The most important family of statistical mechanics models on graphs are the Fortuin-Kasteleyn random cluster measures FK(p,q), giving a random subset of the edges of the graph. E.g., the q=1 case
is independent bond percolation, and the q=2 case gives rise to the Ising model of magnetization. For every q, there is a critical value p_c(q) for p, where the connectivity properties of the system suddenly change completely. On planar lattices, the system at p_c(q) is conjectured to be conformally invariant, proved for the q=1,2 cases on some lattices by Stas Smirnov.

There is a natural Markov chain with stationary measure FK(p,q), updating locally the states of the edges as time goes. For the q=1 case, in several joint projects with Christophe Garban and Oded
Schramm, we understood exactly how the macroscopic connectivity properties are changing with this dynamics. We also found that by doing the updates asymmetrically, hence pushing the system out of criticality, we can describe well the near-critical behavior. In particular, we constructed conformally covariant continuum dynamical and near-critical percolation processes, the scaling limits of the
discrete systems, using the same space-time scaling in the two cases.

In recent work with C. Garban, we have extended the dynamical scaling limit construction to the q=2 case. However, we have found that the near-critical window is now governed by very different mechanisms. In particular, understanding Onsager's classical results on the near-critical Ising model via the conformally invariant critical system remains a huge challenge, which I will try to explain. I will
also present some results and conjectures on the noise and dynamical sensitivity of critical FK(q) models.

I will describe an action of a quantized Heisenberg algebra on the (derived) categories of coherent sheaves on Hilbert schemes of ALE spaces (crepant resolutions of C^2/G). This action essentially lifts the actions of Nakajima and Grojnowski on the cohomology of these spaces. (Joint with Tony Licata.)

In the last of this series of seminars on nonlocally related systems and nonlocal symmetries, we will discuss the interesting situation for linear PDE systems and the situation for PDE systems with three or more independent variables. In the case of three or more independent variables, it will be seen that it is necessary to introduce gauge constraints to obtain potential symmetries. Open problems will be discussed as usual.

We prove results about the L^p-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L^p, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,...,N} of sizes alpha N and beta N then A+B contains an arithmetic progression of length at least about exp(c (alpha beta log N)^{1/2}). Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least about exp(c (alpha log N/(log(beta^{-1}))^3)^{1/2}). (Joint work with Ernie Croot and Olof Sisask.)

We will state some theorems in arithmetic combinatorics, and give outlines of the proofs.
One such theorem bounds the absolute density in the integers of sums of relatively dense sets of prime numbers in terms of the relative density. In discussing the proof of this result, we will recall the concept pseudorandomness of a set previously developed by Green and Green-Tao. These ideas are used to reduce the problem to an analogous statement for relatively dense subsets of the multiplicative subgroup of integers modulo a large integer N, and this statement may be proved using a combinatorial argument. This result is joint work with Mariah Hamel.
We will also present some variants of the Erdös-Szemerédi sum-product problem which have the following flavor: A set of complex numbers whose elements produce few products will necessarily produce many sums. To demonstrate the proofs, we will introduce the concept of the multiplicative dimension of a finite set of complex numbers.
We will view these results in the context of connections between a set's structure and arithmetic.

I will sketch some recent results and open questions in the project of determining which orbit closures of a symmetric subgroup acting on the flag variety of a reductive group have smooth or rationally smooth closure. I will focus on the classical case and the combinatorial notion of pattern avoidance.

A permutation $\pi$ is realized by the shift on $N$ letters if there is
an infinite word on an $N$-letter alphabet whose successive shifts by
one position are lexicographically in the same relative order as
$\pi$.
Understanding the set of permutations realized by shifts, as well as
other one-dimensional dynamical systems, is important because
it provides tests to distinguish deterministic sequences from random ones.
In this talk I will give a characterization of permutations realized
by shifts, and also by a natural generalization of them, where instead
of $N$ we have a real number $\beta$.

In the talk I will survey some results, techniques, and open problems, concerning finite group actions on (i) spheres, (ii) products of spheres and (iii) non-compact space forms modelled on indefinite quadratic forms.

Given a finite graph we wish to order to its vertices in some random way that depends on the graph. If we impose some natural constraints on the order, it turns out that there is only one possible distribution on
the order. I will describe connections to newly developed theory of limits of graphs, and a surprising new proof of a result about embedding metric spaces.

I will introduce the concept of p-local compact groups, which
are combinatorial structures that generalize p-local ﬁnite groups and p-compact
groups, along with some examples. Then I will describe some partial results
towards showing the existence of embeddings of these objects in p-completed
classifying spaces of unitary groups.

We show that the threshold for a subset sampled uniformly from the
range of a random walk on Z^d_n (the d dimensional torus of size n) d>=3, to become indistinguishable from a
uniformly chosen subset of Z^d_n is 1/2 the cover time. As a consequence of
our methods, we show that the total variation mixing time of the random
walk on the lamplighter graph of Z^d_n d>=3, has a cutoff with threshold at
1/2 the cover time. We give a general criterion under which both of these
results hold; other examples for which this applies include bounded degree
expander families, the intersection of an infinite super-critical percolation
cluster with an increasing family of balls, the hypercube, and the
Cayley graph of the symmetric group generated by transpositions. The
proof also yields precise asymptotics for the decay of correlation in the
uncovered set. This is joint work with Yuval Peres.

We will discuss a surprising relationship between stationary Markov chains, stationary Markov random fields and measures with nearest neighbour Gibbs potential on the integer lattice. We will also see why this does not extend to higher dimensions.

In this work we will consider several questions concerning the asymptotic nature of arithmetic functions. First, we conduct an asymptotic analysis of the behavior of the composition of Carmichael's lambda-function and Euler's totient function. Second, we establish an asymptotic formula for a sum of a generalized divisor function on the Gaussian numbers. And third, for complex-valued multiplicative functions that are sufficiently close to 1 on the primes and bounded on prime powers, we determine the average value over a short interval.

MATX 1118 (Note the special location and special time)

Fri 8 Apr 2011, 1:00pm-2:00pm

Abstract

In recent years, there has been an increasing interest in studying constrained variational problems with a fractional diffusion. One of the motivations comes from mathematical finance: jump-diffusion processes where incorporated by Merton into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock's prices, in contrast with the classical lognormal diffusion model of Black and Scholes. These models allow to take into account large price changes, and they have become increasingly popular for modeling market fluctuations, both for risk management and option pricing purposes.

In a joint paper with Luis Caffarelli we study the parabolic version of the fractional obstacle problem, i.e. where the elliptic part of the operator is given (at least at the leading order) by a fractional laplacian. We prove optimal spatial regularity and almost optimal time regularity of the solution, recovering in particular the optimal regularity for the stationary case. To obtain this result, we crucially exploit the fact that the solution coincides with the obstacle at the initial time, which corresponds to the fact that (for the backward operator) the stock's price coincides with the payoff at the final time.

The research solves the problem of Lie group analysis on the Buckley-Leverett equation
and provides the construction of some invariant solutions.
We also view the analytical solutions of a diffusion-convection equation (Rapport-L model)
with capillary effects.

Abstract:
What are the basic skills in Mathematics that can affect success in first-year Calculus and beyond?
We will give an overview of some of the results gathered from the Basic Skills Test and other similar
tests. We will show what questions had the highest degree of correlation with performance in
differential Calculus and discuss common students' errors. Data from a variety of courses will
be presented, ranging from Math 110 to Math 180, 104, and 184, but also including a few
examples from Science One. Finally, we will show that some of the weaknesses in
basics skills continue beyond first-year courses by discussing recent data from Math 220.

After reviewing some basic constructions with multiplier ideals on complex algebraic varieties, we recall the definition of multiplier ideals in positive characteristic and highlight the failure of some desirable properties to carry over in this setting. This leads us to a related measure of singularities coming from commutative algebra -- the test ideal -- which seems to exhibit better behavior than the multiplier ideal in positive characteristic. While test ideals were first introduced in the theory of tight closure, our goal in this talk will be to describe a new and algebro-geometric characterization of test ideals using regular alterations. This characterization is also holds for multiplier ideals in characteristic zero (but not in positive characteristic!!!), providing a kind of uniform description with new insight and intuition. Time permitting, we will use this result to give an analogue of Nadel Vanishing in positive characteristic.

We develop and analyze mixed discontinuous Galerkin finite element methods for the numerical approximation of incompressible magnetohydrodynamics problems.Incompressible magnetohydrodynamics is the area of physics that is concerned with the behaviour of electrically conducting, resistive, incompressible and viscous fluids in the presence of electromagnetic fields. It is modelled by a system of nonlinear partial differential equations, which couples the Navier-Stokes equations with the Maxwell equations.

In the first part of this thesis, we introduce an interior penalty discontinuous Galerkin method for the numerical approximation of a linearized incompressible magnetohydrodynamics problem. All the variables are discretized using completely discontinuous finite element spaces. Under minimal regularity assumptions, we carry out a complete a-priori error analysis and prove that the energy norm error is optimally convergent in the mesh size in general polyhedral domains, thus guaranteeing the numerical resolution of the strongest magnetic singularities in non-convex domains.

In the second part of this thesis, we propose and analyze a new mixed discontinuous Galerkin finite element method for the approximation of a fully nonlinear incompressible magnetohydrodynamics model. The velocity and the magnetic field are now discretized by partially continuous elements. In addition to correctly capturing magnetic singularities, the method yields exactly divergence-free velocity approximations, and is thus energy-stable. We show that the energy norm error is convergent in the mesh size in possibly non-convex polyhedra, and derive slightly suboptimal a-priori error estimates under minimal regularity and small data assumptions.

Finally, in the third part of this thesis, we present two extensions of our discretization techniques to time-dependent incompressible magnetohydrodynamics problems and to Stokes problems with non-standard boundary conditions.

All our discretizations and theoretical results are computationally validated through comprehensive sets of numerical experiments.

Given an infinite stack of arrows (each pointing left or right) at each
vertex of Z, we can define a walk on Z that moves by following and consuming
arrows. If we switch a left arrow to a right arrow, what happens to
the walk? The answer to this question gives some interesting results when
applied to 1-dimensional random walks (such as multi-excited random
walks), and projections of higher-dimensional random walks.
(Joint work with Tom Salisbury)

Let X be a homogeneous space of a connected linear algebraic group G over a number field k with geometric stabilizer H. We may and shall assume that G is a quasi-trivial k-group (a quasi-trivial k-group is a generalization of a simply connected k-group). Assuming that the geometric stabilizer H is either connected or abelian, we give a sufficient condition for the Hasse principle and weak approximation for X in terms of the action of the absolute Galois group of k on the character group of H.

I shall explain in the talk what are a quasi-trivial group, the Hasse principle, weak approximation, etc.

The overlapping cycles shuffle mixes a deck of n cards by moving either the nth card or (n-k)th card to the top of the deck, with probability half each. Angel, Peres and Wilson determined the spectral gap for the location of a single card and found the following surprising behaviour. Supppose that k is the closest integer to cn for a fixed c in (0,1). Then for rational c, the spectral gap is on the order of n^{-2}, while for poorly approximable irrational numbers c, such as the reciprocal of the golden ratio, the spectral gap is on the order of n^{-3/2}. We show that these bounds also apply, up to logarithmic factors, to the mixing time for all the cards. The talk is based on work in progress with Olena Bormashenko and Sukhada Fadnavis.