UIUC Dept. of Mathematics Seminar Calendar

     August 2012           September 2012          October 2012    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
           1  2  3  4                      1       1  2  3  4  5  6
  5  6  7  8  9 10 11    2  3  4  5  6  7  8    7  8  9 10 11 12 13
 12 13 14 15 16 17 18    9 10 11 12 13 14 15   14 15 16 17 18 19 20
 19 20 21 22 23 24 25   16 17 18 19 20 21 22   21 22 23 24 25 26 27
 26 27 28 29 30 31      23 24 25 26 27 28 29   28 29 30 31         
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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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Bill Mance (Ohio State)
Explicit constructions of $\mu$-normal numbers
Abstract: 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

Tuesday, February 14, 2012

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, February 14, 2012

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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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David Aulicino (Maryland)
Teichmueller Disks with Completely Degenerate Kontsevich-Zorich Spectrum
Abstract: 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

Thursday, March 1, 2012

Tuesday, March 6, 2012

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, March 6, 2012

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Ilya Gekhtman (University of Chicago)
Dynamics of Convex Cocompact Subgroups of Mapping Class Groups
Abstract: 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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Alex Wright (University of Chicago)
Arithmetic and Non-Arithmetic Teichm�ller Curves
Abstract: Teichm�ller curves are isometrically immersed curves in the moduli space of Riemann surfaces. Their study lies at the intersection of dynamics, Teichm�ller theory, and algebraic geometry. I will begin by summarizing known results on Teichm�ller 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 Teichm�ller curves.

Tuesday, March 13, 2012

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, March 13, 2012

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Jeffrey Paul Wheeler (University of Pittsburgh)
The Polynomial Method of Alon, Ruzsa, and Nathanson
Abstract: 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

Tuesday, April 3, 2012

Tuesday, April 10, 2012

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, April 10, 2012

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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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Slawomir Solecki (Department of Mathematics, University of Illinois at Urbana-Champaign)
Point realizations of Boolean actions
Abstract: 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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Francesco Cellarosi (IAS/MSRI)
Ergodic Properties of Square-Free Numbers
Abstract: 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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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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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

Monday, October 1, 2012

Wednesday, October 3, 2012

Monday, October 8, 2012

Monday, October 15, 2012

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, October 15, 2012

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Scott Kaschner (IUPUI)
Superstable Manifolds of Invariant Circles and Co-dimension 1 B�ttcher Functions
Abstract: 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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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

Monday, November 5, 2012

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, November 5, 2012

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Van Cyr (Northwestern)
Nonexpansive $\mathbb Z^2$ subdynamics and Nivat�s conjecture
Abstract: 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

Monday, November 26, 2012

Tuesday, November 27, 2012

Algebra, Geometry and Combinatorics
2:00 pm in 345 Altgeld Hall, Tuesday, November 27, 2012

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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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Slawomir Solecki (UIUC)
Closed subgroups generated by generic measure automorphisms
Abstract: 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