Seminars archive

The objective of this dissertation is to study the relationship between network-based centrality measures and epidemic outcome. Determining the key players in contagion processes can inform disease-prevention strategies. We analyze a time-stamped, person-to-person contact network based on human mobility movements within a busy, urban hospital. Movement patterns identified a small number of locations as hubs of activity. Linear algebraic techniques were used to compute a recently proposed temporal centrality measure applied to the empirical network; comparisons with traditional centrality measures were performed to determine if the inclusion of temporal information provides additional insights. Linear regression techniques were employed to describe the relationships between the quantities of interest. We find that while temporal centrality can at times identify key players not captured by traditional measures, it does not necessarily outperform non-temporal measures with respect to predicting epidemic outcome. Strategic removal of connections between highly central nodes resulted in an exponential decrease in the structural connectivity of the network, but this did not translate to a reduction in epidemic outcome. We conclude that contagion on temporal networks is extremely robust to changes in the network, and while network-based centrality can help to identify key players in an epidemic process, more work needs to be done to build an epidemic-containment strategy based on the information afforded by network-based analyses.

THIS SEMINAR HAS BEEN CANCELED. WE APOLOGIZE FOR THE SHORT NOTICE. We introduce the fascinating correspondence problem also known as image registration. Roughly spoken, the goal is to automatically establish correspondences between points in different projections of a scene. In particular, in medical imaging, this problem is very important and used for applications such as motion correction or data fusion. Several examples displaying different facets of the problem are discussed. A mathematical framework for the correspondence problem is outlined. Starting point is a variational formulation, where a joint energy is to be minimized on an appropriate set. Modular building blocks such as distance measures and regularizers are briefly discussed and related to particular applications. Finally, a brief outlook on constrained image registration is presented. Constraints are used to improve the modeling by restricting the admissible set in a smart way.

A celebrated theorem of Mazur asserts that the order of the torsion part of the group of rational points of an elliptic curve over Q is absolutely bounded; it is conjectured that the same is true for abelian varieties over number fields K, though very little progress has been made in proving it. The natural geometric analog where K is replaced by the function field of a complex curve---known as the geometric torsion conjecture---is equivalent to the nonexistence of low genus curves in congruence towers of Siegel modular varieties. In joint work with J. Tsimerman, we prove the conjecture for abelian varieties with real multiplication. We'll discuss a general method for bounding the genus of curves in locally symmetric varieties using hyperbolic geometry to bound Seshadri constants and apply it to some related problems.

The Graham-Pollak theorem states that, if we wish to decompose the complete graph Kn into complete bipartite subgraphs, then we need at least n-1 of them. What happens for hypergraphs? In other words, if we wish to decompose the complete r-graph on n vertices into complete r-partite r-graphs, how many do we need? In this talk we report on recent progress on this question. This is joint work with Luka Milicevic and Ta Sheng Tan.

I will introduce the notion of a finite hypergroup. It turns out that certain subfactors (unital inclusions of von Neumann algebras with trivial center) can naturally be seen as such a fixed point. Chiral conformal field theory can be axiomatized as local conformal nets of von Neumann algebras.The orbifold of a conformal net is the fixed point with respect to a finite group of automorphisms. We define a generalized orbifold to be the fixed point of a conformal net under a proper hypergroup action. The fixed point is finite index subnet and it turns out that all finite index subnets are generalized orbifolds. A holomorphic conformal net is a conformal net with trivial representation category. For example, every positive even self-dual lattice gives such a conformal net. The representation category of a generalized orbifold of a holomorphic net is the Drinfeld center of a categorification of the hypergroup. Based on arXiv:1506.02606.

A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares of quadratic forms. We show more generally that every nonnegative quadratic form on a real projective variety X of minimal degree is a sum of dim(X) + 1 squares of linear forms. This provides a new proof for one direction of a recent result due to Blekherman, Smith, and Velasco. We explain the geometry behind this generalization and discuss what is known about the number of equivalence classes of sum-of-squares representations. (Joint work with G. Blekherman, R. Sinn, and C. Vinzant)

Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of Gal(k/k) on pi_1(X), where k is a finite or p-adic field. As a sample application of our techniques, we show that if A is a non-constant Abelian variety over C(t), such that A[l] is split for some odd prime l, then A has at least four points of bad reduction.

This paper is devoted to the study of some fundamental properties of the sewing homeomorphism induced by a Jordan domain. In chapter 2, we mainly study two important conformal invariants: the extremal distance and the reduced extremal distance. Gives the estimate of extremal distance in the unit disk and the comparison of these two conformal invariants. In chapter 3 and 4, we give several necessary and sufficient conditions for the sewing homeomorphism of a Jordan domain to be bi-Lipschitz or bi-Holder, by using harmonic measure, extremal distance and reduced extremal distance. Furthermore, in chapter 5, we obtain some equivalent conditions for a Jordan curve to be a quasicircle. In chapter 6, we use the Robin capacity to define a new index and use this new index to characterize unit circle.

The main objective of this study is to build successful investment strategies and devise optimal portfolio structures by exploiting the power of forecast of our financial crisis indicators based on random matrix theory. While using daily data constituted of the components of a major equity index like the Standard and Poor’s 500 or the Shanghai-Shenzhen CSI 300, the financial crisis indicators used in this paper are of two kinds. Firstly we consider the financial crisis indicators based on measuring the Hellinger distance between the empirical distribution of the eigenvalues of the correlation matrix of those index components and a distribution of reference built to either reflect a calm or agitated market situation. Secondly, we consider the financial crisis indicators based on the study of the spectral radius of the correlation matrix of the index components where the coefficients have been weighted in order to give more importance to the stock components that satisfy a chosen characteristic related to the structure of the index, market conditions or the nature of the companies which are part of the index. For example, we will attempt to give more importance in the computation of the indicators to the most traded stocks, the stocks from the companies with the highest market capitalization or from the companies with an optimal debt to capital ratio (financial leverage). Our optimal investment strategies exploit the forecasting power of the financial crisis indicators described above in order to produce a ‘buy’, ‘sell’ or ‘stay put’ signal every day that is able to anticipate most of the market downturns while keeping the number of false positives at an acceptable level. Such tools are very valuable for investors who can use them to anticipate market evolution in order to maximize their profit and limit their losses as well as for market regulators who can use those tools to anticipate systemic events and therefore attempt to mitigate their effects.