## Programs & Events

##### Complex and Arithmetic Dynamics

Jan 30 - May 4, 2012

The goal of this program is to bring together researchers in complex dynamics, arithmetic dynamics, and related fields, with the purpose of stimulating interactions, promoting collaborations, making progress on fundamental problems, and developing theoretical and computational foundations on which future work will build. Complex dynamics is the study of iteration of holomorphic self-maps of a complex space. Fundamental examples of such maps arise as algebraic self-maps of algebraic varieties. Starting with the fundamental results of Fatou and Julia, complex dynamics has evolved into a well established field with many deep theorems and many important unresolved questions. Arithmetic dynamics refers to the study of number theoretic phenomena arising in dynamical systems on algebraic varieties. Many global problems in arithmetic dynamics are analogues of classical problems in the theory of Diophantine equations or arithmetic geometry, including for example uniform bounds for rational... (more)

##### Organizing Committee

- Rob Benedetto
- Laura DeMarco
- Mikhail Lyubich
- Juan Rivera-Letelier
- Joseph Silverman
- Lucien Szpiro
- Michael Zieve

##### Complex and p-adic Dynamics

Feb 13 - 17, 2012

This workshop will bring together researchers working in classical complex dynamics and in the newer area of p-adic (nonarchimedean) dynamics. It will promote interactions between the two groups by highlighting the similarities and differences between complex and p-adic dynamics.

In particular, it will address Berkovich space, whose introduction has greatly enhanced the exchange of ideas between complex and p-adic dynamics.

##### Organizing Committee

- Matthew Baker
- Rob Benedetto
- Charles Favre
- Kevin Pilgrim
- Juan Rivera-Letelier

##### Global Arithmetic Dynamics

Mar 19 - 23, 2012

This workshop will examine global arithmetic dynamics from the perspectives of number theory, algebraic geometry, and model theory. It will introduce aspects of this topic to a larger audience, and clarify connections between different perspectives. In addition, there will be extensive discussion periods in which participants can collaborate on theoretical and computational aspects of the topic.

##### Organizing Committee

- Xander Faber
- Michelle Manes
- Lucien Szpiro
- Thomas Tucker
- Michael Zieve

##### Special Colloquium: "Spheres"

Apr 5, 2012

A historical survey on progress in understanding topological spheres and differentiable spheres.

Thursday, April 5, 2012 from 4:00pm-5:00pm. There will be a reception following the talk.

##### Special Colloquium: "Smooth Topology-Preserving Approximations of Rough Domains"

Apr 12, 2012

The talk concerns bounded domains with continuous boundary. We study how the corresponding "good direction", with respect to which the boundary is locally a graph of a continuous function, varies in a neighborhood of the boundary, and thus show how such domains can be approximated both from the inside and the outside by topologically equivalent smooth domains. The good directions form a globally-defined field that carries some topological information about the domain, which we explore. Finally we describe a surprising consequence of the study, that the domain has portions of the boundary with better regularity. This is joint work with Arghir Zarnescu.

This is a joint colloquium co-sponsored by Applied Mathematics, Mathematics, and ICERM. Thursday, April 12, 2012 from 4:30-5:30pm in the lecture hall at ICERM. There will be a reception at 4pm preceding the talk.

##### Moduli Spaces Associated to Dynamical Systems

Apr 16 - 20, 2012

This workshop will bring together dynamicists, number theorists, and algebraic geometers to study the geometry and arithmetic of dynamical moduli spaces. The set `Rat _{d}^{n}` of rational degree

`d`self-maps of

`P`has a natural structure as an affine variety. The dynamical moduli space

^{n}`M`is the quotient of

_{d}^{n}`Rat`by the conjugation action of the group

_{d}^{n}`PGL`. Problems to be investigated include the geometry of

_{n+1}`M`, the distribution of special maps such as post-critically finite maps in

_{d}^{n}`M`, dynamical modular curves associated to one-parameter families of maps with a marked point of period

_{d}^{n}`N`, and degeneration of families of maps and the associated points on the boundary of moduli space. A tutorial session will be held the week before this workshop.

##### Organizing Committee

- Laura DeMarco
- Adam Epstein
- Sarah Koch
- Curtis McMullen
- Joseph Silverman

##### Special Colloquium: "The Prevalence of Chaos"

May 3, 2012

Dynamics is the study of systems which evolve with known rules. We want to understand whether the long term behavior is predictable or chaotic. For example, is the solar system stable? Very often, this amounts to studying recursively defined sequences. After presenting various dynamical systems and showing what chaos is, I will focus on sequences of the form

`Ζ`with

_{n+1}= Ζ^{2}_{n}+ c`Ζ ∈ C`and

`c ∈ C`.

For such sequences, the locus of chaos is known as a Julia set. I will present a result (*joint with Arnaud Cheritat*) that there exist parameters `c ∈ C` for which chaos prevails: with positive probability, a randomly chosen initial point `Ζ _{0}` will have unpredictable behavior.

This is a joint colloquium co-sponsored ICERM and the Clay Mathematics Institute. Thursday, May 3, 2012 from 4:00pm-5:00pm. There will... (more)

##### Heterostructured Nanocrystalline Materials

May 30 - Jun 1, 2012

The theme of this workshop is the computation, modeling, and mathematical analysis of heterostructured nanocrystalline materials. This includes quantum dots, nanowires, graphene, and grain boundaries. These various phenomena will be discussed in the context of modeling and computation on different scales ranging from density functional theory to continuum mechanics. The workshop will also address various techniques that allow one to combine models on different scales to yield efficient computational methods.

##### Organizing Committee

- Tim Schulze
- Vivek Shenoy
- Peter Smereka

##### Research Experiences for Undergraduate Faculty (REUF)

Jun 4 - 8, 2012

This workshop, sponsored by AIM, ICERM, and the NSF, will introduce undergraduate faculty to research opportunities in several fields of mathematics that will equip them with the tools to mentor students in undergraduate research in mathematics. Lectures at the workshop will provide background information and introduce open problems. The majority of the workshop will be spent working on problems, reporting on progress, and formulating plans for future work.

The workshop will be hosted at ICERM.

Preference will be given to faculty who teach and advise substantial numbers of minority students.

Mathematical topics at the workshop will include algebra, graph theory, and number theory.

For more information email workshops@aimath.org or go to http://aimath.org/ARCC/workshops/reuf4.html.

##### Organizing Committee

- Leslie Hogben
- Roselyn Williams
- Ulrica Wilson

##### NSF/CBMS Conference: Finite Element Exterior Calculus (FEEC)

Jun 11 - 15, 2012

FEEC is a recent advance in the mathematics of finite element methods that employs differential complexes to construct stable numerical schemes for several important types of application problems. It has aroused great interest because it both presents interesting mathematical problems and shows great potential for application in computational science and engineering. The concentrated sequence of lectures in this program will provide participants with an understanding of the mathematical tools required to fully grasp the concepts in FEEC. ICERM is pleased to host this NSF-CBMS Regional Research Conference.

Douglas Arnold will deliver 10 lectures, one in the morning and one in the afternoon of each day of the workshop. Richard Falk and Anil Hirani will both give one presentation.

Limited funding is available to support participants.

##### Organizing Committee

- Alan Demlow
- Johnny Guzman
- Dmitriy Leykekhman

##### Summer@ICERM 2012: Geometry and Dynamics

Jun 18 - Aug 10, 2012

Imagine spending eight-weeks on the beautiful Brown University campus in historic Providence, RI, working in a small team setting to solve mathematical research problems developed by faculty experts in their fields.

Imagine creating career-building connections between peers, near peers (graduate students and postdocs), and academic professionals.

Imagine spending your summer in a fun, memorable, and intellectually stimulating environment.

Now, imagine having this experience with support for travel within the U.S., room and board paid, plus a $3,000 stipend*.

The Summer@ICERM 2012 program is designed for a select group of 10-12 undergraduate scholars. Students in the 2012 program worked in small groups of two or three, supervised by a faculty advisor and aided by a teaching... (more)

##### Organizing Committee

- Pat Hooper
- Sergei Tabachnikov