Mathematics

Discrete Morse Theory and Reidemeister Torsion

Mark Schubel (UIUC Physics)

Abstract: Through Morse theory we are able to get valuable insights into the topology of a manifold by studying differentiable functions on that space. In this talk we will consider a discrete version of Morse theory by first introducing a discrete Morse function and gradient vector field. This notion can be generalized to combinatorial vector fields and used to derive the Morse inequalities. We will then define the discrete version of a flow generated by a vector field and discuss the relationship of Reidemeister torsion to these flows. This talk is based on two papers by Robin Forman: Combinatorial Vector Fields and Dynamical Systems; and Morse Theory for Cell Complexes.

Schmidt’s game, nondense orbits, and the set of badly approximable systems of linear forms

Ryan Broderick (Northwestern)

Abstract: We will discuss an infinite game which produces a class of full-dimension sets that is closed under countable intersection and under taking images by C^1 diffeomorphisms. In particular, the countable intersection of C^1 diffeomorphic images of such a set meets any sufficiently regular fractal in a set of full Hausdorff dimension. We present joint work with L. Fishman and D. Simmons, where we have shown that, for any surjective endomorphism of a torus, the set of nondense orbits belongs to the above-mentioned class. We also show that the set of badly approximable systems of linear forms has this property. We will then discuss recent work with D. Kleinbock which uses the game, along with techniques from homogeneous dynamics, to produce complements to the above results concerning `uniform’ versions of the sets.

An introduction to coding theorems for quantum capacity

Gao Li (UIUC Math)