UT - math Calendar

      March 2017                April 2017                 May 2017      
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                           30

4:00 pm Monday, April 17, 2017
Colloquium: Analysis based methods for solving linear elliptic PDEs numerically

by Gunnar Martinsson (University of Colorado at Boulder and Oxford University) in RLM 5.104

For some of the simplest linear elliptic PDEs, explicit solution formulas are known. Consider for instance, the Poisson equation Lu=f, where L is the Laplace operator. When the domain is free space R^d, and appropriate decay conditions are enforced at infinity, the unique solution is given as a convolution between the load function f and the free space fundamental solution of the Laplace operator. An explicit solution formula such as this can be turned into a powerful numerical tool once it is coupled with an efficient algorithm for rapidly evaluating the integral operator, such as, e.g., the Fast Multipole Method (FMM). The talk describes a set of generalizations to this "FMM based approach", where we numerically build approximations to solution operators for a broad class of elliptic PDEs, such as boundary value problems involving "general" domains, and/or differential operators with variable coefficients. These new so called "fast direct solvers" have many advantages, including improved stability and robustness, and dramatic improvements in speed in certain environments. The talk will also briefly describe methods based on randomized projections for rapidly executing common linear algebraic computations such as, e.g, low-rank approximation, factorization of matrices, solving linear systems, etc. These randomized methods are used to accelerate the direct solvers for elliptic PDEs, but have also proven highly competitive in machine learning, data analysis, etc.

3:00 pm Wednesday, April 19, 2017
Candidacy Talk: The generalized double semion TQFT

by Arun Debray in RLM 10.176

In this talk, I’ll discuss the low-energy effective field theory of the generalized double semion lattice model. This model, a twist of Kitaev’s toric code, was introduced by Freedman-Hastings, who prove that the state spaces agree with those of untwisted ℤ/2-Dijkgraaf-Witten theory in even dimensions, and are in general distinct in odd dimensions. Working from their criterion, I provide a description of the entire (nonextended) field theory in terms of Stiefel-Whitney classes and use it to reprove their results as well as a few new ones, including characterizing the low-energy effective field theory in 3 dimensions and showing that in even dimensions, the theory is distinct from untwisted and twisted Dijkgraaf-Witten theories. This is a candidacy talk.

3:00 pm Wednesday, April 19, 2017
Random Structures: Coupling, geometry and hypoellipticity

by Sayan Banerjee (University of North Carolina, Chapel Hill) in RLM 8.136

Coupling is a way of constructing Markov processes with prescribed laws on the same space. The coupling is called Markovian if the coupled processes are co-adapted to the same filtration. We will first investigate Markovian couplings of elliptic diffusions and demonstrate how the rate of coupling (how fast you can make the coupled processes meet) is intimately connected to the geometry of the underlying space. Next, we will consider couplings of hypoelliptic diffusions (diffusions driven by vector fields whose Lie algebra spans the whole tangent space). Constructing successful couplings (where the coupled processes meet almost surely) for these diffusions is a much more subtle question as these require simultaneous successful coupling of the driving Brownian motions as well as a collection of their path functionals. We will construct successful Markovian couplings for a large class of hypoelliptic diffusions. We will also investigate non-Markovian couplings for some hypoelliptic diffusions, namely the Kolmogorov diffusion and Brownian motion on the Heisenberg group, and demonstrate how these couplings yield sharp estimates for the total variation distance between the laws of the coupled diffusions when Markovian couplings fail. Furthermore, we will demonstrate how non-Markovian couplings can be used to furnish purely analytic gradient estimates of harmonic functions on the Heisenberg group by purely probabilistic means, providing yet another strong link between probability and geometric analysis. This talk is based on joint works with Wilfrid Kendall, Maria Gordina and Phanuel Mariano.

4:00 pm Wednesday, April 19, 2017
Groups & Dynamics: Borel complexity of generic points of dynamical systems with a weak specification property

by Dylan Airey (UT) in RLM 10.176

H. Ki and T. Linton showed that for any integer $b \geq 2$ , the set of normal numbers in base $b$ are $\Pi^0_3$ -complete. Other authors have since used this result to find the complexity of other sets of normal numbers. Let $(X,T,\mu)$ be a dynamical system with $X$ a Polish space, $T$ continuous, and $\mu$ a shift invariant probability measure. We generalize Ki and Linton's result by showing that if $(X,T)$ has more than one invariant measure and satisfies a weak form of the specification property then the set of $\mu$ -generic points is $\Pi^0_3$ -complete. As a consequence we establish that the sets of normal numbers with respect to a wide class of expansions other than the $b$ -ary expansion, for example the continued fraction expansion or the $\beta$ -expansion for irrational $\beta$ , are also $\Pi^0_3$ -complete.

1:00 pm Friday, April 21, 2017
Math/ICES Center of Numerical Analysis Seminar: Integral equation methods for the Laplace-Beltrami problem

by Michael O'Neil (Courant Institute) in POB 6.304

The reformulation of many of the classical constant-coefficient PDEs of mathematical physics (e.g. Laplace, Helmholtz, Maxwell, etc.) as boundary integral equations is a standard mathematical tool, which, when coupled with iterative solvers and fast algorithms such as fast multipole methods (FMMs), allows for the nearly optimal-time solution of these PDEs. Extending these methods to variable coefficient PDEs, especially those defined along surfaces, is not straightforward and ongoing research. In this talk, we will address the problem of solving the Laplace-Beltrami problem along surfaces in three dimensions. The Laplace-Beltrami problem is a variable coefficient PDE, with applications in electromagnetics, fluid-structure interactions, and surface diffusions. Our resulting integral equation is ready for acceleration using standard FMMs for Laplace potentials; several numerical examples will be provided.

1:00 pm Friday, April 21, 2017
Number Theory Seminar: Modular forms, Galois representations and Galois cohomology

by Chandrashekhar Khare (UCLA) in RLM12.166

In 1916, Ramanujan made three conjectures about his $\tau$ -function in a paper entitled ``Some arithmetical functions''. The Ramanujan $\tau$ -function is defined via $\Sigma_n \tau(n)q^n:=q\Pi_n(1-q^n)^{24}$ . I will argue that Ramanujan's conjectures and observations played a key role in work over the next few decades relating modular forms to Galois theory and motives. They led Serre to formulate his conjecture on (odd) extensions of the rationals with Galois group $\mathrm{GL}_2(\mathbb F_p)$ . This conjecture was proved in 2009, using the ``modularity lifting'' techniques which Wiles developed in the course of his proof of Fermat's Last Theorem. I will end by explaining how ``modularity lifting'' techniques offer new perspectives on higher dimensional generalizations of a classical conjecture of Leopoldt.