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.

A metric space is multiended if it admits a bounded subset whose complement has at least two unbounded connected components. For instance, the line is multiended but higher-dimensional Euclidean spaces are not. In the late sixties, Stallings has given a remarkable characterization of those finitely generated groups whose Cayley graph is multiended; the only such torsion-free groups are free products of two nontrivial groups! A key part of the proof is the construction of a action on a tree; in the seventies, the study of general group actions on trees was achieved by Bass and Serre. In the meantime, the study of multiended Schreier graphs was started by Abels and Houghton, and a remarkable connection with nonpositively curved cube complexes was discovered by Sageev twenty years later. While outstanding applications of cube complexes have been made since then, I will try to focus on the question of understanding which finitely generated groups admit a multiended Schreier graph. Tea at 3:30 pm in SC 1425.

A character on a group is a positive definite function which takes the identity to 1 and is constant on conjugacy classes. Characters on a finite group gives an essential tool for understanding the representation theory of the group and motivated by this Thoma in 1964 initiated the study of characters on infinite groups. In 1966 Kirillov classified all characters on GL_n(k), and SL_n(k) for k a field and n \geq 2, excluding the case of SL_2(k). A number of other classification results have since been obtained for other groups by Ovcinnikov, Vershik, Kerov, and more recently by Bekka, Dudko, and Medynets, however the classification for SL_2(k) has not been completed. In my talk I will present the classification for SL_2(k) and SL_2 of some other rings and give some applications of these results. This is based on joint work with Andreas Thom.

nos-o-co-mi-al  (adjective)  - originating or occurring in a hospital
“get down R0, know your place, do not torment the human race”
primum non nocere – first, do no harm
Nosocomial epidemics are an increasing threat to society. The microbes that cause these epidemics are increasingly resistant to antibiotics. In the US they cause more than 100,000 deaths each year and that number is increasing. R0 (pronounced R-naught) is a quantity derived from mathematical models that predict the course of an epidemic. It is obtained from various parameters that determine the transmission dynamics of the infection. If R0 < 1, then the epidemic will abate. If R0 > 1, then the epidemic will worsen. The Hippocratic oath is the promise of physicians to not make the condition of a patient worse. It is the fundamental precept of medicine. I will tell you how all these are connected. I will also tell you how to avoid being infected by a nosocomial infection. Nosocomial infections occur in specific locations in the US, and only in these locations. You will never suffer a nosocomial infection if you do not go to one of these locations. I will tell you where these locations are.