For a Lie groupoid G with a twisting σ (a PU(H)-principal bundle over G), we use (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of the Connes-Skandalis longitudinal index theorem.

Ever wonder how you can safely send your credit card number over the internet? The answer is RSA, the first widely used public-key cryptographic communications system. Using only elementary techniques from number theory, RSA allows you to send secure communications over public channels without a pre-arranged code. In this talk, we discuss the difference between public-key and private-key cryptography, and cover some basic ideas from number theory. Then we will show how to use RSA to encode and decode messages, and explain why this process works and why it is so difficult to crack.

I will talk about the asymptotic geometry of Teichmuller space equipped with the Weil-Petersson metric. In particular, I will give a criterion for determining when two points in the asymptotic cone of Teichmuller space can be separated by a point; motivated by a similar characterization in mapping class groups by Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by Behrstock-Charney. As a corollary, I will explain a new way to uniquely characterize the Teichmuller space of the genus two once punctured surface amongst all Teichmuller space in that it has a divergence function which is superquadratic yet subexponential.

The problem of finding integer solutions to Diophantine equations is one that has fascinated mathematicians for thousands of years. Although we now know (thanks to the work of Davis, Matiyasevich, Punam and Robinson resolving Hilbert's 10th problem in the negative) that it is impossible to do this in general, it ought to be possible to say a great deal in special cases. For example, when the equation in question defines an elliptic curve, a remarkable conjecture due to Birch and Swinnerton-Dyer implies that the behaviour of the solutions is governed by the properties of an analytic object (whose very existence is a deep problem in and of itself), namely the L-function attached to the elliptic curve. In this talk, I shall explain some of the ideas that go into the formulation of the Birch and Swinnerton-Dyer conjecture, and I shall discuss some aspects of what is currently known about the conjecture. I shall not assume any previous knowledge of this topic. Tea at 3:30 pm in SC 1425.

Positive solutions of nonlinear parabolic problems can have very complex behavior. However, assuming certain symmetry  conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is `stable'. More specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. We discuss problems on bounded and unbounded domains and, by possibly suprising counterexamples, we show optimality of our  assumptions. As an application, we formulate new results on convergence of solutions to a single equilibrium.

This will be a colloquium-style talk giving a survey of certain classical results concerning the Galois structure of rings of integers. Starting with an elementary result in Galois theory (the normal basis theorem), one is led to a remarkable connection between the Galois module structure of rings on integers on the one hand, and the behaviour of certain analytical objects called Artin L-functions. I shall discuss this connection and how it fits into a much broader picture than one might have at first suspected.

March 12-16. The aim of the workshop is to bring together specialists in various areas of mathematics and Computer Science studying large networks: from Cayley graphs of groups to social networks and robotics. For details see http://www.math.vanderbilt.edu/~msapir/galn/main.html

For many quantitative applications of Floer theories, one is required to compute the homology with respect to some filtration and in practice this can be difficult.  In this talk I will outline a strategy for turning certain filtered Hamiltonian Floer homology computations into contact homology computations. The proof of this strategy requires a general compactness theorem, which includes `stretching the neck' for Hamiltonian Floer trajectories, and generalizations of Bourgeois--Oancea's work relating symplectic homology with contact homology.  This is joint work in progress with Y. Eliashberg and L. Polterovich, and is part of a larger project with L. Diogo and S. Lisi.

This is first part of my talk on my recent joint work with Guoliang Yu on problems of positive scalar curvature on manifolds through noncommutative geometric methods. Suppose $M$ is a complete Riemannian manifold with positive scalar curvature toward infinity. Let $M_\Gamma$ be a Galois covering of $M$ by a discrete group $\Gamma$. Then one can define a higher index class $\ind(D_\Gamma)\in K_0(C_r^\ast(\Gamma))$ of the Dirac operator $D_\Gamma$ over $M_\Gamma$. Now suppose $M_0$ and $M_1$ are two manifolds with positive scalar curvature towards infinity and moreover their $\Gamma$-coverings coincide towards infinity. Let $M_2$ be the manifold obtained by gluing $M_0$ and $M_1$ along some ends. We prove the following relative higher index theorem: \[  \ind(D_2) = \ind(D_0) - \ind(D_1)  \in K_0(C_r^\ast(\Gamma) ),\] where $D_i$ is the Dirac operator on $(M_i)_\Gamma$.

The American Heritage Dictionary defines gerrymandering as the act of “dividing a geographic area into voting districts so as to give unfair advantage to one party.” The problem of gerrymandering has led to the development of several mathematical measures of shape compactness, some of which have been used in court cases to argue for or against the legality of congressional redistricting plans. In this talk, we will show how the notion of convexity can be used to detect irregularly shaped districts. We will explore both theoretical and empirical aspects of this convexity-based measure of shape compactness.

Quasi-morphisms on the group of Hamiltonian diffeomorphisms are a convenient way to package and see various rigidity phenomenon in symplectic topology. Their general construction due to Entov and Polterovich uses Hamiltonian Floer homology and requires a certain computation in the quantum homology ring. In this talk I will explain how symplectic reduction provides a sort of geometric functoriality that allows quasi-morphisms to descend along symplectic reductions without further quantum homology computations. The proof involves a lemma about pulling quasi-morphisms back along `quasi-homomorphisms. As an application, I will present a family of closed symplectic manifolds whose group of Hamiltonian diffeomorphisms have infinite dimensional spaces of quasi-morphisms.

Symplectic geometry is the study of symplectic manifolds endowed with a closed nondegenerate 2-form. Unlike Riemannian manifolds, symplectic manifolds have no local invariants. Instead, they feature a rich global theory which is commonly referred to as "symplectic topology". The main idea that I wish to convey in this talk is that spaces of free loops play a fundamental role in symplectic topology. On the one hand, algebraic invariants of symplectic manifolds are often extensions to free loop spaces of invariants familiar from differential topology. On the other hand, this is a manifestation of the close and somewhat mysterious relationship between symplectic rigidity phenomena and dynamical properties of Hamiltonian systems. Tea at 3:30 pm in SC 1425.

(Feb. 17-18) The goal of this mini-workshop is to explore connections between Hamiltonian Dynamics and Symplectic Topology, focusing on recent developments in these areas. For more information, see the workshop web site http://www.math.vanderbilt.edu/~gurelzb/Shanks2012/ws.html

I will present some of my work on reduced cohomology and property (T) for totally disconnected groups and dense countable subgroups. The primary application of this work is to show property (T) for a class of totally disconnected groups arising from a conjecture of Margulis and Zimmer regarding the classification of all commensurated subgroups of lattices in higher-rank Lie groups. The key idea in our work is to expand on Kleiner's work on the energy of a cocycle (the idea of which goes back to Mok) and derive a very general result about energy and reduced cohomology. This is joint work with Yehuda Shalom.

There is evidence that cancer develops when cells acquire a sequence of mutations that alter normal cell characteristics. This sequence determines a hierarchy among the cells, based on how many more mutations they need to accumulate in order to become cancerous. When cells divide, they exhibit telomere loss and differentiate, which defines another cell hierarchy, on top of which is the stem cell. We propose a mutation-generation model, which combines the mutation-accumulation hierarchy with the differentiation hierarchy of the cells, allowing us to take a step further in examining cancer development and growth. The results of the model support the hypothesis of the cancer stem cell’s role in cancer pathogenesis: a very small fraction of the cancer cell population is responsible for the cancer growth and development. Also, according to the model, the nature of mutation accumulation is sufficient to explain the faster growth of the cancer cell population. However, numerical results show that in order for a cancer to develop within a reasonable time frame, cancer cells need to exhibit a higher proliferation rate than normal cells.

(Feb. 17-18) The goal of this mini-workshop is to explore connections between Hamiltonian Dynamics and Symplectic Topology, focusing on recent developments in these areas. For more information, see the workshop web site http://www.math.vanderbilt.edu/~gurelzb/Shanks2012/ws.html

This is the second part of my talk. Now let $X$ be closed manifold with Riemannian metric $g_0$ of positive scalar curvature and let $\Psi$ be a diffeomorphism $X  \to  X$. The induced metric $g_1 = \Psi^\ast g_0$ also has positive scalar curvature. Endow $X\times \mathbb R$ with the metric $g_t + (dt)^2$ where g_t =  g_0 for t=<0,  g_t = g_1 for t >=1, and  any smooth homotopy from $g_0$ to $g_1$ for 0< t <1. Then $X\times \mathbb R$ has positive scalar curvature towards infinity. Consider a $\Gamma$-covering $X_\Gamma$ of $X$ and the Dirac operator $D_\Gamma$ over $X_\Gamma\times \mathbb R$.  We use the relative higher index theorem above to show that the image of $\ind(D_\Gamma) $ inside $  K_0(C_r^\ast(\Gamma\rtimes_{\Psi} \mathbb Z) )$ coincides with  $\ind(D_{X_\Psi}) \in  K_0(C_r^\ast(\Gamma\rtimes_{\Psi}\mathbb Z))$, where $X_\Psi$ is the mapping cylinder of $X$ under $\Psi$ and $D_{X_\Psi}$ is the Dirac operator over $X_\Gamma\times \mathbb R$ as  a $(\Gamma\rtimes_{\Psi} \mathbb Z)$-covering of $X_\Psi$.

Ever wonder where the numbers in Pascal's Triangle come from, or why they work out so nicely?  In this talk, we'll look at some basic techniques of counting by combinations and permutations and see how they lead us to some interesting results including the binomial theorem, which is the source of the numbers in Pascal's famous triangle.

Gromov was first to notice a connection between the homotopic properties of asymptotic cones of a finitely generated group and algorithmic properties of the group. I will show three methods for understanding the fundamental group of an asymptotic cone of a group and give examples of groups for each method.

The Mobius Function mu(n) is minus one to the number of prime factors of n, if n has no square factors and is zero otherwise. Understanding the randomness (often referred as the Mobius randomness principle) in this function is a fundamental and very difficult problem.We will explain a precise dynamical formulation of this randomness principle and report on recent advances in establishing it and its applications. Tea at 3:30 pm in SC 1425.

Infinite index subgroups of integer matrix groups like SL(n,Z) which are Zariski dense in SL(n), arise in geometric diophantine problems (eg Integral Apollonian packings), as monodromy groups associated with families of algebraic varieties, as reflection groups... One of the key features needed when applying such groups in number theoretic problems is that the congruence graphs associated with these groups are "expanders". We will introduce and explain these ideas and review some recent developments and applications.

In 1956, Tutte proved that all 4-connected planar graphs are Hamiltonian by extending an earlier result by Whitney which made the same conclusion about 4-connected planar triangulations.  Since then, Thomas and Yu proved the same result for 4-connected projective planar graphs in 1994.  The Hamiltonicity of 4-connected toroidal graphs, however, is still unknown.  Grünbaum and Nash-Williams have conjectured that these graphs are in fact Hamiltonian, and this talk looks at possible steps towards proving this conjecture.  In particular, we examine hamiltonicity properties of graphs, closely related to 4-regular bipartite toroidal graphs, that pose particular problems for establishing the conjecture.

Both the arc operad and planar algebras are built from graphs on surfaces. Although similar in nature they were constructed with different aspects of geometry and applications in mind. With hindsight there is a striking similarity in one particular application. To elucidate this possible connection, we will proceed by introducing the arc operad and its discretization. The latter gives rise to actions on the Hochschild complex of a (Frobenius) algebra. These operations include Deligne's conjecture, its cyclic version and string topology. This discretization is also what is closely related to planar diagrams as we shall discuss.

To any homeomorphism T of the Cantor set K, we can associate a discrete group, consisting of those homeomorphisms of K that are piecewise powers of T, called the topological full group of T. I'll survey a few recent results about full topological groups of Cantor minimal systems.

Your voting power is the probability that your vote is decisive in an election.  Does the Electoral College fairly allocate voting power among US citizens?  How can voters form coalitions to maximize their voting power?  Is any political system inherently unstable?  We will survey some of the major ideas and consider the practical implications for our political system.

Generalizing the Margulis Normal Subgroup Theorem, Margulis and Zimmer conjectured that any subgroup of a lattice in a higher-rank Lie group which is commensurated by the lattice is (up to finite index) of a standard form. I will present some of my work on property (T) for totally disconnected groups and countable dense subgroups and explain how it provides "half" of the solution to the conjecture. This is joint work with Yehuda Shalom.