I'll give an introduction to the notions of dimension, trace and character in homotopical algebra and geometric representation theory, following an ongoing project with David Nadler. I hope to convey what a "geometric dimension formula" entails and how variants of many theorems arise from this perspective. (Some examples include Riemann-Roch theorems, Atiyah-Bott fixed point theorem, Lefschetz trace formula, Frobenius and Weyl character formulas, Arthur-Selberg trace formula, and the Fundamental Lemma.)

The twisted torus knots are a class of knots which can be considered as curves in the standard genus 2 Heegaard surface F for the 3-sphere. Among the twisted torus knots, one can find knots that are primitive/primitive, knots that are primitive/Seifert and knots that are fibered. In this talk, we will discuss some results about twisted torus knots that have at least one of the three properties above. This is a thesis defense.

Note the new room for the seminar this semester.

This is a thesis defense.

A state sum is an expression approximating the partition function of a n-dimensional quantum field theory on a closed n-manifold from a triangulation of that manifold. To introduce state sums into the context of completely local 2-dimensional topological field theories (TFTs), we first describe a mechanism for incorporating triangulations of surfaces into the cobordism (infinity,2)-category. This serves to produce a state sum formula for any fully extended 2-dimensional TFT possibly with extra structure. In the second part of this work, we follow the Cobordism Hypothesis in classifying fully extended 2-dimensional G-equivariant TFTs for a finite group G. These are oriented theories in which bordisms are equipped with principal G-bundles. Combining the mechanism mentioned above with our classification result, we derive Turaev's state sum formula for such theories. This is a thesis defense.

Continuum damping is a common property of linear infinite dimensional Hamiltonian systems. We examine one such system, the Caldeira-Leggett model. We derive a transformation which diagonalizes the system allowing us to solve it exactly. In the process, we are able to interpret the damping as an analogue of Landau damping in the linearized Vlassov-Poisson system.

In Joint work with David Gay, we give existence and uniqueness theorems for the 2-dimensional analogue of a Morse function. The main interest is in dimension n=4. I'll define terms, state the theorems, sketch a proof or two, and talk about examples of homotopy 4-spheres.

(Joint work with Su-ion Ih) Let be a number field with algebraic closure , and let be a finite set of primes of containing all the infinite ones. Let be an elliptic curve, be a finitely generated subgroup of , and be the division group attached to . Fix an effective divisor of with support containing either (i) at least two points whose difference is not torsion, or (ii) at least one point not in . We prove that the set of ``integral division points on ," i.e., the set of points of which are -integral on relative to is finite. We also prove the -analogue of this theorem, thereby establishing the 1-dimensional case of a general conjecture we pose on integral division points on semi-abelian varieties.

In the Strominger-Yau-Zaslow description of mirror symmetry, singularities of the torus fibration lead to difficulties in the construction of mirror spaces and the computation of algebraic structures associated to these spaces. We will discuss one such algebraic structure, the Floer cohomology of Lagrangian sections of the torus fibration, in a space with a simple type of singularity, the Landau-Ginzburg mirror of CP^2 relative to a conic and a line, along with some natural generalizations. Of particular interest will be the canonical basis for the Floer cohomology groups (and hence of sheaf cohomology groups on CP^2) that our construction gives rise to.

This is a thesis defense.

http://www.ma.utexas.edu/harrington11/

The problem of estimating uncertainties in climate prediction is not well defined. While one can express its solution within a Bayesian statistical framework, the solution is not necessarily correct. One must confront the scientific issues for how observational data is used to test various hypotheses for the physics of climate. Moreover, one also must confront the computational challenges how estimates the posterior distribution without the help of a statistical emulator of the forward model. I will present results of a recently completed estimate of the uncertainty in specifying 15 parameters important to clouds, convection, and radiation of the Community Atmosphere Model. We learned that the maximum posterior probably is not in the same region of parameter space as the minimum log-likelihood. We have interpreted these differences to the existence of model biases and the potential that the minimum log-likelihood, which are often the desired solutions to data inversion problems, are over-fitting the data. Such a result highlights the need for a combination of scientific and computational thinking to begin to address uncertainties for complex multi-physics phenomena.

For a class of irrational triangle billiards, having very fast rational approximation of angles, we find an explicit upper estimate for a splitting time of a parallel beam of trajectories. The method involves study of some circle pseudo-dynamics and a search for large monochromatic arithmetic progressions. Relation to the search of periodic orbits will be also discussed.

Frequently in topology it is useful to cut a space into simpler pieces and then see how they glue back together to get information about the original space. Handle decompositions provide a nice way of breaking up a manifold into basic blocks. I will introduce these decompositions, draw examples, discuss some Morse theory that underlies this decomposition, and then talk about applications in low-dimensions.

http://www.ma.utexas.edu/harrington11/