Graduate Student Seminar 07-08

Abstract: In this talk I will present congruence and divisibility properties of two different classes of combinatorial sums. The first class involves products of powers of two binomial coefficients; we will see that even though in general there is no evaluation in closed form, the sums behave in certain respects like single binomial coefficients. This is evident through a result similar to Wolstenholme's theorem, and through the fact that under certain conditions the sums are divisible by all primes in specific intervals. The second class of combinatorial sums is the alternating version of a well-known sum that was used in the theory of Bernoulli numbers. This new sum is evaluated modulo an odd prime, and as an application it is shown that the n-th Bernoulli polynomial cannot have multiple roots, which solves a long-standing conjecture.

Abstract: Evolutionary bioinformatics is a subfield of bioinformatics addressing issues in molecular evolution, phylogenetics, evolutionary biology and genomics. After a general introduction to maximum likelihood and evolutionary models in phylogenetic inference, I will first describe our work on statistical modelling of the covarion evolutionary process. The covarion process is one that rates of evolution at sequence sites change in different parts of the tree. This is distinct from the rates-variation-across-sites (RAS) process that evolutionary rates are constant across the tree. The changing rates of evolution along the tree under the covarion process are formulated as a Markov model of rate switching between different rate classes. The models, implemented in our new phylogenetic inference package (PROCOV), gave significantly better likelihoods than using the RAS model for all 23 protein data sets we tested. In one case the covarion models were found to support a different optimal topology than the RAS model, highlighting the importance of covarion modelling. I will then briefly discuss our study on topological estimation bias with covarion evolution. We simulated four-taxon sequence data under covarion models and estimated the phylogeny under the RAS model. We found that ignoring the covarion process would lead to an unusual long-branch-repulsion type of bias in phylogenetic estimation. The third part of the talk will focus on genomic adaptation to environmental temperature. In particular, I will show whether the G+C content of the genome and structural RNA genes will change with optimal growth temperature in prokaryotes. I finish the talk by proposing a few topics for future research.

Abstract: An introduction to surreal numbers. The talk will parallel the introduction to surreal numbers given by Knuth in the book Surreal Numbers. If times permits, I'll talk about how surreal numbers generalize to games (as in combinatorial game theory).

Abstract: The motivation behind classification analysis is to produce a procedure that will classify objects accurately while providing knowledge of the predictive structure of the data. Tree based classifiers provide a simple but powerful solution to the classification problem. They're flexible, provide built-in variable selection and have an easily interpretable graphical representation. I will begin with a simple example using the famous Fisher/Anderson iris data. I will then show how the classification tree method can be applied using covariates derived from amino acid sequences to classify proteins into structural groups.

Abstract: I will give a simple introduction to some concepts in projective geometry. I'll talk about spheres, real and complex projective spaces, and possibly projective geometry with a finite number of points. There will be more pictures than proofs.

Abstract: Besides its mathematical aesthetics, representation theory has vast and surprising meaning in physics, chemistry, probability and number theory, potentially underlying some deep questions about the nature of reality. For example, as explained by J.M.G. Fell (University of Pennsylvania) ``It seems plausible that the kinds of 'irreducible' particles that can exist in a quantum-mechanical universe should be correlated with the possible irreducible representations of its underlying symmetry group. If this is so, then it should be a physically meaningful project to classify all the possible representations of that group.'' The Mackey Machine takes a giant leap toward this goal, i.e. describing all the irreducible unitary representations of a group. Here we'll look at some of the reasoning behind Mackey's method for the case when the group is almost abelian. This relies heavily on his powerful Imprimitivity Theorem, the proof of which was influenced by von Neumanns proof that Heisenberg and Schrodinger's formulation of QM are equivalent! ``I used to say: "Everything is Representation Theory". Now I say: "Nothing is Representation Theory". -Israel Gelfand (Gelfand seminar and workshop, Rutgers University, Sept. 10, and Nov. 19, 2001)

Abstract: The history of geometry has gone through 3 major generalizations from Euclidean geometry. The first two came from the ideas of Riemann and Klein, which at first seemed unrelated, then came the final generalization which brought together the two, what is now called Cartan geometry. The idea of this talk will be to explain how one can roll a manifold on another to study the structure of the underlying manifold. The key example, will of course be in 3-dimensions, so one can have some visuals. Using this idea, which came from Elie Cartan in the early 1920's, we can say that all of geometry can be studied by use of a connection on a principal bundle.

Abstract: I will introduce The Firefighter Problem, a deterministic discrete time model of the spread and containment of fire on a graph. Although there are a number of different objectives that can be pursued, this talk will focus on two objectives: maximizing the number of 'saved' vertices when containing a fire; and minimizing the number of firefighters used at each step in containing a fire.

Abstract: A paper in the Mathematics Magazine v.77(5) drew my attention a while ago. It contains a classification of the subgroups of the rational numbers when seen as an additive group. The study does not require any high level algebra, and with my modest background I shall state the theorem and guide you through the nice and elegant proof backing up the result.