Departmental Colloquium

The notion of simplicial volume was introduced by Gromov and Thurston in the early 1980's, and measures how efficiently a manifold can be "triangulated over the reals". I'll explain various results that follow from positivity of this invariant. I will also outline the proof (joint with B. Schmidt) of the following conjecture of Gromov: every closed locally symmetric space of non-compact type has positive simplicial volume.

Given a system of polynomial equations in several variables and in 1 parameter, does there exist a "rational solution", i.e., a family of solutions which is a rational function (fraction of polynomials) in the parameter? Do there exist enough rational solutions to approximate every power series solution in the parameter to arbtirary order? The first problem, or rather the problem of answering the first problem, is Hilbert's 10th problem for $\mathbb{C}(t)$. It is expected there is no algorithm to answer the first problem. The second problem, the "Weak Approximation Problem", conjecturally has a very simple answer: there are enough rational solutions precisely if after substituting a general value for the parameter, the corresponding system is "rationally connected", i.e., every pair of solutions are common members of a family of solutions which are the output of a rational function. I will discuss the topological and number theoretic motivation of this conjecture, the evidence for the conjecture due to Hassett -- Tschinkel, Hassett, Knecht and Colliot-Th\'el\`ene -- Gille, and a new approach of Mike Roth and myself putting this conjecture in the larger context of "algebro-geometric analogues of topological obstruction theory".

In this talk we develop the notion of generalized vacillating tableaux from molecular contact structures. We prove a bijection between tangled diagrams (tangles) and vacillating tableaux, allowing for exact and asymptotic enumeration. We show how tangles connect the concepts of partitions and enhanced partitions, which are shown to be in bijection with a specific subclass of tangles. We then show how tangles embed in a generalization of the Brauer algebra and analyze the structure of the latter.

We survey recent advances toward basic algorithmic questions in real algebraic geometry. In particular, we consider the two questions and their algorithmic complexity: (1) Does a system of sparse polynomial equations have a real root? (2) What is the topology of a real algebraic set defined by a set of sparse polynomial equations? A special case of Question (2) is counting real roots, and even here optimal upper bounds are still an open question. So we also review some concrete examples. Along the way, we will see an unusual connection to the famous Masser-Oesterle ABC-Conjecture, and an extension of Smale's 17th Problem (on approximating complex roots of polynomial systems). We assume no background in complexity theory or algebraic geometry.

A complex projective variety is a subset of complex projective space defined by a set of homogeneous polynomials. In this talk I will discuss recent results that describe the geometry of these varieties. The case of varieties of dimension 1, also known as Riemann surfaces, is classical. The geometry of smooth varieties of dimension 2 was understood by the Italian school of Algebraic Geometry at the beginning of the 20-th century. The Minimal Model Program is an attempt to generalize these results to higher dimension. The 3 dimensional case was understood in the 1980's by celebrated work of Mori and others. In this talk I will discuss joint work with Birkar, Cascini and McKernan towards extending the Minimal Model Program to arbitrary dimension. In particular I will discuss the following: Theorem: The canonical ring of any smooth projective algebraic variety is finitely generated. (Note that this Theorem was independently proven by Y.-T. Siu.)

Nonstandard analysis is a relatively new area of mathematics in which infinitesimal numbers can be defined and manipulated rigorously like real numbers. To demonstrate the power of the subject, the problem of shock wave jump conditions is studied for a one-dimensional compressible gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. To use conservation laws, pre-distributions of the Dirac delta measure are applied whose supports are contained within the shock thickness. Furthermore, piecewise differentiable pre-distributions of the Heaviside function are applied which vary from zero to one across the shock wave. It is shown that if the equations of motion are expressed in non-conservative form then the relationships between the jump functions for the flow parameters may be found unambiguously. The analysis yields the classical Rankine-Hugoniot jump conditions for an inviscid shock wave. Examples developed include a normal shock wave and a radially symmetric shock wave.