# Complex Analysis and Dynamics Seminar

## Fall 2015 Schedule

#### Sep 4: Diogo Pinheiro (Brooklyn College) Optimal Control of Dynamical Systems with Multiple Sources of Uncertainty

I will discuss optimal control problems associated with dynamical systems containing different sources of randomness. I will focus on the derivation of generalized dynamic programming principles, as well as on the corresponding Hamilton-Jacobi-Bellman equations. Joint work with N. Azevedo (University of Lisbon, Portugal).

#### Sep 11: Rodrigo Trevino (NYU) Quasicrystals, Ergodic Theory and Cohomology

I will talk about some recent results on deviation of ergodic averages for systems coming from aperiodic tilings and aperiodic point sets which are self affine (the Penrose tiling is an example of these). These rely on fun interactions between ergodic theory and some cohomology theories associated with aperiodic point sets. Time permiting I will discuss applications to problems of diffraction, counting problems, and futue directions. This is joint work with S. Schmieding (Maryland).

#### Sep 18: Enrique Pujols (IMPA, Brazil and Graduate Center of CUNY) The Prisoner’s Dilemma: The Emergence of Cooperation

Note: This is part of the seminar "Dynamical Systems: Ideas in Applications" and will be held in Room 9204.

#### Oct 16: Konstantin Mischaikow (Rutgers University) Regulatory Networks

Note: This is part of the seminar "Dynamical Systems: Ideas in Applications" and will be held in Room 4102.

#### Oct 23: Richard Schwartz (Brown University) The Projective Heat Map

One of the simplest polygon iterations works by taking a polygon and replacing it by the new polygon whose vertices are the midpoints of the sides of the original one. This construction is related to heat flow and is well understood. In this talk I will explain a projectively natural version of the midpoint map, and I'll explain how you analyze what it does to pentagons. The analysis involves a computer-assisted analysis of a fairly high degree rational map in 2 real variables.

#### Oct 30: Ethan Akin (City College of New York) The Prisoner's Dilemma

For the iterated Prisoner's Dilemma, there exist Markov strategies which solve the problem when we restrict attention to the long term average payoff. When used by both players these assure the cooperative payoff for each of them. Neither player can benefit by moving unilaterally any other strategy, i.e. these are Nash equilibria. In addition, if a player uses instead an alternative which decreases the opponent's payoff below the cooperative level, then his own payoff is decreased as well. Thus, if we limit attention to the long term payoff, these "good strategies" effectively stabilize cooperative behavior. We characterize the good strategies and relate them to the recent exciting work of Press and Dyson. The talk should be accessible to advanced undergraduates.

#### Nov 13: Michael Yampolsky (University of Toronto) Renormalization and Rotational Attractors of Two-Dimensional Dissipative Dynamical Systems

We study dissipative rotational attractors in two settings: Siegel disks of Henon maps and minimal attractors of diffeomorphisms of the annulus. Jointly with D. Gaydashev, we extend renormalization of Siegel maps and critical circle maps to small 2D perturbations, and use renormalization tools to study the geometry of the attractors. In the Siegel case, jointly with D. Gaydashev and R. Radu we prove that for sufficiently dissipative Henon maps with semi-Siegel points with golden-mean rotation angles, Siegel disks are bounded by (quasi)circles. In the annulus case, jointly with D. Gaydashev, we prove that for bounded type rotation number, critical annulus maps have a minimal attractor which is a $C^0$, but not smooth, circle – answering a question of E. Pujals.

#### Nov 20: Lai-Sang Young (Courant Institute, NYU) Dynamics of the Visual Cortex

Note: This is part of the seminar "Dynamical Systems: Ideas in Applications" and will be held in Room 9204.

#### Dec 4: Grace Work (University of Illinois) Gap Distribution for Slopes of Saddle Connections on the Octagon

Following a strategy developed by Athreya and Cheung, we compute the gap distribution of the slopes of saddle connections on the translation surface associated to the regular octagon by translating the problem to a question about return times of the horocycle flow to an appropriate Poincare Section. This same strategy was used by Athreya, Chaika, and Lelievre to compute the gap distribution on the Golden L. The octagon is the first example of this type of computation where the Veech group has two cusps.

#### Dec 11: Tarik Aougab (Brown University) Short Representatives and Simple Covers of Closed Geodesics

A. Basmajian showed that if $g$ is a closed geodesic with self-intersection number at least $k$ on a hyperbolic surface, then $g$ has length at least $\log(k)/2$, and this is sharp. In the other direction, we give a sharp upper bound for the length of the Teichmuller minimizer for any curve with at most $k$ self-intersections: if $g$ is a closed geodesic with at most $k$ self-intersections, then there exists a hyperbolic surface on which $g$ admits a representative of length at most the square root of $k$ (up to constants depending on the topology of the surface). As an application and complementing a recent result of Gupta-Kapovich, we obtain sharp bounds on the growth of the function which, given a surface $S$ and $k>0$, outputs $D= D(k)$ such that if $g$ is a closed geodesic on $S$ with at most $k$ self-intersections, then there exists a cover of $S$ of degree at most $D$ such that $g$ lifts to a simple curve on the cover. This work is joint with J. Gaster, P. Patel, and J. Sapir. Time permitting, we will discuss connections to other lifting functions, and possible generalizations of Basmajian's lower bound in the setting of hyperbolic 3-manifolds (joint with J. Gaster).

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