Academic Year 2011-12
UF TOPOLOGY AND DYNAMICAL SYSTEMS SEMINAR
Fall 2011: Tuesdays 5th Period (11:45am - 12:35pm)
368 Little Hall
Spring 2012: Tuesdays 5th Period (11:45am - 12:35pm)
368 Little Hall
Schedule of Talks for Fall Semester
August 30
Organization meeting
September 13
Arkady Leiderman (University of Beer Sheva, Israel), Basic families of functions and embeddings of free locally convex spaces
Abstract. Let X be a completely regular topological space.
The free locally convex space on X is a locally convex space L(X)
for which X forms a Hamel basis and such that
every continuous mapping from X to a locally convex space E
extends uniquely to a continuous linear operator from L(X) to E.
In our talk we survey the results which are related to the following
problem:
characterize all topological spaces X such that
L(X) can be embedded into L[0,1] as a closed linear subspace,
where [0,1] is a usual unit segment.
September 20
Phil Boyland, Homological rotation sets for two-torus homeomorphisms I
September 27
Phil Boyland, Homological rotation sets for two-torus homeomorphisms II
October 4
Rob Newton, Category computations for connected sums I
Abstract. In this talk, I will use some known results on Lusternik-Schnirelmann
category and some homotopy properties to establish a fine set of bounds on
the LS category of a connected sum.
October 11
Rob Newton, Category computations for connected sums II
October 18
Jed Keesling, The Hilbert-Smith Conjecture: New techniques for an old problem Abstract. This talk will give the history of Hilbert's Fifth Problem and the
remarkable techniques and results that have arisen from work on the problem. There still remains
a significant portion of the problem that has not been solved. The techniques that we have developed
and work done will also be described.
October 25
Albert Fathi (Lyon), Smooth time functions for stably causal and non stably causal manifolds II
Abstract. We will give details about the topics presented in the Colloquium. In particular,
we will insist on the dynamical flavor of the whole setting, explaining connections with
Lyapunov functions and chain recurrence.
November 1
Lou Block, Topological conjugacy, transitivity, and patterns
Abstract. For a continuous map f of the compact interval to itself, let h(f) denote the topological entropy of
the map. We will recall some basic properties of topological entropy in the first part of the talk. For each
periodic orbit of the map, the entropy of the periodic orbit may also be defined. It is known that h(f) is the
supremum of the entropies of the periodic orbits of f. In this talk, we consider the following question. When is
the supremum a maximum? We answer this question in the case where f is topologically transitive. This talk is
based on joint work with Dennis Ledis.
November 8
Thanos Gentimis, On the cohomology of the Higson compactification of the hyperbolic spaces
Abstract. This talk has two parts. In the first part we will describe the $C_0$ coarse structure and the corresponding compactification on topological spaces, especially the hyperbolic space of dimension n. We will also introduce the idea of the $l_\infty$ Cohomology and some general properties of it. In the second part, we will define the Lipschitz obstruction theory and use it to show that $H^k(\overline{\mathbb H}^n$) is zero for all $k$,$n$ where the denoted compactification is the $C_0$ compactification. We will end with a brief history of the problem and a partial result for the standart Higson compactification.
November 15
Thanos Gentimis, On the cohomology of the Higson compactification of the hyperbolic spacesII
November 22
James Maissen, Invariant subsets of Peano continuua under free p-adic actions
Abstact. In my talk I will discuss and demonstrate that there is only one free action on the space of irrationals, criteria for lifting homotopies of maps from the quotient space under the free action, and how to realize invariant objects such as solenoids and Menger curves about arbitrary points of a Peano continum that has a p-adic group acting freely upon it (subject to a small, reasonable condition as to what sets can locally separate the continuum).
November 29
Sergii Kutsak, Essential manifolds with additional structures
Abstact. Let M be a closed, connected, orientable manifold of
dimension n and let π be the fundamental group of M. Let f:M → K(π,1) be
a classifying map for the universal cover M.
Gromov called a manifold M inessential if f cannot be
homotoped to the (n - 1)-skeleton of of K(π,1). M is called essential
otherwise. Given a class of manifolds C we denote by EC the subclass that consists
of rationally essential manifolds. We consider the
classes of algebraic manifolds, of Kahler
manifolds, of symplectic manifolds, of
symplectic manifolds with the hard Lefschetz property, and the
class of cohomologically symplectic manifolds.
For every class of manifolds C we denote by C(π,n) a subclass of
n-dimensional manifolds with fundamental group π. I will discuss the
following conjecture
proposed by Dranishnikov and Rudyak:
CONJECTURE. For any of the above classes for n > 2 the condition
EC(π,2n) ≠ ∅ implies that EC(π,2n-2) ≠ ∅.
We will prove for the above classes a weaker version of the conjecture
that deals with symplectically aspherical manifolds.
December 6
Sergii Kutsak, Essential manifolds with additional structures II
January 10
Organization meeting for spring semester.
January 17
Alexander Dranishnikov, Some classic formulae in dimension theory.
January 24
Alexander Dranishnikov, Some classic formulae in dimension theory. II
January 31
Jed Keesling, Hilbert space and the Hilbert-Smith Conjecture.
Abstract.
Hilbert's Fifth Problem still survives as the Hilbert-Smith Conjecture. The Hilbert-Smith Conjecture asks whether it is possible for a compact non-Lie group to act faithfully on a manifold. For dimension one and two the answer is known. For dimension three it has just been solved by John Pardon.
The talk will show some constructions that are possible in Hilbert space that could lead to a general solution to the problem. The constructions are interesting in themselves and can be applied to the space of homeomorphisms of a manifold.
February 7
Jed Keesling, Hilbert space and the Hilbert-Smith Conjecture. II
February 14
Michael Zarichnyi, Convex bodies of constant width and Eilenberg-MacLane spaces.
Abstract. A convex compact set X in a Euclidean space is said to be of constant width d, if X-X is the ball of diameter d centered at the origin. The Lie group S1 acts naturally on the hyperspace of convex bodies of constant width 1 inscribed into a unit square and we show that the orbit space of this action contains Eilenberg-MacLane spaces that are infinite-dimensional manifolds. The related hyperspaces of rotors, bodies of constant relative width, and rosettes of constant width are also considered.
February 21
Yuli Rudyak, The degree of map and Lusternik-Schnirelmann category.
Abstract. Abstract. In 1999 the author conjectured that, for a map of degree 1,
the LS category of the range does not exceed that of the domain.
We discuss this conjecture and, in particular, prove that
the conjecture holds in rational category.
February 28
Yuli Rudyak, The degree of map and Lusternik-Schnirelmann category II.
March 13
Phil Boyland, A parameterized Brown-Barge-Martin construction.
Abstract. Barge and Martin (1990) gave a construction of an embedding
of the inverse limit L of a map f on a spine S inside a manifold
M and a homeomorphism F on M so that L is a global attractor and
F restricted to L is the natural extension of f. The main ingredient
in the construction is M. Brown's (1960) much-cited theorem on the
inverse limit of a near-homeomorphism. The construction has been very
useful in generating interesting examples in dynamical systems.
We give a parameterized version which we use to make dynamically
interesting families of homeomorphisms of the two-torus. Because
our contribution is really just a remark, most of the lectures
will be exposition including Ancel's (1987) short, elegant proof
of Brown's Theorem which we adapt to the parametrized case.
March 20
Phil Boyland, A parameterized Brown-Barge-Martin construction. II
March 27
Tulsi Srinivasan, On the Lusternik-Schnirelmann category.
April 3
David Groisser, Some nice disgusting spaces. Abstract. The space of affine shapes of k points in Rn, Shkn,
is defined to be the quotient of (Rn)k by the diagonal action of
Aff(n), the group of invertible affine transformations of Rn. These
spaces, which arise naturally in several image-processing applications,
have very poor separation properties: they are T0 but not T1. This
leads to some unsettling and unappealing features: compact sets need not
be closed, and there is only one point that is a closed set.
Nonetheless, the affine-shape spaces have some very pretty topological
features. Each of these spaces naturally stratifies as a disjoint union
of Grassmannians. There is a natural ordering on the strata, but there
can be "dimensional anomalies", with a highest-dimensional stratum
occurring somewhere between the top and bottom strata. The strata are
attached to each other in a way that cannot be described through
closures alone; a non-T1 notion we call "blur", which is in some sense
dual to closure, is necessary as well. Grassmannians truly abound in
Shkn: the blur of a point is a disjoint union of Grassmannians (not
the same Grassmannians as the strata, but contained in them in
interesting ways); the closure of a point is a disjoint union of
Grassmannians; the intersection of the blurs of two point or the
closures of two points is a disjoint union of Grassmannians. We will
discuss as many of these features as time permits.
April 10
Kevin Knudson, Relative Malcev completions.
April 17
Ugur Abdulla, Sharkovski's Theorem and universality in chaos.
Abstract. Recently a fascinating order was revealed for the distribution of periodic
windows in chaotic regime for the discrete dynamical systems. In this talk
we attempt to analyze this phenomenon through a fine classification of
periodic orbits of continuous endomorphisms on the real line.
This page was last modified by J. Keesling on April 13, 2012.