Organization meeting

Lou Block, Topological Groups, Duality, and Solenoids


Abstract. We discuss the duality theory (due to Pontryagin and Van Kampen) for topological groups. We use this theory and a Theorem of Wladimiro Scheffer to prove a new result about solenoids. This talk is based on joint work with Dennis Ledis and Jed Keesling. The talk will be accessible to students familiar with the basic definitions of topology and group theory.

Yuli Rudyak, Topological Complexity and Configuration Spaces


Abstract. n the paper On Higher Analogs of Topological Complexity (January 2010) the author introduced the concept of higher topological complexity of a path-connected space. This concept was developed after Farber's notion of Topological Complexity (2003) related to a basic problem of robot motion planning algorithm We develop further the properties of the newly discovered concept, relating it to multiples of the Lusternik-Schnirelmann category and an analog of cup-length. We compute the numerical value of topological complexity for products of spheres, closed 1-connected symplectic manifolds (e.g. complex projective spaces), and quaternionic projective spaces.

In addition, we explore the symmetrized version of the topological complexity and introduce a new symmetrization which is a homotopy invariant. We obtain a strong upper bound for symmetrized version of the topological complexity for spheres. This is attained by introducing and studying a new concept---that of cellular stratified spaces. This concept allows us to import techniques from the theory of hyperplane arrangements in order to construct efficient simplicial complexes modelling (up to equivariant homotopy) configuration spaces of distinct points on spheres. .

Sergii Kutsak, Construction of an essential algebraic manifold with given fundamental group

Sasha Dranishnikov, On rationally essential manifolds

Ibai Basabe, Symmetric and high-dimensional analogues of Topological Complexity


Abstract. In practice the motion planning problem might be subject to a number of restrictions imposed by the particular problem under consideration. For instance, no motion at all might be required when the initial and final configurations agree (motion optimality). Another usual restriction is that the motion from configuration A to configuration B should be the time-reverse movement of that from B to A (symmetry). These and other restricted instances were studied in Farber and Grant's Symmetric motion planning, which is the first main reference for this talk. The speaker will also consider Iwase and Sakai's Topological complexity is a fibrewise L-S category, from where the optimality condition can be identified as the problem for having homotopy invariance. Another natural restriction in motion planning arises when repeated motion planning cycles are required, in each of which partial prescribed configurations should be attained (e.g. in an industrial manufacturing proces!

These types of restrictions were considered in Farber and Grant's Symmetric motion planning and Rudyak's On higher analogs of topological complexity. The latter reference introduces the concept of high-dimensional TC, which is the second main goal for this talk. The analysis of the symmetric version of TC requires a good working control of the algebraic topology of unordered configuration spaces B(X,n). When X is a manifold, the mod-2 cohomology of B(X,n) was first worked out in Haefliger's Points multiples dÕune application et produit cyclique rŽduit, and a summarized account is given in Farber and Grant's Symmetric motion planning. The ordered case essentially comes from Milnor and Stasheff's Characteristic classes, and serves as a convenient comparison point for the Totaro spectral sequence which computes the cohomology algebras of configuration spaces.

Jed Keesling, Semi-conjugacies and mappings on tori


Abstract. Suppose that f is a mapping of an n-torus to itself. If the associated endomorphism g is hyperbolic, then f is semi-conjugate to g. Suppose, on the other hand, that f and g are maps of an n-torus to itself with the endomorphism associated to g hyperbolic. Then if f and g have different degrees, then f cannot be conjugate to g. We will discuss these results and give some proofs.

Dennis Ledis, Conjugacies and semi-conjugacies of the tent map


Abstract. I will give criteria for a family of maps to be semi-conjugate to the tent map and find conditions for maps to be conjugate. I will talk about a few corollaries, some mentioned in Dr Keesling's talk last week.

Sasha Dranishnikov, On asymptotic dimension of Thompson's group

Dave Wilson, A topological approach to Iterated Function Systems


Abstract. The goal of this talk is to introduce the topic of iterated function systems (IFS) from a topological viewpoint. The ideas presented will be applied to three classes of mappings: affine, bilinear, and projective. These three classes of mappings are fundamental in Michael BarnsleyÕs fractal techniques for morphing digital pictures and images. Following the lead of Charles Conley and Mike Hurley, we begin the mathematical discussion by defining the terms topological contraction and topological attractor for an iterated function system. After indicating propositions which guarantee that maps in these classes will be topological contractions on certain convex bodies, we describe examples which show that care must be exercised for projective transformations. Since the idea of a coding map is used in BarnsleyÕs techniques, we define what it means for an attractor to have a coding map. We also indicate theorems that guarantee when an affine or projective IFS has an attractor with a coding map. Examples are given which show that a similar theorem for a bilinear IFS will be more complicated. Thoughts are provided to indicate how these ideas might relate to a Conley Index for and IFS and the shape of an IFS.

Dave Wilson, A topological approach to Iterated Function Systems II

Kevin Knudson, The Friedlander-Milnor Conjecture


Abstract. I will discuss the conjecture of E. Friedlander and J. Milnor concerning the cohomology of algebraic groups made discrete. Known cases will be explained along with recent work of F. Morel which reduces the conjecture to proving homotopy invariance in the homology of linear groups.

Jed Keesling, Extending topological group actions to compactifications

Sasha Dranishnikov, Cohomology of Higson compactifications


Abstract. This will be an introduction to Thanos' talk to be given in the near future. The subject now has a twenty year old history. Still there are many unanswered questions in the area. The progress is rather slow there, and every minor breakthrough is an event.

Schedule of Talks for Spring Semester

Jed Keesling, Extending topological group actions to compactifications II

James Maissen, Extending group actions


Abstract. Given a compact group acting on a metric space X. If C is a metric compactification of X such that each element of the group has a continuous extension then the group action extends continuously. The speaker will reiterate the example provided during the last talk by Professor Keesling demonstrating the need for the group to be compact and the compactification to be metric.

Yuli Rudyak, On homotopy dimension of configuration spaces

Sasha Dranishnikov, Cell-like maps and dimension

Dennis Ledis, Topological entropy of maps with fixed finite invariant sets


Abstract. Given a finite invariant set P, we may consider the piece-wise monotone map, L, that "connects the dots." It is known how to calculate entropy of such maps. I will present some background and some ideas about families of maps that are conjugate to L.

Jindra Zapletal, Borel equivalence relations


Abstract. This is an advertisement for a one-year sequence that I plan to offer beginning with Fall 2011. The theory of Borel equivalence relations experienced explosive growth in the last two decades. It compares various classification problems in mathematics according to their complexity, constructing their roadmap in the process. As a result, a number of classification programs were in effect terminated. I will explain the basics of the theory and outline the contents of the course.

Albert Fathi (Ecole Normale Superieure de Lyon), Smooth Lyapunov functions

Sasha Dranishnikov, On Dold-Thom's theorem


Abstract. The Dold-Thom theorem allows us to define the homology via homotopy groups. I'll give a sketch of an alternative proof of it.

Jed Keesling, Toeplitz sequences, adding machines, and extending topological group actions


Abstract. It is well-known that no subshift of the Bernoulli shift can be an adding machine. There are subshifts that can be close to adding machines by being semiconjugate to adding machines. These are created using Toeplitz sequences.

Lou Block and I gave a characterization of adding machines that has proven useful in identifying when the closure of an orbit is in fact an adding machine. It turns out that this is useful in determining when an adding machine action on a separable metric space can be extended to an action on a metric compactification of the space.

In this talk we will discuss details and give examples.

Jed Keesling, Toeplitz sequences, adding machines, and extending topological group actions II

Yuli Rudyak, Higher Topological Complexity of Closed Surfaces