Complex Analysis and Dynamics Seminar

Department of Mathematics
Graduate Center of CUNY

Fridays 2:00 - 3:00 pm
Room 5417
Organizers: Ara Basmajian, Patrick Hooper, Jun Hu, and Saeed Zakeri

Past seminars:

Fall 2006, Spring 2007
Fall 2007, Spring 2008
Fall 2008, Spring 2009
Fall 2009, Spring 2010
Fall 2010, Spring 2011
Fall 2011, Spring 2012
Fall 2012, Spring 2013
Fall 2013, Spring 2014
Fall 2014, Spring 2015
Fall 2015, Spring 2016

Fall 2016:

Sep 2: No seminar

Sep 9: No seminar because of Workshop on Analytic Aspects of Higher Teichmuller Theory

Sep 16: Maciej Capinski (AGH University of Science and Technology, Krakow, Poland)
Beyond the Melnikov Method: A Computer Assisted Approach

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. In our approach, we do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are ‘small enough,’ as is the case in the classical Melnikov approach.

Sep 23: David Aulicino (Brooklyn College of CUNY)
Weak Mixing for Translation Surfaces with Intermediate Orbit Closures

Work of Avila and Forni established weak mixing for the generic straight line flow on generic translation surfaces, and the work of Avila and Delecroix determined when weak mixing occurs for the straight line flow on a Veech surface. Following the work of Eskin, Mirzakhani, Mohammadi, which proved that the orbit closure of every translation surface has a very nice structure, one can ask how the orbit closure affects the weak mixing of the straight line flow. In this talk all of the necessary background on translation surfaces and weak mixing will be presented followed by the answer to this question. This is a joint work in progress with Artur Avila and Vincent Delecroix.

Sep 30: No seminar

Oct 7: No seminar

Oct 14: No seminar

Oct 21: Mark Bell (University of Illinois at Urbana-Champaign)
Slowly Converging Pseudo-Anosovs

A classical property of pseudo-Anosov mapping classes is that they act on the space of projective measured laminations with north-south dynamics. This means that under iteration of such a mapping class, laminations converge exponentially quickly towards its stable lamination. We will discuss a new construction (joint with Saul Schleimer) of pseudo-Anosovs where this exponential convergence has base arbitrarily close to one and so is arbitrarily slow.

Oct 28: Enrique Pujals (IMPA and the Graduate Center of CUNY)
Two-dimensional Blaschke Products: Degree Growth and Ergodic Consequences

For dominant rational maps of compact, complex, Kahler manifolds there is a conjecture specifying the expected ergodic properties of the map depending on the relationship between the rates of growth for certain degrees under iteration of the map. In the present talk, we will discuss the case of two-dimensional Blaschke products, observing that they fit naturally within this conjecture, having examples from each of the three cases that the conjecture gives for maps of a surface.
The results to be discussed are included in different works with Mike Shub and Roland Roeder.

Nov 4: Christian Wolf (CCNY and Graduate Center of CUNY)
Ground States and Mutual Ergodic Optimization

Nov 11: Jonah Gaster (Boston College)
New Bounds for `Homotopical Ramsey Theory' on Surfaces

Farb and Leininger asked: How many simple closed curves on a finite-type surface $S$ may pairwise intersect at most $k$ times? Przytycki has shown that this number grows at most as a polynomial in $|\chi(S)|$ of degree $k^{2}+k+1$. We present narrowed bounds by showing that the above quantity grows slower than $|\chi(S)|^{3k}$. In particular, the size of a maximal 1-system grows sub-cubically in $|\chi(S)|$. The proof uses a bound for the maximum size of a collection of curves of length at most $L$ on a hyperbolic surface homeomorphic to $S$. Specializing to the case that $S$ is an $n$-holed sphere and $k=2$, we use the coloring computations of Gaster-Greene-Vlamis to show that this bound can be improved to $O(n^5 \log n)$. This is joint work with Tarik Aougab and Ian Biringer.

Nov 18: Zhiqiang Li (Stony Brook University)

Nov 25: No seminar

Dec 2: Anja Randecker (University of Toronto)

Dec 9: Lien-Yung Kao (University of Notre Dame)

Dec 16: Yun Yang (Graduate Center of CUNY)

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