In 1980s, Yau conjectured that the existence of Kaehler Einstein metric on Fano manifold is related to an algebraic geometric  condition of ``stability''.  The recent work with Donaldson, Sun Song confirmed this conjecture.  In the talk, we will  review history of this problems as well as this subject, and we also will review earlier work of  G. Tian and others on this problems.  We will outline the strategy of proof, which involves deforming through metrics  with cone singularities.  If time permits, we will give more details about various aspects of the proof. Tea at 3:30 pm in SC 1425.

Spectral triple is the fundamental object of the metric aspects of Connes' noncommutative geometry. A spectral metric space is a spectral triple (A, H, D) with additional properties guaranteeing that the Connes metric on the state space of A induces the weak*-topology. It is, in fact, the noncommutative analog of a complete metric space. Let (A,H,D) be a spectral metric space and G be a group of automorphisms of A. In this talk I will consider the problem of whether there is a natural spectral triple for the crossed product algebra C*(G,A) that can characterize the metric properties of the dynamical system (G,A). I will discuss a solution to this problem when a single automorphism of A generates G as an equicontinuous family of quasi-isometries. I will also address the converse problem, namely, when a spectral metric space for the crossed product gives rise to one for A. When the action is not equicontinuous (e.g., when the action is uniformly hyperbolic), following the philosophy of Diffeomorphism-Invariant Geometry of Connes and Moscovici, we suggest replacing the dynamical system (G,A) by a dynamical system (G,B), where G acts isometrically. The algebra B is called the metric bundle associated with (G,A). Some candidates for the metric bundle B will be introduced. This talk is based on a joint work with Jean Bellissard and Matilde Marcolli.

A partial differential equation model is provided for the two-slit experiment of quantum mechanics. The state variable of the equation is the probability density function of particle positions. The equation has a local diffusion term corresponding to stochastic variation of particles, and a nonlocal dispersion term corresponding to oscillation of particles in the transverse direction perpendicular to their forward motion. The model supports the ensemble interpretation of quantum mechanics and gives descriptive agreement with the Schrodinger equation model of the experiment.

Perhaps the most outstanding open problem in mathematical relativity is the so called Cosmic Censorship conjecture, which roughly asserts that singularities in the evolution of spacetime must always be hidden inside black holes, and moreover that spacetime must eventually settle down to a stationary final state. Based on heurisitc physical arguments, R. Penrose derived a series of geometric inequalities relating total mass, area of the event horizon, electricromagnetic charge, and angular momentum, all of which serve as necessary conditions for the validity of Cosmic Censorship. In this talk, we will detail recent advances in the rigorous mathematical formulation and proofs of some of these inequalities. Tea at 3:30 pm in SC 1425

A group is called topologizable if it admits a non-discrete Hausdorff group topology. In this talk I will discuss some recent results (joint with A. Klyachko and A. Olshanskii)  and open questions about topologizable and non-topologizable groups.

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.

 

Given a graph G, the hitting time from u to v is the expected number of steps in a simple random walk u v_1 v_2 v_3 ... v where no v_i equals v. The average hitting time (AHT) of u is the arithmetic mean of the hitting times from u to every other vertex in G. When examining the spread of information through a social network represented as a connected digraph, the AHT can be seen as a proxy measure for a vertex's centrality in this diffusion process.  Since social networks are best represented as small-world graphs, we used the graph construction methods described by Watts and Strogatz and by Barabasi and Albert, comparing them to simple random graphs built using the Viger-Latapy algorithm. The AHT distribution varies widely between different graph structures and generation methods, although in every case is extremely closely tied to vertex degree.  Varying the order or average degree of a graph has consistent and predictable effects on the parameters of the AHT distribution, as does varying the control variables for the generation method in question.  In particular, graphs with strong preferential attachment behavior demonstrated an isolated group of vertices with extremely low AHT, as well as greater clustering in discrete "layers" defined by similar AHT values.

A group has Property FW if every action on a set commensurating a subset fixes a subset at bounded distance. This is a combinatorial weakening of Kazhdan Property T (and strengthening of Serre's Property FA), which was characterized in a similar (measurable) fashion by Robertson and Steger. I will discuss Property FW in various contexts, and notably for lattices in Lie groups.

A series of informal talks, following the book by Peter J. Olver.