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for events the day of Monday, September 16, 2013.

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Monday, September 16, 2013

10:00 am in 145 Altgeld Hall,Monday, September 16, 2013

Topological Hamiltonian and contact dynamics, part I: an introduction

Stefan Mueller (UIUC Math)

Abstract: In classical mechanics, the dynamics of a Hamiltonian vector field models the motion of particles in phase space, and the dynamics of a contact vector field play a similar role in geometric optics (in the mathematical model of Huygens' principle). Topological Hamiltonian dynamics and topological contact dynamics are relatively recent theories that explore natural questions regarding the regularity of such dynamical systems (on an arbitrary symplectic or contact manifold). In a nutshell, Hamiltonian and contact dynamics admit genuine generalizations to non-smooth dynamical systems with non-smooth generating (contact) Hamiltonian functions. The talk begins with examples that illustrate the central ideas and lead naturally to the key definitions. The main technical ingredient is the well-known energy-capacity inequality for displaceable subsets of a symplectic manifold. We use it to prove an extension of the classical 1-1 correspondence between isotopies and their generating Hamiltonians. This crucial result turns out to be equivalent to certain rigidity phenomena for smooth Hamiltonian and contact dynamical systems. We then look at some of the foundational results of the new theories. The end of the talk touches upon sample applications to topological dynamics and to Riemannian geometry, which will be explored further in a second talk.

1:30 pm in 345 Altgeld Hall,Monday, September 16, 2013

The Geometry and Topology of Algebraic Decision Trees

Jeff Erickson (UIUC Computer Science)

Abstract: I’ll describe several classical techniques for proving lower bounds on the depth of algebraic decision trees. The story begins with the folklore counting argument that sorting $n$ numbers requires $\Omega(n \log n)$ comparisons, and continues through component counts, volumes, Euler characteristics, and Betti numbers of convex polytopes, subspace arrangements, and semi-algebraic sets. In particular, we will revisit some of the results from Yuliy’s talk last week. I’ll also describe some inherent limitations of these techniques and several related open problems.

4:00 pm in 241 Altgeld Hall,Monday, September 16, 2013

Anomalous modulus of continuity for the theta process and logarithm laws for geodesics.

Francesco Cellarosi (Illinois)

Abstract: Theta sums are particular exponential sums with deep number-theoretical and physical connections. The planar curves obtained by linearly interpolating their partial sums are sometimes called 'curlicues' because of their rich geometric structure of spirals arranged in an approximate multi-fractal structure. Analogously to the construction of the Brownian Motion starting from simple symmetric random walks, I will briefly explain how to obtain a random process (the Theta Process) using equidistribution of horocycles under the action of the geodesic flow on a suitable hyperbolic manifold. Among the properties of this process, I will discuss the anomalous modulus of continuity of typical realizations of this process (different from that of a typical Brownian path), and derive this property using a logarithm law for geodesics due to Kleinbock and Margulis. This implies, in particular H\"older continuity of typical realizations for any exponent less than 1/2. Joint work with Jens Marklof (Bristol)

5:00 pm in 241 Altgeld,Monday, September 16, 2013

Uniqueness of group measure space decomposition for Popa's HT factors (after Adrian Ioana): Part II

Bogdan Udrea (UIUC Math)

Abstract: We continue to discuss Ioana's results on the HT factors of Popa.