Seminars and Colloquiums
for the week of December 4, 2017
Elise A. Weir, UTK, Monday
Jan Rosinski, UTK, Tuesday
Nick Dexter, UTK, Thursday
Ryan Unger, Thursday
3:00 pm – 3:30 pm
Hosted By: Shane Sawyer
Monday, December 4th
Title: The Dimension of the Restricted Hitchin Component for Triangle Groups
Speaker: Elise A. Weir, UTK
Room: Ayres 113
A triangle group T(p,q,r) is the group of rotational symmetries of a tiling of the hyperbolic plane by geodesic triangles. We will begin by discussing a component of the representation variety of T(p,q,r) in PSL(n,R) called the Hitchin component, noteworthy in part because representations inside are all discrete and faithful.
For n = 3, the dimension of the Hitchin component for hyperbolic triangle groups follows from a special case of work by Choi and Goldman. More recently, Long and Thistlethwaite determined its dimension for general n >= 3. Our results retain this broader n-dimensional context, but focus >in on those representations contained in the subgroup G = SO(m,m+1) or >G = Sp(2m). In particular, we will give a formula for the dimension of >this "restricted" Hitchin component for hyperbolic T(p,q,r) within G >for all n >>= 3.
Tuesday, December 5th
STOCHASTICS/ PROBABILITY SEMINAR
Title: Strong path-wise approximation of Gaussian and Levy processes, extension of Ito-Nisio theorem, and continuity of Ito map
Speaker: Jan Rosinski, UTK
Room: Ayres 113
N. Wiener (1923) gave two representations of Brownian motion as random trigonometric series, and showed that these series converge path-wise uniformly. This made Brownian motion a well-defined mathematical object and explained the nature of white noise. Nowadays, such types of series representations are also used to simulate stochastic processes. The mode and type of convergence in the series is crucial when we want to approximate output of a stochastic system (such as the Ito map) by the simulated input. The uniform convergence of the input processes is often insufficient for the convergence of the outputs, which triggered the development of rough path theory. We take a different approach to this problem by extending Ito-Nisio theorem to non-separable function spaces. We apply this approach to show strong path-wise convergence in series expansions of Levy processes, which yields the continuity of Ito map. This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen of Aarhus University.
Thursday, December 7th
COMPUTATIONAL and APPLIED MATHEMATICS (CAM) SEMINAR
Title: Joint-sparse approximation for high-dimensional parameterized PDEs
Speaker: Nick Dexter, UTK
Room: Ayres 113
We consider the problem of reconstructing a set of sparse vectors, sharing a common sparsity pattern, from incomplete measurements. This problem is often referred to as the problem of joint-sparse recovery, and has long been known to be a tractable approach for sparse approximation of solutions to a wide class of linear inverse problems.
We establish connections between the joint-sparse recovery problem and the problem of approximating solutions to high-dimensional parameterized PDEs, discussing methods of regularization enforcing sparsity, error estimates, and algorithms for solution. We conclude with detailed numerical experiments, showing the efficiency of our approach relative to several classical methods for approximation.
GEOMETRIC ANALYSIS SEMINAR
Title: The Yamabe Problem IV
Speaker: Ryan Unger
Room: Ayres 404
We complete the proof of the Yamabe conjecture in the positive case by giving some details on conformal normal coordinates, the asymptotic expansion of the conformal Green's function, and the test function estimate in the low-dimensional and LCF cases.
If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar.
If you have questions, or a date you would like to confirm, please contact colloquium AT math DOT utk DOT edu