Seminar Calendar
for Ergodic Theory events the year of Thursday, September 20, 2012.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 17, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, January 17, 2012
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Submitted by jathreya.
 Bill Mance (Ohio State)Explicit constructions of $\mu$-normal numbersAbstract: While there are examples of normal numbers with respect to the b-ary expansions and continued fraction expansion, it has been an open problem to construct normal numbers with respect to many other series expansions. In joint work with Manfred Madritsch we obtain a result that admits explicit constructions of numbers normal with respect to the continued fraction expansion, Luroth series expansion, b-ary expansions, and others.

Thursday, January 26, 2012

5:00 pm   in 241 Altgeld Hall,  Thursday, January 26, 2012
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Submitted by funk3.
 Austin Rochford (UIUC Math)Amenability Across AnalysisAbstract: Part one of a two part talk. We will define amenable groups and explore generalizations of amenability to various areas of analysis. This week we will consider harmonic analysis and ergodic theory.

Tuesday, February 14, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, February 14, 2012
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Submitted by jathreya.
 Albert Fisher (University of Sao Paolo)Invariant measures for adic transformations on Bratteli diagrams Abstract: Vershik's adic transformations are a class of combinatorially defined maps that can be used to topologically and measure-theoretically model a wide variety of dynamical systems, including substitution dynamical systems, cutting-and-stacking constructions and interval exchange transformations. They act on the path space of a Bratteli diagram, defined by a sequence of nonnegative integer matrices and so generalizing subshifts of finite type to nonstationary combinatorics. Let us say a matrix sequence is primitive if for all $k$ there exists $n>k$ such that $M_k... M_n> 0$. Primitivity implies minimality (that every orbit is dense) for adic transformations, but in the nonstationary case, unique ergodicity (that there is a unique unvariant probability measure) does not always follow. In recent work we classify the invariant Borel measures which are finite positive on the path space of some sub-Bratteli diagram, for the bounded alphabet but not necessarily primitive case. This includes some interesting (and naturally occuring) measures which are infinite on every open subset. Our results extend theorems of Bezuglyi, Kwiatkowski, Medynets and Solomyak. This is joint work with Marina Talet, Universite de Provence.

Tuesday, February 21, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, February 21, 2012
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Submitted by jathreya.
 David Aulicino (Maryland)Teichmueller Disks with Completely Degenerate Kontsevich-Zorich SpectrumAbstract: The moduli space of genus $g$ Riemann surfaces is the space of all complex structures on a closed orientable surface of genus $g$ up to orientation preserving diffeomorphisms. The Teichmueller geodesic flow is the flow on the cotangent bundle of the Teichmueller space of surfaces defined by the direction of minimal dilatation and it descends to the cotangent bundle of the moduli space under the action of the mapping class group. It is well-known that the Lyapunov spectrum of this flow is determined by $g$ numbers $$1 = \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_g \geq 0.$$ The Kontsevich-Zorich conjecture, proven by Forni and Avila-Viana, showed that generically all the inequalities are strict with respect to the canonical absolutely continuous measures. However, Forni found an example of a measure on the genus three moduli space, and Forni-Matheus found a measure in genus four, with completely degenerate spectrum, i.e. $$1 = \lambda_1 > \lambda_2 = \cdots = \lambda_g = 0.$$ We prove that these are the only such measures in genus three and four. Furthermore, there are no such measures for $g=2$ and $g \geq 13$. Finally, if there are no square-tiled surfaces in genus five that determine a measure with completely degenerate spectrum, then there are no examples for $g \geq 5$.

Tuesday, February 28, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, February 28, 2012
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Submitted by jathreya.
 Francois Ledrappier (Notre Dame)Entropy rigidity for non-positively curved compact manifolds.Abstract: We consider different asymptotic rates related to the geometry of the universal cover of a compact manifold. We discuss relations between these rates, a characterization of symmetric spaces of non-positive curvature and related problems.

Thursday, March 1, 2012

5:00 pm   in 147 Altgeld Hall,  Thursday, March 1, 2012
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Submitted by funk3.
 Kelly Funk (Department of Mathematics, University of Illinois at Urbana-Champaign)Rigidity Across DynamicsAbstract: We will discuss examples of rigidity and uniform rigidity sequences in ergodic theory. We will also discuss the problem of characterizing these sequences.

Tuesday, March 6, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by jathreya.
 Ilya Gekhtman (University of Chicago)Dynamics of Convex Cocompact Subgroups of Mapping Class GroupsAbstract: Convex cocompact subgroups of mapping class groups are subgroups of the mapping class group whose orbits in Teichmueller space are quasi-convex. We develop an analogue of Patterson-Sullivan theory for the action of subgroups G of Mod(S) on Teichmuller space and its boundary the space of projective measured foliations and use it to compute multiplicative asymptotics for the number of orbit points of G in a ball of radius R in Teichmueller space and the number of pseudo-Anosovs in G with dilatation at most R.

Thursday, March 8, 2012

Joint Group Theory/Differential Geometry/Ergodic Theory
1:00 pm   in 347 Altgeld Hall,  Thursday, March 8, 2012
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Submitted by jathreya.
 Alex Wright (University of Chicago)Arithmetic and Non-Arithmetic Teichmüller CurvesAbstract: Teichmüller curves are isometrically immersed curves in the moduli space of Riemann surfaces. Their study lies at the intersection of dynamics, Teichmüller theory, and algebraic geometry. I will begin by summarizing known results on Teichmüller curves, pointing out some similarities to the study of lattices, for example in PU(n,1). I will then move on to new research involving abelian square-tiled surfaces, Schwarz triangle mappings, and the Veech-Ward-Bouw-Moller Teichmüller curves.

Tuesday, March 13, 2012

Graph Theory and Combinatorics
3:00 pm   in 241 Altgeld Hall,  Tuesday, March 13, 2012
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Submitted by west.
 Jeffrey Paul Wheeler (University of Pittsburgh)The Polynomial Method of Alon, Ruzsa, and NathansonAbstract: We will explore a particular method of tackling problems in Additive Combinatorics, namely the Polynomial Method of Noga Alon, Imre Ruzsa, and Melvyn Nathanson.  Additive Combinatorics can be described as the study of additive structures of sets.   This area is attractive in that it has numerous connections with other areas of mathematics, including Number Theory, Ergodic Theory, Graph Theory, Finite Geometry, and Group Theory and has drawn the attention of many good mathematicians, including Fields Medalist Terence Tao (2006).

Tuesday, March 27, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, March 27, 2012
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Submitted by jathreya.
 Anish Ghosh (University of East Anglia)Measure Rigidity for torus actionsAbstract: Recent years have seen major developments in the theory of measure rigidity for group actions on homogeneous spaces. I will explain some of these developments and present a few recent advances.

Tuesday, April 3, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, April 3, 2012
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Submitted by jathreya.
 Joshua Bowman (Stony Brook)Basins of infinity for polynomial maps of $\mathbb{C}^2$Abstract: Basins of attraction in holomorphic dynamics are well understood in dimension 1, much less so in higher dimensions. We will consider regular polynomial maps of $\mathbb{C}^2$ (maps which extend to endomorphisms of $\mathbb{P}^2$) and describe some tools for studying their basins of infinity. We show that there exist endomorphisms of $\mathbb{P}^2$ whose basins of infinity have infinitely generated second homology.

Tuesday, April 10, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, April 10, 2012
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Submitted by jathreya.
 Maxim Arnold (Department of Mathematics, University of Illinois at Urbana-Champaign)On the identical configurations of abelian sandpile model.Abstract: The Abelian sandpile model was introduced by Bak in the attempt to describe avalanche formations. During the next two decades it was intensively studied and many connections to the similar models was discovered. In particular the set of most often configurations of ASM can be considered as a representation of very natural abelian group. I shall shortly introduce the dynamics corresponding to ASM and state some results concerning identity of this group acting on Sierpinski graph.

Tuesday, April 17, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, April 17, 2012
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Submitted by jathreya.
 Slawomir Solecki (Department of Mathematics, University of Illinois at Urbana-Champaign)Point realizations of Boolean actionsAbstract: We will look at measure preserving Boolean actions of Polish groups and consider the problem, going back to Mackey, of determining when such actions have point realizations. We will explore the boundary line between the groups whose all Boolean actions have point realizability and those that do not have this property. One result, joint with Kwiatkowska, states that Boolean action of Polish groups of isometries of locally compact separable metric spaces can always be point realized. On the other hand, a very recent result with Moore, states that the group of all continuous functions from an uncountable compact space to the circle does not have the point realizability property. In several respects, this last group is quite different from other groups that were shown earlier, by Vershik, Becker and Glasner-Weiss, not to have the point realizability property. Connections with the solution to Hilbert's 5-th problem, with the concentration of measure phenomena, and with the Cameron-Martin theorem will be mentioned.

Tuesday, April 24, 2012

Joint Ergodic Theory/Number Theory Seminar
11:00 am   in 347 Altgeld Hall,  Tuesday, April 24, 2012
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Submitted by jathreya.
 Francesco Cellarosi (IAS/MSRI)Ergodic Properties of Square-Free NumbersAbstract: We study binary and multiple correlations for the set of square-free numbers and we construct a dynamical systems naturally associated to them. We prove that such dynamical system has pure point spectrum and it is therefore isomorphic to a translation on a compact abelian group. In particular, the system is ergodic but not weakly mixing, and it has zero metric entropy. The latter results were announced recently by Peter Sarnak and our approach provides an alternative approach. Joint work with Yakov Sinai.

Monday, September 10, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, September 10, 2012
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Submitted by fcellaro.
 Francesco Cellarosi   [email] (UIUC)Statistical Mechanics of k-free Numbers and Smooth Sums Estimates. Abstract: I will present a generalization of a previous work by Ya.G. Sinai and myself concerning certain sparse sets of k-free integers, equipped with a complex measure. After rescaling, it turns out that the distribution of these integers is approximated by self-convolutions of the Dickman-De Bruijn distribution. Using the language of Statistical Mechanics, this result gives a thermodynamical limit for our ensembles. As an application, we get asymptotic estimates of certain smooth sums over smooth k-free integers.

Monday, September 17, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, September 17, 2012
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Submitted by fcellaro.
 Andrew Parrish (UIUC)Convergence of Sparse Subset Averages of $L^1$ Functions.Abstract: The behavior of time averages when taken along subsets of the integers is a central question in subsequence ergodic theory. The existence of transference principles enables us to talk about the convergence of averaging operators in a universal sense; we say that a sequence $\{a_n\}$ is universally pointwise good for $L^1$, for example, if the sequence of averages \begin{equation*} \frac{1}{N} \sum_{n=0}^{N-1} f \circ \tau^{-a_n}(x) \end{equation*} converges a.e. for any $f\in L^1$ for every aperiodic measure preserving system $(X, \mathcal{B}, \mu, \tau)$. Only a few methods of constructing a sparse sequence that is universally pointwise $L^1$-good are known. We will discuss how one can construct families of sets in $\mathbb{Z}^d$ which are analogues of these sequences, as well as some challenges and advantages presented by these higher-dimensional averages. Joint work with P. LaVictoire (University of Wisconsin, Madison) and J. Rosenblatt (UIUC).

Monday, September 24, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, September 24, 2012
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Submitted by jathreya.
 Joe Rosenblatt (University of Illinois, Department of Mathematics)Directional Behavior for Two-variable Commuting ActionsAbstract: Directional ergodicity and directional weak mixing of the action of two commuting transformations S and T can be analyzed by looking at extensions in which S and T are embedded in a two real variable flow. For a suitable class of extensions, the directional behavior observed is determined not by the extension itself, but by intrinsic spectral properties of the original action by S and T.

Monday, October 1, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, October 1, 2012
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Submitted by fcellaro.
 Joseph Vandehey (UIUC)When T-normality equals S-normalityAbstract: Determining whether a point is normal for a given transformation T is a very difficult problem, yet there are several results that state that being T-normal is the same as being S-normal for two different transformations T and S. For example if a number is base-2-normal then it must also be base-4-normal, and vice-versa. We will present some new investigations into this topic, including a relation between normality for regular continued fractions and odd continued fractions. This will be an informal talk based on work in progress.

Wednesday, October 3, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm   in 245 Altgeld Hall,  Wednesday, October 3, 2012
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Submitted by laugesen.
 Florin Boca   [email] (UIUC Math)Angular distribution of lattices points and related geometric probability problemsAbstract: We will discuss some problems arising from the study of various discrete periodic configurations of points in Euclidean or Hyperbolic spaces. The associated distribution of angles is of particular interest to us. Ideas and methods from Number Theory, Ergodic Theory and Dynamical Systems that play an important role in the study of this kind of problems will be outlined.

Monday, October 8, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, October 8, 2012
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Submitted by fcellaro.
 Joseph Rosenblatt (UIUC)Optimal Norm Approximation in Ergodic TheoryAbstract: Classical ergodic averages give norm approximations but these averages are usually not the best one can do among all possible averages. We consider what the optimal approximation can be in terms of the transformation and the function.

Monday, October 15, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, October 15, 2012
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Submitted by fcellaro.
 Scott Kaschner (IUPUI)Superstable Manifolds of Invariant Circles and Co-dimension 1 Böttcher FunctionsAbstract: Let $f:X\dashrightarrow X$ be a dominant meromorphic self-map, where $X$ is a compact connected Hermitian manifold of dimension $n > 1$. Suppose there is an embedded copy of $\mathbb P^1$ that is invariant under $f$, with $f$ holomorphic and transversally superattracting with degree $a$ in some neighborhood. Suppose also that f restricted to this line is given by $z\rightarrow z^b$, with resulting invariant circle $S$. The regularity of the local stable manifold $\mathcal W^s_{\scriptsize{loc}}(S)$ is dependent on $a$ and $b$. Specifically, I will show that when $a\geq b$, $\mathcal W^s_{\scriptsize{loc}}(S)$ is real analytic, and the condition $a\geq b$ cannot be relaxed without adding additional hypotheses.

Monday, October 22, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, October 22, 2012
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Submitted by fcellaro.
 Francesco Di Plinio (IUB)A discrete model for the Hilbert transform along a smooth vector field in the plane.Abstract: Given a Calderon-Zygmund convolution kernel on R, we study the correspondent maximal directional singular integral T_V along directions in a finite set V with N elements. This operator can be regarded as a discrete version of the Hilbert transform along a planar vector field, the object of a conjecture of Stein, which in turn is related to differentiation along smooth vector fields. We are interested in the sharp dependence on N of the L^p and weak L^2 norms of T_V. We prove sharp bounds for both lacunary and Vargas sets of directions. The former case answers a question posed by M. Lacey. The latter includes uniformly distributed directions and the finite truncations of the Cantor set. This partially answers a conjecture of J. Kim. We make use of both classical harmonic analysis methods and new product-BMO based time-frequency analysis techniques, which could further prove useful in the study of multilinear multiparameter operators with modulation symmetry and of the conjectures of Zygmund and Stein. Joint work with Ciprian Demeter.

Monday, October 29, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, October 29, 2012
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Submitted by fcellaro.
 Javier Parcet (Instituto de Ciencias Matemáticas, Madrid)Twisted Hilbert Transforms and Classical Harmonic Analysis

Monday, November 5, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, November 5, 2012
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Submitted by fcellaro.
 Van Cyr (Northwestern)Nonexpansive $\mathbb Z^2$ subdynamics and Nivat’s conjectureAbstract: For a finite alphabet $A$ and $\eta: \mathbb Z\to A$, the Morse-Hedlund Theorem states that $\eta$ is periodic if and only if there exists $n\in\mathbb N$ such that the block complexity function $P_\eta(n)$ satisfies $P_\eta(n)\leq n$. In dimension two, a conjecture of M. Nivat states that if there exist $n, k\in\mathbb N$ such that the $n\times k$ rectangular complexity function, $P_\eta (n, k)$, satisfies $P_\eta(n, k)\leq nk$, then $\eta$ is periodic. There have been a number of attempts to prove Nivat’s conjecture over the past 15 years, but the problem has proven difficult. In this talk I will discuss recent joint work with B. Kra in which we associate a $\mathbb Z^2$-dynamical system with $\eta$ and show that if there exist $n,k\in\mathbb N$ such that $P_\eta(n,k)\leq nk$, then the periodicity of $\eta$ is equivalent to a statement about the expansive subspaces of this action. The main result is a weak form of Nivat’s conjecture: if there exist $n, k\in\mathbb N$ such that $P_\eta(n,k)\leq \frac{1}{2}nk$, then $\eta$ is periodic.

Monday, November 12, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, November 12, 2012
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Submitted by fcellaro.
 Yiannis Konstantoulas (UIUC)Exponential decay for multiple correlations of $SL(n,\mathbb R)$ actions.Abstract: Mozes's theorem implies that a measure preserving, mixing $SL(n,\mathbb R)$ action on a probability space $X$ is mixing of all orders. In this work we study multiple correlation integrals for $n\geq 3$ and prove explicit rates of convergence to the product of the integrals for good spaces of functions on $X$; the results do not depend on Mozes's theorem and thus provide an independent proof of it for $SL(n,\mathbb R)$ actions.

Monday, November 26, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, November 26, 2012
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Submitted by fcellaro.
 Jayadev Athreya (UIUC)Counting special trajectories for right-angled billiards and pillowcase covers, I.Abstract: In joint work with A. Eskin and A. Zorich, we derive pointwise weak quadratic asymptotics for counting special trajectories for billiards in polygons whose angles are integer multiples of 90 degrees. We describe a proof using ergodic theory on the moduli space of meromorphic quadratic differentials on CP. We will also start to describe how to explicitly compute the constants in our formulas by computing volumes of moduli spaces, which will be continued in our AGC talk on Tuesday.

Tuesday, November 27, 2012

Algebra, Geometry and Combinatorics
2:00 pm   in 345 Altgeld Hall,  Tuesday, November 27, 2012
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Submitted by bwyser.
 Jayadev Athreya   [email] (UIUC Math)Counting special trajectories for right-angled billiards and pillowcase covers, II.Abstract: In joint work with A. Eskin and A. Zorich, we compute volumes of moduli spaces of meromorphic quadratic differentials on CP, via enumerating pillowcase covers. One motivation comes from understanding the (weak) quadratic asymptotics for counting special trajectories for billiards in polygons whose angles are integer multiples of 90 degrees. This talk is a continuation of, but will be independent from, my Ergodic Theory seminar on Monday. In particular, you do not need to have attended the Ergodic Theory talk to understand this talk.

Monday, December 3, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, December 3, 2012
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Submitted by fcellaro.
 Slawomir Solecki (UIUC)Closed subgroups generated by generic measure automorphismsAbstract: I will show that for a generic measure preserving transformation $T$, the closed group generated by $T$ is a continuous homomorphic image of a closed linear subspace of $L_0(\lambda, {\mathbb R})$, where $\lambda$ is Lebesgue measure, and that the closed group generated by $T$ contains an increasing sequence of finite dimensional toruses whose union is dense. These results strengthen earlier results by de la Rue, de Sam Lazaro and Ageev, and are related to a conjecture by Glasner and Weiss. I will survey earlier work done on closed subgroups generated by generic measure automorphisms.

Monday, December 10, 2012

Ergodic Theory
4:00 pm   in 241 Altgeld Hall,  Monday, December 10, 2012
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Submitted by fcellaro.
 Kelly Yancey   [email] (UIUC)On Rigid HomeomorphismsAbstract: We will start by discussing the history of generic results in the setting of measure-preserving automorphisms in ergodic theory. Then we will specifically discuss weakly mixing homeomorphisms that are uniformly rigid and give generic type results for this class of homeomorphisms defined on the two torus and the Klein bottle. We will end with a discussion of spaces which do not admit weakly mixing, uniformly rigid homeomorphisms.