Connections Between Complex Dynamics, Statistical Physics, and Limiting Spectra of Self-similar Group Actions---Reading Material for the learning talks

Connections Between Complex Dynamics, Statistical Physics, and Limiting Spectra of Self-similar Group Actions

Indiana University Purdue University Indianapolis from August 15 to 19, 2016.

Funded by the National Science Foundation, the IUPUI School of Science, and the IUPUI Office for Academic Affairs.

Material for the learning talks.

The objective of these talks is for everyone to learn how certain problems from the Ising Model and self-similar group theories are related to iteration of rational maps (often in two variables).

For the most part, the learning talks will present sections of the following three papers (and of earlier works leading up to them):

[GZ] The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum (by Grigorchuk and Zuk)

[FJ] Brolin's Theorem for Curves in Two Complex Dimensions (by Favre and Jonsson)

[BLR] Lee-Yang Zeros for the DHL and 2D Rational Dynamics, I. Foliation of the Physical Cylinder. (by Bleher, Lyubich, and Roeder).

Mathematical Physics:
A good general reference is Baxter's Book (free online).

Dianne Holcomb (U. Arizona): The Ising Model.
Section 1.1 and 2.1-2.4 of BLR and possibly Appendix E.1. Also, Onsager's solution for zero external magnetic field.

Mikel de Viana (Georgia Tech): Fisher Zeros on the Diamond Heirarchical Lattice.
Paper by Derrida, DeSeze, and Itzkson. Possibly also p. 1-4 and Figure 1.3 from Lee-Yang-Fisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation.

Danielle Larcomb (U. Denver): Exponential Random Graph Models.
Summary of the key definition and ideas from the paper by Chatterjee and Diaconis.

Self-Similar Groups Good general references: Nekrashevich survey, Zuk survey.

Hongming Nei (Indiana University): Self-similar groups and Iterated Monodromy Groups.
Material through Section 3.1, skipping Example 2.5.4, of Nekrashevich survey.

Yuki Takahashi (UC Irvine): Calculation of the spectrum for the action of the Lamplighter group.
Sections 6 and maybe 7 of [GZ].

Scott Kaschner (Butler U): Rational map in two variables arising when trying to compute the spectrum of the action of the IMG for z^2-1.
This talk will be based on some calculations Scott, Rodrigo, and Roland did a few years ago. (No notes are available to post.)

Several Complex Variables Dynamics (and connections):
Good general references are Paper by Hubbard and Papadopol , Sibony's survey and Dinh and Sibony's later survey.

Stefano Silvestri (IUPUI): Compactness Theorem for Subharmonic Functions.
Definition of subharmonic functions and proof of Theorem 4.1.9 from The analysis of Linear partial differential operators I by Hormander. (See also Theorem 1.18 from my informal notes.)

Joanna Furno (IUPUI): Application of the Compactness Theorem for Subharmonic Functions to Brolin's Theorem.
Hartogs' Lemma and Application to Complex Dynamics slides by Ivan Chio.

Sara Lapan (Northwestern and UC Riverside): What is a closed positive (1,1) current and explanation of the introduction of [FJ]. Slides from Sara's talk.
For definitions involving currents, see Section 3 of Paper by Hubbard and Papadopol or the appendices from Sibony's survey and Dinh and Sibony's later survey.

Olivier Remy (École Polytechnique): Lee-Yang zeros for the DHL.
Summary of [BLR].

Ivan Chio (IUPUI): Lee-Yang-Fisher zeros for the DHL
Summary of Lee-Yang-Fisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation.

Further Instructions For the Speakers:
We would like very much for these initial learning talks to do the following:

provide a background for each of the topics generally
provide background for specific connections between the topics
have a colloquium-style atmosphere
welcome questions and comments from the audience
be accessible to a first-year graduate student
be approximately forty minutes in length (and no more than fifty).

Preparing a good talk on your assigned topic may require some discussions with us by email or skype--we're all very friendly and open to discussion. Please contact us about whatever you need.

Connections Between Complex Dynamics, Statistical Physics, and Limiting Spectra of Self-similar Group Actions

Indiana University Purdue University Indianapolis from August 15 to 19, 2016.

Funded by the National Science Foundation, the IUPUI School of Science, and the IUPUI Office for Academic Affairs.

Material for the learning talks.

The objective of these talks is for everyone to learn how certain problems from the Ising Model and self-similar group theories are related to iteration of rational maps (often in two variables).

Back to workshop main page.