Events for the week of May 12 2008

Abstract. I hope to explain to the non-expert why the ideas of Iwasawa theory, especially the interplay between arithmetic problems, the complex world, and the p-adic world, can be used to study a variety of theoretical and computational problems in number theory, which seem otherwise inaccessible.

Abstract. The Iwasawa algebra $L(G)$ of a compact p-adic Lie group $G$ seems to be one of the most intersting examples of a ring which is non-commutative once $G$ is non-commutative. When $G$ admits a closed normal subgroup $H$ such that $G/H$ is isomorphic to $\bf Z_p$ the additive group of p-adic numbers), I will explain how $L(G)$ admits a canonical Ore set $S$, and how localization with respect to $S$ enables one to give a vast generalization of the 'main conjectures' of classical commutative Iwasawa theory.

Abstract. Microorganisms swimming in viscous fluids inhabit a world quite different from the one we are used to experiencing. In this talk, we will discuss some properties and recent results of fluid-based locomotion on very small scales in fluids and geometries where nonlinear behavior naturally arises. We will first discuss experimental and theoretical results on the hydrodynamic attraction of swimming cells by solid surfaces. We will then present theoretical work on locomotion in viscoelastic fluids.

Abstract. The asymptotic relationship between mean-field theory and actual d-dimensional spin-systems is investigated from a (very traditional) thermodynamic perspective. Results are derived for all d ≥ 2 but even for d = 2, (and to a certain extent d = 1) it is proved that mean-field theory will provide an asymptotic guideline to the phase structure of actual systems. In particular, for attractive pair interactions that are sufficiently "spead out" it is shown that the energy, free energy and, in particular, the block magnetization (as defined on scales that are large compared with the lattice spacing but small compared to the range of the interaction) will only take on values near to those predicted by the associated mean-field theory. While this applies for systems in all dimensions, the significant applications are for d = 2 where it is shown: (a) If the mean-field theory has a discontinuous phase transition featuring the breaking of a discrete symmetry than this sort of transition will occur in the actual system. Prominent examples include the two?dimensional q = 3 state Potts model. (b) If the mean-field theory has a discontinuous transition accompanied by the breaking of a continuous symmetry, the thermodynamic discontinuity is preserved even if the symmetry breaking is forbidden in the actual system. E.g. the two-dimensional O(3) nematic liquid crystal. Additional results include layered systems, phase transitions that are not accompainied by the breaking of any symmetry and the analysis of triple points.

Abstract. In this talk we treat the following question: given a fixed t, how many monochromatic copies of t-cliques must one find in any two-colouring of the edges of complete graph on n vertices (for n large)? This is an old question of Erdos, and he proved bounds that essentially mirror the known bounds for Ramsey′s theorem. In particular, for the upper bound, he showed that one has at least n^t / r(t)^t ≥ n^t / 4^(t^2) monochromatic t-cliques. <br><br> Our main result is a large improvement on this lower bound, increasing it to n^t / C^(t^2), where C ≈ 2.18 is an explicitly defined constant. The proof involves the construction of a recursion which we believe to be the correct analogue, for multiplicites, of the Erdos-Szekeres proof of Ramsey′s theorem. The solution of this recursion is, however, markedly more complicated than that of its counterpart.

Abstract. I will give an exposition of my recent work with Andrei Rapinchuk in which we have introduced a new notion of 'weak commensurability' of Zariski-dense subgroups. Weak commensurability of arithmetic subgroups of semi-simple Lie groups turns out to have very strong consequences. Weak commensurability is intimately related to the commensurability of the set of lengths of closed geodesics on, and isospectrality of, locally symmetric spaces of finite volume (and with nonpositive sectional curvatures). Using our results we are able to answer Marc Kac’s famous question 'Can one hear the shape of a drum?' for compact arithmetic locally symmetric spaces. Our proofs use algebraic number theory, class field theory, and also some results and conjectures from transcendental number theory.

Abstract. TV-L1 can be used as a functional to regularize images subject to a fidelity term. It can also be used to extract image information on different scales which can give rise to image signatures. Where TV-L1 is defined, it is equivalent to the flat norm. The flat norm however can generalize TV-L1 to more situations. Some implementation and applications will be discussed.

Abstract. I’ll explain the invariants of the title and show how they can be used to classify the Bernoulli shifts over a sofic group up to measure-conjugacy. (Sofic groups include all linear groups). This answers a question of Ornstein and Weiss. The new invariants are similar to the classical Kolmogorov-Sinai entropy. Together with orbit equivalence (OE) and von Neumann equivalence (vNE) results due to Sorin Popa, this immediately yields new classification results for these shifts up to OE and vNE for large classes of groups.

Abstract. We will review some of the results known about this classical problem in mathematical biology. The long-time asymptotics of this model of cell motility due to chemotaxis will be analysed in the critical case. Its connection to free energies and the logarithmic HLS inequality will lead to the proof of infinite time aggregation in the critical case. A similar problem with nonlinear diffusion exhibiting analogous behavior will be studied in any dimension. Its analysis is reminiscent of the techniques used for blow-up in nonlinear Schrodinger equations.

Abstract. Formalizations of arithmetic ordinarily proceed along one of the three avenues: (1) as a first-order theory in which numbers are taken as primitive and their properties laid down by means of particular extra-logical axioms; (2) as a second-order theory in the manner of Frege and Russell, in which numbers are naturally identified with classes of (extensions of) equinumerous concepts, and their properties derived from such a characterization; or (3) as a first-order theory embedded within a set-theoretic apparatus, in the manner of Zermelo or von Neumann, selecting particular representatives for the equivalence classes (with respect to equinumerosity). None of these options is completely satisfying in every respect. Option (1) leaves the nature of number unexplained, and option (3) does not seem general enough, in that the connection of numbers with their cardinal properties (i.e., ultimately, with the act of counting) is lost. Option (2) is perhaps the most conceptually satisfying, in that the characterization of numbers as equivalence classes is well-motivated and completely general. Unfortunately, option (2) also takes us into the hostile landscape of second-order logic. In this talk we show how to overcome these shortcomings. After looking at abstraction principles in general, and with the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction, we propose a formalization of arithmetic at the first order, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier and the abstraction operator.