Programs & Events
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
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 Ζ0 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)
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 Ratdn of rational degree d self-maps of Pn has a natural structure as an affine variety. The dynamical moduli space Mdn is the quotient of Ratdn by the conjugation action of the group PGLn+1. Problems to be investigated include the geometry of Mdn, the distribution of special maps such as post-critically finite maps in Mdn, dynamical modular curves associated to one-parameter families of maps with a marked point of period 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: "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.
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.
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
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
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
Mathematical and Statistical Aspects of Cryptography (in Kolkata, India)
Jan 12 - 14, 2012
This workshop focuses on mathematical and statistical aspects of public key cryptography. The main ingredients from mathematics so far include discrete logarithms and factoring over the integers, generalizations of the discrete logarithm to elliptic curves, hyperelliptic curves and further generalizations, aspects of infinite non-abelian groups, and closest vector problems (CVP) in integer lattices. Cryptanalysis in all of these areas can involve analyses of patterns in vast amounts of data, hence the need for statistical methods. One goal of this workshop, though not the only one, is to focus attention on the problem of quantifying the complexity of lattice-based problems, for example extrapolating the difficulty of solving a CVP in an integer lattice as a function of its dimension and other parameters.
A copy of the presentations given at this workshop is available as a PDF... (more)
Organizing Committee
- Jeffrey Hoffstein
- Jill Pipher
- Bimal Roy
Synchronization-reducing and Communication-reducing Algorithms and Programming Models for Large-scale Simulations
Jan 9 - 13, 2012
As concurrency in scientific computing pushes beyond a million threads and performance of individual threads becomes less reliable for hardware-related reasons, attention of mathematicians, computer scientists, and supercomputer users and suppliers inevitably focuses on reducing communication and synchronization bottlenecks. Though convenient for succinctness, reproducibility, and stability, instruction ordering in contemporary codes is commonly overspecified. This workshop attempts to outline evolution of simulation codes from today's infra-petascale to the ultra-exascale and to encourage importation of ideas from other areas of computer science into numerical algorithms, new invention, and programming model generalization.
Organizing Committee
- David Keyes
- Matt Knepley
- Katherine Yelick
Boltzmann Models in Kinetic Theory
Nov 7 - 11, 2011
The celebrated Boltzmann equation is the foundation of the kinetic theory for dilute collections of particles, which undergo elastic binary collisions. The Boltzmann theory is at the center of a series of multi-scaled physical models that connect microscopic multiparticle models to macroscopic fluid models such as the Navier-Stokes equations:
The first arrow refers the Boltzmann-Grad limit, while the second arrow refers to various hydrodynamic limits which lead to the fundamental equations of fluids. The Boltzmann theory therefore provides a practical tool and machinery for deriving macroscopic models in broad physical applications. Due to its importance, there has been an explosion of mathematical studies, both theoretical and numerical, for the Boltzmann equation.A major open problem that remains is to determine whether or not smooth initial data would lead to a unique global-in-time solution of the Boltzmann... (more)
Organizing Committee
- Kazuo Aoki
- Yan Guo
- Shi Jin
- Lorenzo Pareschi
- Laure Saint-Raymond
NSF Mathematics Institutes Modern Math Workshop (at SACNAS)
Oct 26 - 27, 2011
The eight NSF mathematics institutes are pleased to offer three concurrent sessions immediately preceding the SACNAS annual meeting â one for graduate students and recent PhDs, and two for undergraduate students â to invigorate the research careers of minority mathematicians and mathematics faculty at minority-serving institutions. The âModern Math Workshopâ will highlight presentations on topics drawn from the institutesâ upcoming programs, a keynote speaker, and an informative panel presentation on the 2012-13 programs and workshops. The two undergraduate sessions (applicants will choose one) are appropriate for students of any major interested in learning how mathematics contributes to our understanding of various scientific topics. Activities will include lectures and group work.
All sessions will begin with lunch on Wed. Oct. 26 and include an evening reception. The sessions will continue on Thursday morning and will end at 12:30 pm prior to the SACNAS conference lunch.... (more)
Novel Applications of Kinetic Theory and Computations
Oct 17 - 21, 2011
There are several fundamental applications involving kinetic theory and computations. They range from semiconductor modeling involving kinetic and quantum charged transport, radiative transfer in cosmology, conservative and dissipative phenomena in rarefied gas dynamics in mixtures, and grain and polymer flows.
Issues to be addressed involve the derivation and multi-scale modeling due to different scales of effective constants, spatial heterogeneities and strength of boundary conditions. Because the basic drift-diffusion, hydrodynamic and quantum models may interact through interfaces, a basic understanding of boundary conditions as well as phase transitions are critical. An example of such modeling problem appears naturally in semiconductors devices where the electron and holes density flows through a highly heterogeneous crystal lattice.
It is well established that drift-diffusion models are currently inadequate for the simulations of submicron devices where effective fields... (more)
Organizing Committee
- Irene Gamba
- Axel Klar
- Benoit Perthame
- Christian Ringhofer
- Chi-Wang Shu