universe.