UIUC Dept. of Mathematics Seminar Calendar

      July 2012             August 2012           September 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  5  6  7             1  2  3  4                      1
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    2  3  4  5  6  7  8
 15 16 17 18 19 20 21   12 13 14 15 16 17 18    9 10 11 12 13 14 15
 22 23 24 25 26 27 28   19 20 21 22 23 24 25   16 17 18 19 20 21 22
 29 30 31               26 27 28 29 30 31      23 24 25 26 27 28 29
                                               30

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

Tuesday, January 22, 2013

Monday, January 28, 2013

Monday, February 4, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, February 4, 2013

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Ayşe Şahin (DePaul University)
Low complexity Alpern Lemmas for $\mathbb R^d$ actions.
Abstract: In recent work with B. Kra and A. Quas we showed that every ergodic, measure preserving $\mathbb R^d$ action has a section where the return times consist of $d+1$ rectangular times. This answers a question of Rudolph about the optimal complexity of return times. For $d=2$ using a detailed analysis of the set of invariant measures on tilings of $\mathbb R^2$ by two rectangles we show that this bound is optimal for actions with completely positive entropy. In the zero entropy category we show that there exist mixing $\mathbb R^2$ actions whose orbits can be tiled by 2 tiles. The talk will start by explaining how to interpret return times to a section in terms of tilings of the acting group.

Monday, February 11, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, February 11, 2013

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Albert Fisher (University of S�o Paulo, Brazil)
Ergodic theorems and fractal return-time structure for infinite measure.
Abstract: Certain infinite measure-preserving transformations and flows exhibit fractal-like return times; evidence for this is existence of a dimension for return times, and of an order-two ergodic theorem. Examples include horocycle flows for finitely generated Fuchsian groups of second type, and certain adic transformations and maps of the interval with an indifferent fixed point. We explain the overall framework and then focus on recent results with Marina Talet on renewal processes and indifferent fixed points.

Thursday, February 14, 2013

Logic Seminar
1:00 pm in 241 Altgeld Hall, Thursday, February 14, 2013

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Jeremy Avigad (Carnegie Mellon University)
Computability and uniformity in analysis
Abstract: Countless theorems of analysis assert the convergence of sequences of numbers, functions, or elements of an abstract space. Classical proofs often establish such results without providing explicit rates of convergence, and, in fact, it is often impossible to compute the limiting object or a rate of convergence from the given data. This results in the curious situation that a theorem may tell us that a sequence converges, but we have no way of knowing how fast it converges, or what it converges to. On the positive side, it is often possible to "mine" quantitative and computational information from a convergence theorem, even when a rate of convergence is generally unavailable. Moreover, such information can often be surprisingly uniform in the data. In this talk, I will discuss examples that illustrate the kinds of information that can and cannot be obtained, focusing on results in ergodic theory.

Monday, February 18, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, February 18, 2013

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Mariusz Urbanski (University of North Texas)
Fine Inducing and Equilibrium Measures for Rational Functions of the Riemann Sphere
Abstract: Let $f:\mathbb{C}\to \mathbb{C}$ be an arbitrary holomorphic endomorphism of degree larger than $1$ of the Riemann sphere $\mathbb{C}$.. Denote by $J(f)$ its Julia set. Let $\varphi:J(f) \to\mathbb{R}$ be a H\"older continuous function whose topological pressure exceeds its supremum. It is known that then there exists a unique equilibrium measure $\mu_\varphi$ for this potential. I will discuss a special inducing scheme with fine recurrence properties. This construction allows us to prove three results. Dimension rigidity, i.e. a characterization of all maps and potentials for which $HD(\mu_\varphi)=HD(J(f))$. Real analyticity of topological pressure $P(t\varphi)$ as a function of $t$. Exponential decay of correlations, and, as its consequence, the Central Limit Theorem and the Law of Iterated Logarithm for H\"older continuous observables. Finally, the Law of Iterated Logarithm for all linear combinations of H\"older continuous observables and the function $\log|f'|$. Geometric consequences of the Law of Iterated Logarithm lead to comparison of equilibrium states with appropriately generalized Hausdorff measures on the Julia set $J(f)$.

Monday, February 25, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, February 25, 2013

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Marian Gidea (Northern Illinois University & Institute for Advanced Study)
Perturbations of geodesic flows producing unbounded growth of energy
Abstract: We consider a geodesic flow on a manifold endowed with some generic Riemannian metric. We couple the geodesic flow with a time-dependent potential driven by an external dynamical system, which is assumed to satisfy some recurrence condition. We prove that there exist orbits whose energy grows unboundedly at a linear rate with respect to time; this growth rate is optimal. In particular, we obtain unbounded growth of energy in the case when the external dynamical system is quasi-periodic, of rationally independent frequency vector (not necessarily Diophantine). Our result generalizes Mather's acceleration theorem and is related to Arnold's diffusion problem. It also extends some earlier results by Delshams, de la Llave and Seara

Monday, March 4, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, March 4, 2013

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Francesco Cellarosi (UIUC)
Autocorrelations in Quantum Mechanics and Homogenous Dynamics
Abstract: I will discuss an application of a result by J. Marklof and myself on the limiting distribution of theta sums to the the study of autocorrelations in certain quantum mechanical systems. I will show how, under quantum evolution, particles ''remember'' their initial profiles. Moreover, when studying the simultaneous evolution of two (or more) related particles, there are some ''forbidden regions'' for the joint autocorrelation functions. All the results are based on a result in homogeneous dynamics, namely equidistribution of horocycles under the action of the geodesic flow. The talk will be elementary and no knowledge of quantum mechanics / homogenous dynamics will be assumed.

Monday, March 11, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, March 11, 2013

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Maxim Arnold (UIUC)
Pinball dynamics
Abstract: Theory of small perturbations of completely integrable Hamiltonian systems has a long history. Summarily, one could say that Kolmogorov-Arnold-Moser theorem states that if perturbation is sufficiently smooth, then positive measure of invariant tori survives and thus in particular for planar area-preserving transformations there are no trajectories escaping to infinity. In some physical settings one deals with non-smooth perturbations where KAM-technique does not apply. However, there are no general instruments to work with such systems. I shall formulate natural family of such systems which covers many known examples and shall show how to construct an escaping trajectory in one representative example.

Wednesday, March 13, 2013

Monday, March 25, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, March 25, 2013

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Steven J. Miller (Williams College)
Benford's Law, Values of L-Functions and the 3x+1 Problem
Abstract: Many systems exhibit a digit bias. For example, the first digit (base 10) of the Fibonacci numbers or of $2^n$ equal 1 about 30% of the time; the IRS uses this digit bias to detect fraudulent corporate tax returns. This phenomenon, known as Benford's Law, was first noticed by observing which pages of log tables were most worn from age -- it's a good thing there were no calculators 100 years ago! The first digit of values of L-functions near the critical line also exhibit this bias. A similar bias exists (in a certain sense) for the first digit of terms in the 3x+1 problem, provided the base is not a power of two. For L-functions the main tool is the Log-Normal law; for $3x+1$ it is the rate of equidistribution of $n log_B 2 \mod 1$ and understanding the irrationality measure of $log_B 2$. This work is joint with Alex Kontorovich.

Monday, April 1, 2013

Tuesday, April 2, 2013

Monday, April 8, 2013

Thursday, April 11, 2013

Joint Logic - Ergodic Theory seminar
1:00 pm in 137 Henry Administration Building, Thursday, April 11, 2013

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Sergey Bezuglyi (Institute for Low Temperature Physics, Kharkov, Ukraine)
Homeomorphic measures on Cantor sets and dimension groups
Abstract: Two measures, m and m' on a topological space X are called homeomorphic if there is a self-homeomorphism f of X such that m(f(A)) = m'(A) for any Borel set A. The question when two Borel probability non-atomic measures are homeomorphic has a long history beginning with the work of Oxtoby and Ulam who gave a criterion when a probability Borel measure on the cube [0, 1]^n is homeomorphic to the Lebesgue measure. The situation is more interesting for measures on a Cantor set. There is no complete characterization of homeomorphic measures so far. But, for the class of the so called good measures (introduced by E. Akin), the answer is simple: two good measures are homeomorphic if and only if the sets of their values on clopen sets are the same. I will focus in the talk on the study of probability measures invariant with respect to a minimal (or aperiodic) homeomorphism. These measures are in one-to-one correspondence with traces on a corresponding dimension group. The technique of dimension groups allows us to apply new methods for studying good traces. A good trace is characterized by its kernel having dense image in the annihilating set of affine functions on the trace space. A number of examples with seemingly paradoxical properties is considered. The talk will be based on a joint paper with D. Handelman.

Monday, April 15, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, April 15, 2013

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Ilya Vinogradov (University of Bristol, U.K.)
The distribution of directions in an affine lattice: two-point correlations and mixed moments
Abstract: We consider an affine Euclidean lattice and record the directions of all lattice vectors of length at most T. Marklof and Strombergsson proved that the distribution of gaps between the lattice directions has a limit as T tends to infinity. For a typical affine lattice, the limiting gap distribution is universal and has a heavy tail; it differs markedly from the gap distribution observed in a Poisson process which is exponential. We show that the limiting two-point correlation function of the projected lattice points exists and is Poissonian, and answer a recent question of Boca, Popa, and Zaharescu [arXiv:1302.5067]. The existence of the limit is subject to a certain Diophantine condition. We also establish the convergence of more general mixed moments. Joint work with D. El-Baz and J. Marklof.

Monday, April 22, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, April 22, 2013

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Joseph Vandehey (UIUC)
Continued fractions on the Heisenberg group
Abstract: We will discuss some recent joint work with Anton Lukyanenko developing a theory of continued fractions on the Heisenberg group. These appear to be a very natural higher-dimensional analog of continued fractions on the real line; there are many formulas which have analogs on the Heisenberg group but which are missing from other higher-dimensional continued fraction algorithms (such as Jacobi-Perron). We will discuss the basic development of these continued fractions as well as difficulties associated with proving ergodicity of the associated shift map.

Monday, April 29, 2013

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, April 29, 2013

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Rachel Bayless (University of North Carolina, Chapel Hill)
Entropy of Transformations Preserving an Infinite Measure
Abstract: The entropy of a system measures the amount of information gained with each application of an experiment or transformation, and higher entropy corresponds to more disorder and less predictable systems. Classical measure-theoretic entropy is only well-defined for finite-measure-preserving transformations, and there is no universal analogue for transformations preserving an infinite measure. Three possible definitions have been given independently by Krengel (1967), Parry (1969), and Roy (2009). Although two of these definitions have been around for over 40 years, there exist very few examples where any of these entropies have been computed explicitly. In this talk we provide a method of computing the Krengel entropy for all conservative rational functions which preserve Lebesgue measure on the real line.