Seminars archive

Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.

We construct a Ramsey class whose objects are Steiner systems. In contrast to the situation with general r-uniform hypergraphs, it turns out that simply putting linear orders on their sets of vertices is not enough for this purpose: we also have to strengthen the notion of subobjects used from induced subsystems to something we call strongly induced subsystems. Moreover we study the Ramsey properties of other classes of Steiner systems obtained from this class by either forgetting the order or by working with the usual notion of subsystems. This leads to a perhaps surprising induced Ramsey theorem in which it designs get coloured. This is joint work with Vindya Bhat, Jaroslav Nevsetvril, and Vojtvech Rodl.

AT UGA

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)