Series: Other Talks
We argue that computational models have an essential role in uncovering the principles behind a variety of biological phenomena that cannot be approached by other means. In this talk we shall focus on evolution. Living organisms function according to complex mechanisms that operate in different ways depending on conditions. Darwin's theory of evolution suggests that such mechanisms evolved through random variation guided by natural selection. However, there has existed no theory that would explain quantitatively which mechanisms can so evolve in realistic population sizes within realistic time periods, and which are too complex. Here we suggest such a theory. Evolution is treated as a form of computational learning from examples in which the course of learning depends only on the aggregate fitness of the current hypothesis on the examples, and not otherwise on individual examples. We formulate a notion of evolvability that distinguishes function classes that are evolvable with polynomially bounded resources from those that are not. For example, we can show that monotone Boolean conjunctions and disjunctions are demonstrably evolvable over the uniform distribution, while Boolean parity functions are demonstrably not. We shall discuss some broader issues in evolution and intelligence that can be addressed via such an approach.
Series: Geometry Topology Seminar
In this talk I will describe some relations between embedded graphs, their polynomials and the Jones polynomial of an associated link. I will explain how relations between graphs, links and their polynomials leads to the definition of the partial dual of a ribbon graph. I will then go on to show that the realizations of the Jones polynomial as the Tutte polynomial of a graph, and as the topological Tutte polynomial of a ribbon graph are related, surprisingly, by the homfly polynomial.
Monday, October 20, 2008 - 11:05 , Location: Skiles 255 , Chris Godsil , University of Waterloo , Organizer: Robin Thomas
In its simplest form, the Erdos-Ko-Rado theorem tells us that if we have a family F of subsets of size k from set of size v such that any two sets in the family have at least one point in common, then |F|<=(v-1)\choose(k-1) and, if equality holds, then F consists of all k-subsets that contain a given element of the underlying set. This theorem can also be viewed as a result in graph theory, and from this viewpoint it has many generalizations. I will outline how it can be proved using linear algebra, and then discuss how this approach can be applied in other cases.
Friday, October 17, 2008 - 15:00 , Location: Skiles 255 , Ravi Montenegro , University of Massachussetts , Organizer: Prasad Tetali
The Birthday Paradox says that if there are N days in a year, and 1.2*sqrt(N) days are chose uniformly at random with replacement, then there is a 50% probability that some day was chosen twice. This can be interpreted as a statement about self-intersection of random paths of length 1.2*sqrt(N) on the complete graph K_N with loops. We prove an extension which shows that for many graphs random paths with length of order sqrt(N) will have the same self-intersection property. We finish by discussing an application to the Pollard Rho Algorithm for Discrete Logarithm. (joint work with Jeong-Han Kim, Yuval Peres and Prasad Tetali).
Friday, October 17, 2008 - 14:00 , Location: Skiles 269 , Jim Krysiak , School of Mathematics, Georgia Tech , Organizer: John Etnyre
This will be a continuation of the previous talk by this title. Specifically, this will be a presentation of the classical result on the existence of three closed nonselfintersecting geodesics on surfaces diffeomorphic to the sphere. It will be accessible to anyone interested in topology and geometry.
Series: Stochastics Seminar
Adaptive estimation of linear functionals occupies an important position in the theory of nonparametric function estimation. In this talk I will discuss an adaptation theory for estimation as well as for the construction of confidence intervals for linear functionals. A between class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability and in the construction of adaptive procedures in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single parameter space. Our results thus "geometrize" the degree of adaptability.
Series: School of Mathematics Colloquium
We prove that a smooth compact submanifold of codimension $2$ immersed in $R^n$, $n>2$, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman. These results consist of joint works with Stephanie Alexander and Jeremy Wong, and Robert Greene.
Series: Research Horizons Seminar
In the study of one dimensional dynamical systems it is often assumed that the functions involved have a negative Schwarzian derivative. However, as not all one dimensional systems of interest have this property it is natural to consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class, that is to functions with an eventual negative Schwarzian derivative. The property of having an eventual negative Schwarzian derivative is nonasymptotic therefore verification of whether a function has such an iterate can often be done by direct computation. The introduction of this class was motivated by some maps arising in neuroscience.
Wednesday, October 15, 2008 - 11:00 , Location: Skiles 255 , Yang Kuang , Arizona State University , Organizer:
Chronic HBV infection affects 350 million people and can lead to death through cirrhosis-induced liver failure or hepatocellular carcinoma. We present the rich dynamics of two recent models of HBV infection with logistic hepatocyte growth and a standard incidence function governing viral infection. One of these models also incorporates an explicit time delay in virus production. All model parameters can be estimated from biological data. We simulate a course of lamivudine therapy and find that the models give good agreement with clinical data. Previous models considering constant hepatocyte growth have permitted only two dynamical possibilities: convergence to a virus free or an endemic steady state. Our models admit periodic solutions. Minimum hepatocyte populations are very small in the periodic orbit, and such a state likely represents acute liver failure. Therefore, the often sudden onset of liver failure in chronic HBV patients can be explained as a switch in stability caused by the gradual evolution of parameters representing the disease state.