Departmental Colloquium

We will begin by giving an elementary introduction to the GL(2,R) action on the Hodge bundle (sometimes called Teichmuller dynamics), after which we will give a survey of some of new developments in this field. This will include restrictions on the structure of orbit closures echoing Ratner's Theorems on homogeneous spaces, the discovery by Moller and Filip that orbit closures can be defined purely in terms of algebraic geometry, and new examples of sub-varieties of the Hodge bundle which provide counterexamples to a conjecture of Mirzakhani. The talk will include joint work with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

The integral cohomology groups of a complex algebraic variety are one of the most fundamental invariants associated to the variety. The ranks of these groups are well understood in terms of the equations defining the variety, thanks to Hodge theory. However, the torsion tends to be much more "transcendental" in nature and not easily accessible via algebraic techniques. Torsion cohomology classes have recently played a pivotal role in fundamental advances in many subjects such as number theory, algebraic geometry, and representation theory, so it is important to better understand torsion from an algebraic perspective. In my talk, I'll explain how to bound the torsion explicitly in terms of the equations defining the variety. The bound is a consequence of the construction of a new cohomology theory in p-adic Hodge theory, and the bulk of my talk will be dedicated to explaining why the coefficient ring of this cohomology theory (i.e., its value on a point) makes meaningful mathematical sense of a small piece of the non-existent object "Z tensor Z over F_1". This talk is based on joint work with Matthew Morrow and Peter Scholze.

Normaliz started in the late nineties as a tool for the normalization of affine monoids. Meanwhile it has grown into a versatile package for computations in discrete convex geometry: it computes lattice points in polyhedra, or, from a different perspective, solves linear diophantine systems of inequalities, equations and congruences. This scope has given it a variety of applications, expected and unexpected ones. We will explain the computation goals of Normaliz and some of its algorithms.

Hyperkahler (HK) manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. Compact HK manifolds are one of the building blocks for algebraic varieties with trivial first Chern class and their role in algebraic geometry has grown immensely over the last 20 year. In this talk I will give an overview of the theory of compact HK manifolds and then focus on some of my work, including a recent joint work with R. Laza and C. Voisin.

TBA

Traveling waves appear throughout biology and are often driven by local spatial coupling, e.g., diffusion. The past ten years have revealed traveling waves in cells. These waves can be pulses of biochemical factors (diffusing proteins or metabolites) and also mechanical factors (such as the cell cortex). One example of mechanical traveling wave is offered by cellular blebs, pressure-driven bubbles on the cell surface implicated in cell division, apoptosis and cell motility. Blebs exhibit a range of behaviors including contracting in place, travel around the cell’s periphery, or repeated blebbing, making them biophysically interesting. Mechanical traveling waves are naturally modeled using non-local integro-PDEs, which lack the theoretical tools available for reaction-diffusion waves, obfuscating simple questions such as what determines if a bleb will travel or not, and, if it travels, what determines its velocity? We present results in two parts: First, we develop a simple model of the cell surface describing the membrane, cortex, and adhesions, including the slow timescale cortical healing (treating implicitly the fast timescale of fluid motion). We find traveling and stationary blebs, which we characterize through numerical simulation. In the second part, we review the so-called Maxwell condition for reaction-diffusion systems that determines whether an excitation will travel or recover in place. We present our progress in deriving an analogue of the Maxwell condition for non-local integro-PDEs suitable for our cell surface model. This condition allows the theoretical (simulation-free) elucidation of this dynamic biological phenomenon. Spring 2017