Department of Mathematics and Computer Science



Departmental Events, Spring 2014:

Wednesday, 14 May 2014, 3pm Room TBA: Department Colloquium
Ruth Davidson
Department of Mathematics
North Carolina State University

Distance-based phylogenetic methods near a polytomy
A phylogenetic tree models the common evolutionary history of a group of species. A tree metric is a distance function on a set of species realized by a tree with edge weights. Distance-based phylogenetic algorithms attempt to solve the NP-hard least-squares phylogeny problem by mapping an arbitrary dissimilarity map representing biological data to a tree metric. The set of all dissimilarity maps is a Euclidean space properly containing the space of all tree metrics as a polyhedral fan. Outputs of distance-based tree reconstruction algorithms such as UPGMA and Neighbor-Joining are points in the maximal cones in the fan. Tree metrics with polytomies, or internal vertices of degree higher than three, lie at the intersections of maximal cones. A phylogenetic algorithm divides the space of all dissimilarity maps into regions based upon which combinatorial tree is reconstructed by the algorithm. We use polyhedral geometry to compare the local nature of the subdivisions induced by least-squares phylogeny, UPGMA, and Neighbor-Joining. Our results suggest that in some circumstances, UPGMA and Neighbor-Joining poorly match least-squares phylogeny when the true tree has a polytomy.  This is joint work with Seth Sullivant. 


Monday, 24 February 2014, 11am, Gillet 311: Department Colloquium
Prof. Anthony Gamst
Department of Biostatistics and Bioinformatics
University of California, San Diego

Some Problems in the Analysis of High-Dimensional Models
Models with large numbers of nuisance parameters are common in modern statistics, having applications in laboratory medicine, econometrics, genomics, medical imaging, physics, epidemiology, and many other areas. Classical techniques, including Maximum Likelihood and Bayesian approaches, often produce sub-optimal or even inconsistent estimates of the parameters of interest in these models, while asymptotically unbiased estimating equations work rather generally. We study several such models, identify the sources of bias and spurious correlation which lead to inconsistency or sub-optimality, and compute the minimal smoothness required for the existence of root-n consistent (and efficient) parameter estimates. We also examine simultaneous estimation of nuisance parameters and parameters of interest. These results are related to every-day practice, particularly to the analysis of regression models with many predictors, and some heuristics are given.


Monday, 10 February 2014, 11am, Gillet 311: Department Colloquium
Prof. Erik Guentner
Department of Mathematics
University of Hawai'i

Geometry and noncommutative duals of groups
Nonommutative geometry, in the sense of Alain Connes, proceeds from the observation that properties of a topological space are reflected by properties of the algebra of functions on it.  Further, in cases when the natural topological space is poorly behaved it may be profitably be replaced by a noncommutative C*-algebra, the algebra of 'functions on a noncommutative space'.  In the talk I will survey results relating geometric properties of a discrete group to its harmonic analysis as manifested by the noncommutative dual space of the group.


Thursday, 6 February 2014, 2pm, Gillet 311: Department Colloquium
Prof. Loredana Lanzani
Department of Mathematics
University of Arkansas

Harmonic Analysis Techniques in Several Complex Variables


Monday, 3 February 2014, 11am, Gillet 219: Department Colloquium
Prof. Michael Usher
Department of Mathematics
University of Georgia

The geometry of the Hamiltonian diffeomorphism group
An important object associated to any symplectic manifold is its infinite-dimensional group of "Hamiltonian diffeomorphisms," consisting of those diffeomorphisms which arise as time-evolution maps in a generalization of Hamilton's formulation of classical mechanics. Rather unusually for an infinite-dimensional Lie group, the Hamiltonian diffeomorphism group admits a bi-invariant metric induced by a norm on its Lie algebra, discovered by Hofer, which can be viewed as giving a coordinate-independent measurement of the "energy" of any Hamiltonian diffeomorphism.  I will discuss some progress in understanding this still-rather-mysterious metric, concerning for instance whether it is always unbounded and how it interacts with submanifolds, and will also touch on some open questions.


Monday, 27 January 2014, 11am, Gillet 219: Department Colloquium
Prof. David Savitt
Department of Mathematics
Brown University

Galois representations
The absolute Galois group of the field of rational numbers is a fundamental object of study in number theory. I will begin by giving a tour of the representation theory of this group, with an emphasis on representations in characteristic p. In the second half of the talk I will describe my recent work with Gee, Liu, and others on generalizations of the weight part of Serre's conjecture.


Friday, 24 January 2014, 11am, Gillet 219: Department Colloquium
Prof. Bianca Viray
Department of Mathematics
Brown University

The local to global principle for rational points
Let X be a connected smooth projective variety over Q. If X has a Q point, then X must have local points, i.e. points over the reals and over the p-adic completions Q_p. However, local solubility is often not sufficient. Manin showed that quadratic reciprocity together with higher reciprocity laws can obstruct the existence of a Q point (a global point) even when there exist local points. We will give an overview of this obstruction (in the case of quadratic reciprocity) and then show that for certain surfaces, this reciprocity obstruction can be viewed in a geometric manner. More precisely, we will show that for degree 4 del Pezzo surfaces, Manin's obstruction to the existence of a rational point is equivalent to the surface being fibered into genus 1 curves, each of which fail to be locally solvable. This talk will be suitable for a general audience.


Friday, 24 January 2014, Carman Hall, 11am: CMACS Talk
Prof. Bud Mishra
Courant Institute of Mathematical Sciences
New York University







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