This workshop will bring together computer scientists and mathematicians to examine the P v. NP problem and its variants from the perspectives of algebra, geometry, and number theory, and to introduce the mathematical aspects of these questions to a larger audience. Diverse researchers working on different aspects of these problems will clarify connections between different approaches.

There will be two main topics: Analogues of P v. NP (e.g., Valiant's conjectures, the Mulmuley-Sohoni Conjecture, the BSS model, and other computational models); and Algebraic, Number Theoretic, and Geometric Aspects of P v. NP (e.g., Holographic algorithms, characterizations of NP in terms of sheaf cohomology, sparse polynomials, and other arithmetic approaches).

The workshop will emphasize the "work" aspect, so there will be few scheduled lectures, with extensive discussion periods, and follow-up lectures scheduled impromptu as needed.

Cluster algebras are commutative algebras with a distinguished set of generators grouped into overlapping subsets of fixed cardinality; the generators and the relations among them are not given from the outset, but are produced by an iterative process of successive mutations. These algebras were developed to explain the "Laurent phenomenon", in which certain a priori rational functions defined by these mutations turn out to always be Laurent polynomials. Cluster algebras encode a surprisingly widespread range of phenomena in settings as diverse as quiver representations, Teichmüller theory, invariant theory, tropical calculus, Poisson geometry, and polyhedral combinatorics. This workshop will explore the connection between cluster algebras and various topics in statistical physics, including the dimer model on surfaces, integrable systems such as the KP equation, and certain dynamical systems (Y- and Q-systems) which play an important role in the theory of the thermodynamic Bethe... (more)

Kinetic theory plays a central role in many areas of mathematical physics, from nanoscales to continuum mechanics. It is an indispensable tool in the mathematical description of applications in physical and social sciences, from its origin in dilute gases, to wide applications such as semi-conductors, polymers, cells, plasma, galaxies, traffic networking, and swarming. The number of particles is typically more than 10²⁰.

On the one hand, kinetic models provide more detailed and accurate description of regimes where hydrodynamic equations are either invalid or simply not available. On the other hand, because modern computers are still inadequate in simulating the molecular or even quantum dynamics in emerging industrial needs in micro- and nanotechnology, kinetic equations provide models that can capture important features of microscopic or quantum phenomena with a manageable computational cost. Kinetic theory is at the core of multiscale modeling, which connects fundamental... (more)

40 Years and Counting: 2011 is the 40th anniversary of the Association for Women in Mathematics (AWM). With this conference, AWM continues to celebrate the progress of women in mathematical professions and to recognize individual achievements. Join us this fall on the Brown University campus in historic Providence, RI.

Vlasov-type models deal with continua of particles where the electric charges dominate the collisions, so that the collisions are ignored. They occur in physical plasmas, including astrophysical plasmas and fusion reactors.

There are many examples of astrophysical plasmas of this type, such as the solar wind. When a fusion reactor is very hot, the relevant times scales are so short that collisions can be ignored. Vlasov theory also models systems where the dominant force is gravity, such as clusters of stars or galaxies.

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)

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)

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.

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)

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)