Monday
November 30, 2009

12 noon- 1 pm
Room 230

Swaminathan Sethu Raman
Volumes of Sums of Squares polynomials and their generalizations
Ever since Motzkin showed by explicit construction, positive polynomials that are not sums of squares, there has been a lot of interest in comparing the volumes of positive polynomials and sums of squares. I shall talk about methods used to bound the volumes of the above families of polynomials and some extensions. I shall also discuss relaxation methods for global optimization of multivariate polynomial functions using sums of squares and semidefinite programming.

Andy Hammerlindl
Computation of Invariant Manifolds
In this talk, I'll briefly introduce dynamical systems, and hyperbolicity for smooth dynamical systems.  In such systems, we can look at stable and unstable invariant manifolds through a point, which tell us of the behaviour of the system.
I'll quickly outline how the existence of these invariant manifolds is proven, and how this relates to their numerical computation.  I'll showcase some of the interesting behaviour the manifolds can possess and, time allowing, show how rigorous computation of the invariant manifold can give lower bounds on the entropy (a technical notion related to the amount of chaos) of these systems.

Chris Conidis
Computability Theory and Reverse Mathematics
We will give a brief introduction to computability theory, and then outline some applications of computability theory to reverse mathematics. Reverse mathematics is a branch of mathematical logic that (roughly speaking) attempts to assign strengths to mathematical theorems based upon the (Turing) degrees of unsolvability in which their proofs can be carried out effectively.

Monday
September 28, 2009

12 noon- 1 pm
Room 230

Monday
October 19, 2009

12 noon- 1 pm
Room 230

Jonathan Hauenstein
Homotopy continuation and numerical algebraic geometry
This talk will provide a general overview of homotopy continuation for computing the isolated solutions of a given polynomial system and describe some of the numerical challenges that arise.  Building from the ability to compute isolated solutions, I will briefly describe how to numerically compute all irreducible solution components of a polynomial system.  The talk will conclude with an algorithm for computing the local dimension of a solution. Many examples will be provided throughout the talk.

Monday
October 26, 2009

12 noon- 1 pm
Room 230

Cristóbal Rojas
Algorithmic randomness: A dynamical point of view

The basic idea of algorithmic randomness is that an individual algorithmic random sequence should satisfy all the ``effective'' probability laws.  There are several different possible definitions, depending on the kind of the considered probability laws and their ``degree of  effectivity''.  A very natural source of probability laws is the ergodic theory of dynamical systems.  In this talk we present some recent results relating algorithmic randomness to the statistical properties of the trajectories of points in ergodic dynamical systems. We do this in the framework of general computable probability spaces (the effective version of the spaces where usual ergodic theory takes place). First we extend algorithmic randomness to this setting and develop some useful tools. Then, we introduce a ``dynamical'' notion of randomness: typicality. Roughly, a point is typical for some ergodic dynamics if it follows the statistical behavior of the system (given by Birkhoff's pointwise ergodic theorem) with respect to every bounded continuous function used to follow its trajectory (or equivalently, every computable function). The main result is the following characterization: in any computable probability space, a point  is Schnorr random if and only if it is typical for every mixing computable dynamical system.

Monday
November 2, 2009
12 noon- 1 pm
Room 230

Chin How Jeffrey Pang
Set-valued analysis: Generic continuity of semi-algebraic maps and generalized differentiability
I will give a brief overview of set-valued analysis and semi-algebraic variational analysis, and present two results. The first result is on the generic continuity of semi-algebraic set-valued maps, and is joint work with Aris Daniilidis. The second result is on a new notion of generalized differentiability of set-valued maps using positively homogeneous maps. Extensions of classical results in view of this notion of generalized differentiability include Lipschitz continuity, the Aubin property, the Mordukhovich criterion, metric regularity and linear openness.

Monday
November 9, 2009
12 noon- 1 pm
Room 230

Michael Coons
The Riemann hypothesis: celebrating 150-epsilon years
The Riemann hypothesis is about to have its sesquicentennial birthday. In celebration of this rather unfortunate (for those that would like a proof) event, I will discuss the history of this conjecture and related results from the theory of prime numbers. Results will be presented using the framework of multiplicative functions.

Monday
November 16, 2009

Workshop on Computational Differential Geometry, Topology, and Dynamics

NO SEMINAR

Monday
November 23, 2009

*11:30 - 12:30 pm*
Room 230

Alexander Grigo
Stable and Random Motion in Billiards and Related Systems

The study of mechanical systems has a several centuries long history. During the previous century there have been major results concerning the stability (KAM theory) and instability (hyperbolicity) in such dynamical systems. A particularly nice class of ``mechanical'' systems for which rigorous results are known are geodesic flows on compact manifold without boundary, and billiards. I will discuss several results on stable and random dynamics in billiards and related systems. The talk will not require any prior knowledge on billiards.