The Dynamics Seminar is currently on hiatus.
Previous Talks:
2/26/15
SPEAKER: Azer Akhmedov
TITLE: Soficity, amenability and C* algebras (Part III)
2/19/15
SPEAKER: Azer Akhmedov
TITLE: Soficity, amenability and C* algebras (Part II)
2/12/15
SPEAKER: Azer Akhmedov
TITLE: Soficity, amenability and C* algebras (Part I)
ABSTRACT. In the series of 3 talks, I plan to survey some recent very exciting developments in the areas of interactions in between group theory, C* algebras and dynamics. As a follow up to Ben Duncan's talks from the last December, we will see examples of three way interactions in between these areas, e.g. how a problem in group theory can be reformulated in the language of C* algebras and then tackled by the tools from dynamics (as opposed to more traditional and direct interactions in between group theory and dynamics).
The first talk is a gentle and brief (but self contained) introduction into the theory of sofic groups and amenable groups. I will mention several important problems in these areas.
2/5/15
SPEAKER: Mike Cohen
TITLE: Topics in large-scale geometry
ABSTRACT: This talk is a follow-up to a series of talks I gave last fall. I'll remind the audience of recent advancements in the study of large-scale geometry of metrizable topological groups, and I'll give a brief survey of an ongoing project of mine to classify the large-scale geometry of certain large diffeomorphism groups. I'll describe what I know and what I don't know, and mention some ideas for trajectories for future research, including some possible connections with ergodic theory.
12/10/14
SPEAKER: Ben Duncan
TITLE: Algebras associated to dynamical systems (Part II)
12/3/14
SPEAKER: Ben Duncan
TITLE: Algebras associated to dynamical systems (Part I)
ABSTRACT: In the first talk we will consider a compact Hausdorff space X and a continuous self map f: X-> X. To the pair (X,f) we will associate an operator algebra and consider the question of how the algebra encodes the dynamics up to conjugacy. We will then consider (into the second talk) some generalizations of this and added complications/questions that the extended constructions inherit. The first talk will be fairly concrete, the second talk is more likely to be speculative/preliminary.
11/19/14
SPEAKER: Liz Sattler
TITLE: Sub-Fractals, Sub-Shifts and Entropy (Part II)
11/12/14
SPEAKER: Liz Sattler
TITLE: Sub-Fractals, Sub-Shifts and Entropy (Part I)
ABSTRACT: We will continue our discussion on calculating the Hausdorff dimension of sub-fractals. We will introduce entropy and discuss the calculation of entropy of sub-shifts of finite type (associated with a sub-fractal). We will explore different methods for calculating entropy, including the use of linear recurrence relationships and the zeta function.
10/29/14
SPEAKER: Liz Sattler
TITLE: Structure of Sub-Fractals (Part II)
10/22/14
SPEAKER: Liz Sattler
TITLE: Structure of Sub-Fractals (Part I)
ABSTRACT: We will define sub-fractals and relate the set with a sub-shift on an appropriate symbolic space. We will discuss existing results for estimating the Hausdorff dimension of sub-fractals constructed with a hyperbolic iterated function system, and the connections with sub-shifts of finite type on a symbolic space. We will work with a few examples to demonstrate these results and explore other type of sub-fractals associated with different sub-shifts on the symbolic space.
10/8/14
SPEAKER: Azer Akhmedov
TITLE: Isometric Embeddings of Trees into Banach Spaces
ABSTRACT: We will concentrate on the questions about embedding finite (infinite) trees isometrically (quasi-isometrically) into various Banach spaces. It turns out, for many Banach spaces (e.g. Hilbert spaces), one cannot embed rooted binary trees strictly isometrically. I will present a proof of J.Bourgain's result about the distortion of embeddings of finite trees into uniformly convex Banach spaces. (this very nice and short proof is due to B.Kloeckner). Bourgain's theorem states that in any embedding of a finite rooted binary tree into a uniformly convex Banach space, the distances get distorted by at least a factor of C(n,\delta) where n is the size of the tree and \delta represents certain convexity coefficient of the Banach space. We will derive a lower bound for C(n,\delta ). As a consequence of this, we will see that non-Abelian free groups do not embed quasi-isometrically into certain (large class of) Banach spaces.
10/1/14
SPEAKER: Azer Akhmedov
TITLE: Large Scale Geometry of Banach Spaces
ABSTRACT: I will continue the topic from the previous talk (by Mike Cohen) discussing the following questions on the geometry of Banach spaces:
Q1. Can a small group act by isometries on a large space with a dense orbit? (e.g. can a finitely generated solvable group act isometrically on an infinite-dimensional separable Hilbert space with a dense orbit?)
Q2: Can one embed a finitely generated negatively curved group (e.g. a non-Abelian free group) quasi-isometrically into an infinite-dimensional Banach space?
Question 1 has been raised by Y.Shalom in 1990s, and has been answered positively by A.Valette et al. a decade later. Question 2 essentially dates back to P.Enflo and J.Bourgain; the latter proved spectacular results in both positive and negative directions.
9/24/14
SPEAKER: Mike Cohen
TITLE: On large-scale geometry for non-locally compact metrizable groups (Part II)
9/17/14
SPEAKER: Mike Cohen
TITLE: On large-scale geometry for non-locally compact metrizable groups (Part I)
ABSTRACT: In geometric group theory one studies the quasi-isometry class of a finitely generated (countable) group G, as determined by the left-invariant word metric on G induced by some choice of finite generating set. This quasi-isometry class is determined uniquely regardless of the choice of generating set. If G is instead a locally compact Polish group, and compactly generated, then similarly the word metrics induced by any given compact generating set are mutually quasi-isometric, and thus a unique class is determined. But in the case of "massive" non-locally compact Polish groups, which are never compactly generated, is there a well-defined notion of quasi-isometry class? In this expository talk, I'll discuss recent work of C. Rosendal which identifies a canonical quasi-isometry class for a large family of non-locally compact groups. I'll give some examples and mention a number of open problems, with an emphasis on groups of homeomorphisms and diffeomorphisms of compact orientable manifolds.
7/16/14
SPEAKER: Anne Kerian
TITLE: The crosscap number of a knot
ABSTRACT: This talk is an introduction to the study of non-orientable spanning surfaces of knots and crosscap number. It will include a comparison with the well developed theory of orientable spanning surfaces and a brief summary of known results about crosscap number.
7/2/14
SPEAKER: Azer Akhmedov
TITLE: A geometric classification of subgroups of PL(I)
ABSTRACT: We will first review a result of C.Bleak which says that a subgroup of PL(I) is non-solvable iff it admits an infinite tower; some striking algebraic applications of this result will be discussed. Then, we will examine how much of the above geometric characterization still holds in Homeo(I).
6/18/14
SPEAKER: Mike Cohen
TITLE: On Polishability and subgroups of Homeo_+[0,1] (Part II)
6/11/14
SPEAKER: Mike Cohen
TITLE: On Polishability and subgroups of Homeo_+[0,1] (Part II)
5/6/14
SPEAKER: Azer Akhmedov
TITLE. Some Applications of Hölder's Theorem.
ABSTRACT. Hölder's Theorem states that if a group acts freely by orientation preserving homeomorphisms of the line then it is Abelian. I'll present several bright applications (some known and some new) of this theorem in dynamics and group theory. The new results are from a joint work with Michael Cohen.
4/22/14
SPEAKER. Mike Cohen
TITLE. On Polishability and subgroups of Homeo_+[0,1] (Part I)
ABSTRACT. A group G is called Polish if it has a separable complete metric topology which makes the group operations continuous. A Borel subgroup H of a Polish group G is called Polishable if H can be made into a Polish group in a way which realizes its Borel structure inherited from G. I'll give some examples of Polishable and non-Polishable groups, and discuss the notion of Polishability as it pertains to subgroups of Homeo^+[0,1], the increasing self-homeomorphism group of the interval.
4/8/14
SPEAKER. Azer Akhmedov
TITLE. On algebraic properties of PL(I) derived from dynamical properties. (Part II)
4/1/14
SPEAKER. Azer Akhmedov
TITLE. On algebraic properties of PL(I) derived from dynamical properties. (Part I)
ABSTRACT. In this talk, we will review some structural results of subgroups of PL(I) (viewed more like an analogue of the Lie group). In the second half of the talk, we will concentrate on properties relevant to M. Cohen's talk from last week.
3/25/14
SPEAKER. Mike Cohen
TITLE. Automatic Continuity of Homomorphisms
ABSTRACT. In this expositional talk, we'll introduce a class of groups which satisfy a certain purely algebraic condition: the normed-in-the-sense-of-Dudley groups, which include the group of integers and the free groups. Then we'll prove the beautiful and bizarre theorem of Dudley (1961), which states that if H is a complete metric group and G is a group which is normed-in-the-sense-of-Dudley, then every abstract homomorphism from H into G must be continuous, regardless of the choice of topology on G! This is an extreme example of automatic continuity phenomena for topological groups, and demonstrates that a group's algebraic structure can greatly restrict what choices of topology are available for the group.
3/11/14
SPEAKER. Brian Chapman
TITLE. Adic Transformations and the Pascal Adic (Part IV)
3/4/14
SPEAKER. Brian Chapman
TITLE. Adic Transformations and the Pascal Adic (Part III)
2/25/14
SPEAKER. Brian Chapman
TITLE. Adic Transformations and the Pascal Adic (Part II)
2/18/14
SPEAKER. Brian Chapman
TITLE. Adic Transformations and the Pascal Adic (Part I)
ABSTRACT. Adic transformations are dynamical systems on the space of infinite paths in certain infinite directed graphs. I will introduce these systems before talking specifically about the Pascal adic, whose underlying graph is related to Pascal's triangle. The structure of this graph will allow us to investigate various dynamical properties of the Pascal adic system, many of which are combinatorial in nature.
2/11/14
SPEAKER. Liz Sattler
TITLE. Hausdorff Dimension of Tree Fractals
ABSTRACT. Last semester, we talked about finding the Hausdorff dimension of specific types of fractals by using topological pressure and constructing a measure to satisfy a condition that gives a lower bound for the Hausdorff dimension. In this talk, we will apply some of the same techniques to find the Hausdorff dimension of the canopy of a specific tree fractal. We will look at variations of some tree fractals and discuss some of the problems are arise by slightly changing some of the maps that define the fractal.
12/13/13
SPEAKER. Liz Sattler
TITLE. Hausdorff Dimension of Other Fractals
ABSTRACT. Last week, we discussed finding the Hausdorff dimension of a very specific collection of fractals called cookie-cutter sets. This week, we will use the same techniques of defining topological pressure and constructing a measure to find bounds for the Hausdorff dimension of a specific type of iterated function system. We will also look at tree fractals and the "canopy" of tree fractals. We will discuss similar approaches and new problems with finding the Hausdorff dimension of the canopy of a tree fractal.
12/6/13
SPEAKER. Liz Sattler
TITLE. Hausdorff Dimension of Cookie-Cutter Sets
ABSTRACT. Calculating the Hausdorff dimension of fractals can be a challenging task. We know the Hausdorff dimension of some well-known fractals, such as the Cantor set and Sierpinski's triangle, but we cannot generalize the process of calculating the dimension easily. In this talk, we will focus on finding the dimension of cookie-cutter sets (which include Cantor-like sets and more) by using a version of topological pressure. We will discuss the problems that arise by trying to extend this process to tree fractals.
11/21/13
SPEAKER. Dogan Çömez
TITLE. A Brief Survey of Symbolic Dynamical Systems (Part II)
11/15/13
SPEAKER. Dogan Çömez
TITLE. A Brief Survey of Symbolic Dynamical Systems (Part I)
ABSTRACT. Symbolic dynamics has originated in attempts to study (topological, measurable, smooth) dynamical systems with as minimal structural requirements as possible. Although its beginning is this humble, it has developed into a branch of dynamical systems of its own. In this talk, I will provide some fundamental aspects of symbolic dynamics, give some important examples and discuss some recent developments in the field.
11/8/13
SPEAKER. Mike Cohen
TITLE. Borel Selectors
ABSTRACT. This is an expository talk in response to a question from two weeks ago. I will give a brief proof of a version of the Kuratowski/Ryll-Nardzewski selection theorem, which states that on any Polish space there is a Borel-measurable function s from the space of closed subsets of X to X such that s(F)\in F for every nonempty closed set F. As a consequence we will deduce that there exist Borel coset selectors for quotients of Polish groups by a closed subgroup.
11/1/13
SPEAKER. Mike Cohen
TITLE. A Short Theorem, and then a Research Problem (Part II)
10/25/13
SPEAKER. Mike Cohen
TITLE. A Short Theorem, and then a Research Problem (Part I)
ABSTRACT. First I will wrap up the short proof of a result from last week's talk. Afterward I will switch gears completely and propose a research problem. It is a topological groups problem but may be phrased entirely in infinite combinatorial terms. I will explain why I think it is approachable, why a solution would be publishable, and what progress I have made on it.
10/18/13
SPEAKER. Mike Cohen
TITLE. Measure Theory in Non-Locally Compact Groups (Part II)
10/11/13
SPEAKER. Mike Cohen
TITLE. Measure Theory in Non-Locally Compact Groups (Part I)
ABSTRACT. I'll define Polish topological groups (a pervasive object) and explain some of the major difficulties in extending the deep and well-understood theory of locally compact groups to the non-locally compact setting. In particular I'll give a brief proof that non-zero $\sigma$-finite left-invariant regular Borel measures (aka Haar measures) do not ever exist on a non-locally compact Polish group.
As a partial substitute for the missing measures, we will investigate the basic properties of the $\sigma$-ideal of Haar null sets as defined by Christensen. This family of sets gives a satisfactory measure-theoretic "smallness notion" for all Polish groups. I will discuss some open and some closed problems in this field of study.
This talk will serve partially as a detailed introduction to a research talk I plan to give later in the Analysis, Geometry, and Applied Math Seminar. It may expand into an additional lecture next week, or eventually morph into a related topic of a different flavor, depending on audience interest.
10/4/13
SPEAKER. Azer Akhmedov
TITLE. Extension of H\"older's Theorem (Part III)
9/27/13
SPEAKER. Azer Akhmedov
TITLE. Extension of H\"older's Theorem (Part II)
9/20/13
SPEAKER. Azer Akhmedov
TITLE. Extension of H\"older's Theorem (Part I)
ABSTRACT. It is a classical result (due to H\"older) that if G is a subgroup of Homeo(R) such that every nontrivial element acts freely then G is Abelian. A natural question to ask is what if every nontrivial element has at most N fixed points where N is a fixed natural number. In the case of N=1, we do have a complete answer to this question: it has been proved by Solodov, Barbot, and Kovacevic that in this case the group is metaabelian, in fact, it is isomorphic to a subgroup of the affine group Aff(R).
We answer this question for an arbitrary N assuming some regularity on the action of the group. Our main result is the following theorem.
Theorem. Let G be a subgroup of Diff ^{1+\epsilon }(I) such that every nontrivial element of G has at most N fixed points. Then G is solvable. If in addition G is a subgroup of Diff ^{2}(I) then we can claim that G is metaabelian.
I will discuss major ideas of the proof. One of the main ingredients is a result about dynamical transitivity of smooth actions. I'll mention (and briefly discuss) several related open problems from the area of smooth dynamics.