# Seminars & Colloquia Calendar

## Algebraic dynamics from topological and holomorphic dynamics

#### Rohini Ramadas, Harvard

Location: ** Hill 425**

Date & time: Wednesday, 28 February 2018 at 2:00PM - 3:00PM

Let \(f:S^2 \rightarrow S^2\) be an orientation-preserving branched covering from the 2-sphere to itself whose *postcritical set *

\(P := f^n(x) | x\) is a critical point of \(f\) and \(n>0\) is finite. Thurston studied the dynamics of \(f\) using an induced holomorphic

self-map \(T(f)\) of the Teichmuller space of complex structures on \((S^2, P)\). Koch found that this holomorphic dynamical system on Teichmuller space descends to algebraic dynamical systems:

1. \(T(f)\) always descends to a multivalued self map \(H(f)\) of the moduli space \(M_{0,P}\) of markings of the Riemann sphere by the finite set \(P\).

2. When \(P\) contains a point \(x\) at which \(f\) is fully ramified, under certain combinatorial conditions on \(f\), the inverse of \(T(f)\)

descends to a rational self-map \(M(f)\) of projective space \(P^n\). When, in addition, \(x\) is a fixed point of \(f\), i.e. \(f\) is a

`topological polynomial' \(\rightarrow\), the induced self-map \(M(f)\) is regular.

The dynamics of \(H(f)\) and \(M(f)\) may be studied via numerical invariants called dynamical degrees: the k-th dynamical degree of an algebraic dynamical system measures the asymptotic growth rate, under iteration, of the degrees of k-dimensional subvarieties.

I will introduce the dynamical systems \(T(f)\), \(H(f)\) and \(M(f)\), and dynamical degrees. I will then discuss why it is useful to study \(H(f)\) (resp. \(M(f)\)) simultaneously on several compactifications of \(M_{0,P}\). We find that the dynamical degrees of \(H(f)\) (resp. \(M(f)\)) are algebraic integers whose properties are constrained by the dynamics of \(f\) on the finite set \(P\). In particular, when \(M(f)\) exists, then the more \(f\) resembles a topological polynomial, the more \(M(f): P^n \rightarrow P^n\) behaves like a regular map.

Charles Weibel Organizer's Page

Brooke Logan

Wujun Zhang Organizer's webpage

P. Gupta, X.Huang and J. Song Organizer's webpage

Swastik Kopparty, Sepehr Assadi Seminar webpage

Jeffry Kahn, Bhargav Narayanan, Jinyoung Park Organizer's webpage

Yonah Biers-Ariel, Mingjia Yang, and Doron Zeilberger --> homepage

Paul Feehan, Manousos Maridakis, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Jason Saied Seminar webpage

Brian Pinsky, Rashmika Goswami website

Corrine Yap Organizer's webpage

Edna Jones Organizer's webpage

Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

Simon Thomas website

Kasper Larsen, Daniel Ocone and Kim Weston Organizer's page

Joel Lebowitz, Michael Kiessling

Yanyan Li, Haim Brezis

Brooke Ogrodnik website

Stephen D. Miller, John C. Miller, Alex V. Kontorovich, Alex Walker seminar website

Stephen D. Miller

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Yael Davidov Seminar webpage

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

Ebru Toprak, Organizer

Organizer: Luochen Zhao

James Holland; Organizer website

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