Tuesday, October 15, 2013

12:00 pm
- 01:00 pm

DnA Seminar, Han Li (Yale): "Effective Discreteness of the 3 Dimensional Markov Spectrum"

Abstract: Let O denote the set of non-degenerate, indefinite, real quadratic forms in 3-variables. We define for every such quadratic form Q, the Markov infimum m(Q)=inf{|Q(v)|^3/|det(Q)|: v is a nonzero integral vector in R^3}. This normalization makes the infimum invariant after rescaling the quadratic form. The set M={m(Q): Q in O} is called the 3-dimensional Markov spectrum. An early result of Cassels-Swinnerton-Dyer combined with Margulis' proof of the Oppenheim conjecture asserts that M consists of rational numbers, and for every a>0 there are only finitely many numbers in M which are greater than a. In this lecture we will discuss an effective improvement of this result. This is an ongoing joint work with Prof. Margulis. The key ingredient is to study the compact orbits of the SO(2,1) action on SL(3, R)/SL(3, Z), and our method involves techniques from the geometry of numbers, dynamics on homogeneous spaces and automorphic representations.

ESC 638

Tuesday, November 27, 2012

12:00 pm
- 01:00 pm

DnA Seminar, Ron Blei - UCONN: "Measurements of randomness and interdependence"

Abstract: Interdependence and randomness can be calibrated by indices based separately on combinatorial measurements, p-variations, and tail-probability estimates. These notions had naturally originated in a context of harmonic analysis, and appeared later in stochastic settings. I intend in this talk to survey and explain these ideas, and (hopefully) also shed some new light on them. No formal proofs will be given. I will speak heuristically, but will try to be precise.

ESC 638

Tuesday, November 06, 2012

12:00 pm
- 01:00 pm

DnA Seminar, Jon Chaika - University of Chicago: "Diophantine approximation for interval exchange transformations"

Abstract: Interval exchange transformations (IETs) are invertible, piecewise order preserving isometries of the unit interval with finitely many discontinuities. They generalize rotations of the circle. Motivated by this connection several diophantine properties will be presented. In particular, this talk will answer the questions of typically how quickly does the orbit of a point under an IET approach another point or itself. This is joint work with M.<br/>Boshernitzan and D. Constantine.<br/>

ESC 638

Tuesday, October 30, 2012

12:00 pm
- 01:00 pm

DnA Seminar, Giulio Tiozzo - Harvard: "Renormalization and alpha-continued fractions"

Abstract: Alpha-continued fraction transformations are a one-parameter family of maps which arise from generalized continued fraction algorithms. The average speed of convergence of these algorithms (which corresponds to the entropy of the maps) varies wildly with the parameter, and is known to be locally monotone outside a closed fractal set E.<br/>Surprisingly, such a set has the same structure as the real slice of the Mandelbrot set, making it possible to apply ideas from complex dynamics to continued fractions: in this talk, we will investigate the self-similar structure of E and characterize the plateaux occurring in the graph of entropy.<br/>

ESC 638

Tuesday, October 23, 2012

12:00 pm
- 01:00 pm

DnA Seminar, Stephen Shea-St Anslem: "Topological Conjugacy on the Complement of the Periodic Points

Abstract: Let Per(X) denote the periodic points of a subshift X. I will say two subshifts X and Y are essentially conjugate if there exists a topological conjugacy from X\Per(X) to Y\Per(Y). In 1990, Susan Williams presented an example of a sofic shift that is not topologically conjugate to a renewal system. I will show that this example is essentially conjugate to a renewal system. I will also present an example of a renewal system that is essentially conjugate to a shift of finite type but not topologically conjugate to a shift of finite type. Lastly, we will provide a sufficient condition for a renewal system to be essentially conjugate to a shift of finite type.

ESC 638

Tuesday, October 09, 2012

12:00 pm
- 01:00 pm

DnA Seminar, David Constantine, Wesleyan: "Frame flow and rank-rigidity"

Abstract: I will describe the dynamics of the frame flow on a compact manifold of negative curvature, as well as some extensions to certain non-positively curved manifolds. I will then use this to prove that a compact manifold with "higher hyperbolic rank" is actually hyperbolic, under a few curvature assumptions. This fits into a family of rank-rigidity theorems/open problems.

ESC 638

Tuesday, November 15, 2011

12:00 pm
- 01:00 pm

DnA Seminar: The Mathematics of Iterated Paperfolding

Speaker: Michel Dekking, Delft, Wesleyan <br /><br/><br/>Abstract: I will present some old and some new results in a project which has been going on for 35 years. There are bits of algebra (representation theory), bits of topology (The Jordan curve theorem, Hausdorff metric), bits of graph theory (Euler's 1736 theorem), bits of number theory (Gaussian primes, Lvschian numbers), bits of theoretical computer science (automatic sequences) and bits of measure theory (Hausdorff dimension).<br/><br/>NOTE: This talk will be appropriate for undergraduates.<br/>

ESC 638

Tuesday, November 08, 2011

12:00 pm
- 01:00 pm

DnA Seminar: Sequences of Intergers Associated wtih Infinte Measure Preserving Transformations

Speaker: Yuji Ito<br/><br/>Abstract: Infinite measure preserving ergodic transformations cannot, because of ergodicity, admit a finite, equivalent, invariant measure. Because of this fact, there exist various kinds of infinite sequences of integers that one can associate with such transformations, which are isomorphism invariants. I will discuss some of these sequences, such as weakly wandering sequences, dissipative sequences, recurrent sequences and so on and describe some of their properties.

ESC 638

Tuesday, November 01, 2011

12:00 pm
- 01:00 pm

DnA Seminar: Sequences of Integers Associated With Infinite Measure Preserving Transformations

Speaker: Yuji Ito<br/><br/>Abstract: Infinnite measure preserving ergodic transformations cannot, because of ergodicity, admit a finite, equivalent, invariant measure. Because of this fact, there exist various kinds of infinite sequences of integers that one can associate with such transformations, which are isomorphism invariants. I will discuss some of these sequences, such as weakly wandering sequences, dissipative sequences, recurrent sequences and so on and describe some of their properties.

ESC 638

Tuesday, October 18, 2011

12:00 pm
- 01:00 pm

DnA Seminar: How fractal is the sum of two random fractals?

Speaker: Michel Dekking, Delft/Wesleyan<br/><br/>Abstract: The Palis conjecture states that if one takes two Cantor sets, such that the sum of their Hausdorff dimensions is greater than 1, then generically it should be true that their algebraic sum should contain an interval. I shall discuss this conjecture in a probabilistic setting.

ESC 638

Tuesday, October 11, 2011

12:00 pm
- 01:00 pm

Dynamics 'n Analysis Seminar: Weakly Wandering, Exhaustive Weakly Wandering and Strongly Weakly Wandering Sequences for Measure Preserving Transformations

Speaker: Vidhu Prasad (UML)<br/><br/>Abstract: We consider sequences of distinct integers associated to T, an ergodic invertible measure preserving transformation of an infinite measure space. It is well known that such a transformation always has associated to it weakly wandering (WW) sequences, and exhaustive weakly<br/>wandering sequences (EWW). Despite this there are many well-known T<br/>for which not a single EWW is known.<br/><br/>In this talk, we consider a class of sequences for T called strongly weakly wandering (SWW). All SWW sequences are also EWW. Their advantage is that the conditions implying SWW are easier to verify than the conditions implying EWW.<br/><br/>We will examine SWW sequences for type zero ergodic transformations. An example of a type zero transformation is simple random walk on the integers. All concepts will be defined in the talk. If time permits we connect these sequences to tiling the integers with a single infinite tile. This represents joint work with Eigen, Hajian and Ito.<br/>

ESC 638

Tuesday, October 04, 2011

12:00 pm
- 01:00 pm

The dynamics of the Nash map for 2 by 2 Games

Speaker: Bruce Kitchens, IUPUI, BU<br/><br/>Abstract: J. Nash defined a continuous map from a product of simplices, corresponding to the strategies for a finite game, to itself and used it to show that every finite game has a Nash equilibrium. The map is known as the Nash better response map and a point is a fixed point for the map if and only it is a Nash equilibrium for the game. In the case of a 2 by 2 game the map reduces to a continuous, piecewise rational map of the unit square to itself. I will discuss the dynamics of the map for several 2 by 2 games - prisoner's dilemma, matching pennies and chicken. The dynamics of the map divide the games into three classes, those having a dominant strategy, those having elliptic dynamics and those having hyperbolic dynamics. The dynamics of the map for the game of matching pennies will be discussed in detail.

ESC 638