In this talk, we will look at the exciting, recent results showing that most images and other signals can be reconstructed from much less information than previously thought possible, using simple, efficient algorithms. A consequence has been the explosive growth of the new field known as compressive sensing, so called because the results show how a small number of measurements of a signal can be regarded as tantamount to a compression of that signal. The many potential applications include reducing exposure time in medical imaging, sensing devices that can collect much less data in the first place instead of collecting and then compressing, and getting reconstructions from what seems like insufficient data (such as EEG). We will see how replacing the convex optimization problem typically used in this field with a nonconvex variant has the effect of reducing still further the number of measurements needed to reconstruct a signal. A very surprising result is that a simple algorithm, designed only for finding one of the many local minima of the optimization problem, typically finds the global minimum. Understanding this is an interesting and challenging theoretical problem. We will see examples, and discuss algorithms, theory, and applications.

Kostant's remarkable formula (which I will recall) generalizes to smooth Schubert varieties in the flag variety G/B of an algebraic group G. On the other hand, Sara Billey noticed that rationally smooth Schubert varieties in G/B give an analogous formula, though it often doesn't agree with the remarkable formula in the singular case. This motivates the question of which rationally smooth Schubert varieties are smooth. I will show that there is a neat answer hinted at in the title.

I will discuss the scattering problem of mass-critical generalized KdV equation. We will see if the scattering of gKdV fails, then a minimal mass blow-up solution exist on the condition that scattering of mass-critical 1D NLS is true. We use concentration compactness argument in addition to an observation that a certain modulated, rescaled version of NLS solution is approximately gKdV solution for highly oscillatory profile.

Combinatorial games are typically played using a disjunctive sum of distinct components. On each player's turn, they choose one of the available components and make a legal move in that component. Traditionally, a player cannot choose a component where a legal move does not exist. We examine the case where this is allowed and call it the short disjunctive sum of games. In particular, we show that both interpretations are equivalent under normal play rules but differ under mis\`ere play rules. Finally, we show how this interpretation can be extended to the analysis of both normal and mis\`ere play games which have non-standard ending conditions.

We will show that a Del Pezzo surface of degree 6 has two explicitly
defined locally free sheaves, whose endomorphism rings generated the
K-theory of the surface. Time permitting, we will also show that a similar
result holds for the derived category of the surface. Some of this work is
joint with Paul Smith and Sue Sierra.

Essentially all cellular processes depend on spatially localized proteins. Some proteins localize to cell poles, others to the particular cell membranes, and yet others to specific cytoplasmic regions. This localization is often dynamic, with proteins shuttling between different regions. The Smoldyn biochemical simulator helps researchers study this intracellular organization; Smoldyn represents each protein as an individual point-like particle that diffuses, reacts, and interacts with membranes, all in continuous space. It was surprisingly difficult to make these processes quantitative, such as for finding the "binding radius" for bimolecular reactions and the adsorption probability for molecules that adsorb to membranes. Smoldyn has enabled a variety of research projects over the last several years. In one example, Smoldyn simulations showed that yeast cells appear to secrete a protease (called Bar1) which degrades extracellular pheromone so that, paradoxically, they can sense the pheromone gradient more accurately. This helps cells improve their mating success.

Start a totally asymmetric simple exclusion process with a second class particle at 0, particles to its left and holes to its right. If X_{t} is the location at time t of the second class particle, then X_{t} / t converges a.s. to a uniform [-1,1] random variable.

I will prove an analogous result for partially asymmetric exclusion process (with Balázs and Seppäläinen), and explain why this is interesting (with Amir and Valkó).

Abstract: In stable homotopy theory the thick subcategory theorem of Hopkins and Smith classifies thick subcategories of the triangulated category of p-torsion finite spectra. Unstably Bousfield classified nullity classes of p-torsion finite suspensions. We look at analagous results in the derived category of a commutative noetherian ring D(R), and some of its subcategories satisfying suitable finiteness conditions. Hopkins and later Neeman proved that thick subcategories of D_{perf}(R) can be classified by their supports, which are subsets of Spec(R). In analogy to Bousfield's result, we show that nullity classes in D^b_{fg}(R) can be classified by certain increasing functions from Z into Spec(R). By an observation of Keller and Vossieck, it turns out that t-structures are just nullity classes together with a right adjoint of the inclusion. From this we derive an extra condition that the increasing function must satisfy to correspond to a t-structure. When R has a dualizing complex, applying a construction of Deligne and Bezrukavnikov thengives a classification of all the t-sctructures in D^b_{fg}(R)

The next talk for UBC/UMC, the undergraduate mathematics colloquium, will be given by Greg Martin.

Title: Prime numbers: What we know, and what we know we think

Abstract:

Questions about the distribution of prime numbers, and about the existence of prime numbers of special forms, have been stymieing mathematicians for over two thousand years. It's almost necessary to study two different subjects: the theorems about prime numbers that we have been able to prove, and the (vastly more numerous) conjectures about prime numbers that we haven't yet succeeded at proving. In this talk I'll describe many of the open problems (and a few closed ones) concerning the distribution of primes, mentioning when I can some techniques that have been used to attack them.

The Graduate Colloquium will feature two speakers this Thursday, November 5. We'll have first Kael that will talk about "Knots, tensors, and statistical mechanics" and then Terry who will be speaking about "Nonmeasurable sets".

So join us from 12:30 to 1:30 in LSK 462 with the usual free pizza and pop! Abstracts below,

Cheers,
David

------------------------------------------------

Speaker: Kael Dixon
Title: Knots, tensors, and statistical mechanics

In this short talk, I will breifly introduce knot theory and abstract tensor diagrams. Then I will show how combining these ideas gives the Yang Baxter Equation, which originated in statistical mechanics. Don't worry, there's not actually going to be any statistical mechanics in the talk.

------------------------------------------------

Speaker: Terry Soo
Title: Nonmeasurable sets

We will discuss the nonmeasurable sets defined by Vitali, Ulam, and Shelaha-Soifer.

The Graduate Colloquium will feature two speakers this Thursday, November 5. We'll have first Kael that will talk about "Knots, tensors, and statistical mechanics" and then Terry who will be speaking about "Nonmeasurable sets".

So join us from 12:30 to 1:30 in LSK 462 with the usual free pizza and pop! Abstracts below,

Cheers,
David

------------------------------------------------

Speaker: Kael Dixon
Title: Knots, tensors, and statistical mechanics

In this short talk, I will breifly introduce knot theory and abstract tensor diagrams. Then I will show how combining these ideas gives the Yang Baxter Equation, which originated in statistical mechanics. Don't worry, there's not actually going to be any statistical mechanics in the talk.

------------------------------------------------

Speaker: Terry Soo
Title: Nonmeasurable sets

We will discuss the nonmeasurable sets defined by Vitali, Ulam, and Shelaha-Soifer.

While univariate polynomial interpolation has been a basic tool of scientific computing for hundreds of years, multivariate polynomial
interpolation is much less understood. Already the question from which polynomial space to choose an interpolant to given data has no obvious answer.

The talk presents, in some detail, one answer to this basic question, namely the ``least interpolant'' of Amos Ron and the speaker which, among other nice properties, is degree-reducing, then seeks some remedy for the resulting discontinuity of the interpolant as a function of the interpolation sites, then addresses the problem of a suitable representation of the interpolation error and the nature of possible limits of interpolants as some of the interpolation sites coalesce.

The last part of the talk is devoted to a more traditional setting, the complementary problem of finding correct interpolation sites for a given
polynomial space, chiefly the space of polynomials of degree le k for some k, and ends with a particular recipe for good interpolation sites in the square, the Padua points.

Carl de Boor is a Professor Emeritus in Mathematics and Computer Science at the University of Wisconsin-Madison. He won the 2003 U.S. National Medal of Science. An expert in numerical analysis, Dr. de Boor is the author of more than 150 papers and four books. He has earned world recognition for his work on spline functions, mathematical expressions that describe free-form curves and surfaces. In particular, Dr. de Boor developed simpler approaches to complex spline calculations, a contribution that revolutionized computer-aided geometric design. His work is now routinely applied in a range of fields that rely on precise geometry, including the use of special effects in films, and in the aircraft and automotive industries. Dr. de Boor grew up in East Germany and came to the United States in 1959. He received a doctorate from the University of Michigan in 1966 and joined the UW-Madison faculty in 1972. Until 2003, Dr. de Boor was the Steenbock Professor of Mathematical Sciences and the P.L. Chebyshev Professor of Mathematics and Computer Sciences. He was awarded the John von Neumann Prize by SIAM in 1996. In 1993 he was elected to the National Academy of Engineering, and in 1997 to the National Academy of Sciences. Dr. de Boor also is a member of the Deutsche Akademie der Naturforscher (1998) and a foreign member of the Polish Academy of Sciences (2000). He holds honorary doctorates from Purdue University (1993) and the Technion in Israel (2002)

I will describe some recent work in inverse problems from seismology for the acoustic wave equation. Caustics (or multi-pathing) present in the underlying ray geometry force one to consider certain Lagrangian manifolds with singularities and associate classes of Fourier integral operators, in analogy with the classical Hormander theory for smooth Lagrangians.

In this first talk we'll give an advertisement for the rigorous and formal tools of optimal transportation, highlighting their contribution to diffusion equations, simple proofs of Sobolev and isoperimetric inequalities, generalizing the Ricci-bounded-below condition beyond smooth manifolds, and geometrically reinterpreting the Schroedinger equation. We'll then learn about two ideas at the center of these applications: 1) that probability measures can be formally seen as a Riemannian manifold (F. Otto '01) and 2) certain entropy functionals are convex in this geometry (R. M cCann '94). We'll fill out the hour by reviewing the formal Riemannian structure (local geometry) and rigorous aspects of (global) Wasserstein distance.

n the first part of the talk, I will explain why we may need to go beyond the standard framework of transition state theory (TST) to describe activated processes and reactive events, and I will present another framework, termed transition path theory (TPT), that permits to do that. Unlike TST, which gives mainly an expression for the rate of the reactive event, TPT describes more fully the statistical properties of the reactive trajectories (i.e. those trajectories by which the reactive event occurs), in particular in terms of their probability density function and their probability current. In the second part of the talk, I will describe how TPT can be used to design and/or improve numerical methods for computing the pathways and rate of reactive events. I will focus in particular on the string method and milestoning. These techniques will be illustrated via examples from molecular dynamics.

In the first of a two lecture series (to be completed by Dave Anderson immediately following), we present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex $G$-variety $X$ by its associated arc space $J_{\infty} X$, with its induced $G$-action. If $X$ admits an `equivariant affine paving', then we deduce an explicit geometric basis for the equivariant cohomology ring. Moreover, under appropriate hypotheses, we obtain explicit bijections between bases for the equivariant cohomology rings of smooth varieties related by an equivariant, properbirational map. As an initial application, we present a geometric basis for the equivariant cohomology ring of a smooth toric variety.

Let G be an algebraic group acting on a smooth complex variety X. In joint
work with Alan Stapledon, we present a new perspective on the G-equivariant
cohomology of X, which replaces the action of G on X with the induced action
of the respective arc spaces. I will explain how this point of view allows
one to interpret the cup product of classes of subvarieties geometrically
via contact loci in the arc space, at least under suitable hypotheses on the
singularities. As an explicit example, I'll discuss GL_n acting on the space
of matrices.

In this talk the Meshing program Netgen and the Finite Element library NGSolve are introduced. The possibilities of this software package will be highlighted with the help of two modern examples for Advanced Higher Order Finite Element Methods: A Discontinuous Galerkin Formulation for the Time Domain Maxwell Equations and a Hybrid Discontinuous Galerkin Formulation for the incompressible Navier Stokes Equations. Both methods will be shown from a theoretical point of view as well as implementation aspects with Netgen/NGSolve.

In this second talk we'll see how commonly studied PDEs like the heat equation, nonlinear diffusion, thin film equation, and Schroedinger equation can be formally seen as geometric evolutions in the Riemannian geometry of probability measures. The work on Schroedinger equation is due to Max-K. von Renesse ('09).

Given a morphism of vector bundles on a variety, one is often interested in
the locus of points where the morphism drops rank. When the map is
sufficiently general, the classical Giambelli-Thom-Porteous formula
expresses the class of this locus in terms of the Chern classes of the
vector bundles. I will explain how this is related to a more general
problem of computing the T-equivariant cohomology classes of B-orbit
closures in g/p, and motivated by this perspective, describe a new symmetry
for morphisms coming from triality.

In this talk on joint work with Shahab Shahabi, I would like to describe how algebraic parts of periods of cycles on Shimura curves are interpolated by certain p-adic L-functions. In appropriate situations, derivatives of these p-adic L-functions are related to Heegner and Stark-Heegner points on elliptic curves. This generalizes results of Bertolini-Darmon and Shahabi concerning the analogous situations for classical modular curves.

Note for Attendees

Cookies and tea will be served between the two talks.

WMAX 110 (PIMS mini-symposium in PDE); time changed

Thu 12 Nov 2009, 3:00pm-4:00pm

Abstract

I will speak on certain kinetic and macroscopic models of traffic flow. After a review of the concept of a fundamental diagram the high-density regime will be considered, and the emergence of macroscopic models with nolocalities will be discussed. Numerical evidence (and real traffic data) suggest that travelling "braking" waves form and propagate in response to trigger events. A traveling wave ansatz for solutions of the macroscopic models leads to an unusual functional differential equation, for which preliminary studies will be shown.

Let E be an elliptic curve defined over Q. A pair of primes (p,q) is called an amicable pair for E if #E(F_p) = q and #E(F_q) = p. Although rare for non-CM curves, such pairs are relatively abundant in the CM case. I will explain the difference, present conjectures and experimental data for their frequency, discuss some generalisations and related questions, and spend some time on the still-mysterious j=0 case. This talk will afford an opportunity to use cubic reciprocity. This is joint work-in-progress with Joseph H. Silverman.

We describe a fourth order family generalizing the linear-mobility thin film equation on R^n. In joint work with R. McCann we derive formally sharp converence rates to self-similarity, using a link to Denzler-McCann's analysis of a second order diffusion. We then show (joint with Matthes, McCann, Savare) that a certain range of nonlinearity allows the obtaining of rigorous results for the fourth-order evolution in 1 dimension.

Imagine some commodity being produced at various locations and consumed at others. Given the cost per unit mass transported, the optimal transportation problem is to pair consumers with producers so as to minimize total transportation costs. Despite much study, surprisingly little is understood about this problem when the producers and consumers are continuously distributed over smooth manifolds, and optimality is measured against a cost function encoding some geometry of the product space.

This talk will be an introduction to the optimal transportation, its relation to Birkhoff's problem of characterizing of extremality among doubly stochastic measures, and recent progress linking the two. It culminates in the presentation of a criterion for uniqueness of solutions which subsumes all previous criteria, yet which is among the very first to apply to smooth costs on compact manifolds, and only then when the topological type of one of the two underlying manifolds is the sphere.

How plants and animals achieve their forms has been an enduring question in the history of biology, from early descriptions to modern genetic manipulations. Plant shapes are especially challenging, since spatial chemical patterns determine cell type, but also drive (and respond to) tissue growth, a major determinant of overall plant architecture. Increasingly, physical and mathematical scientists are becoming involved in the unique problems of mechanics, transport, and pattern formation in plants. My work uses Turing-type reaction-diffusion models to drive localized surface growth, in 3D. I have been able to generate many of the shapes seen in plants, fitting results to data from single-celled algae and more recently to conifer embryos. These shapes can be understood in terms of transitions between solutions to the reaction-diffusion equations in response to domain change. I will describe some of the open computational problems to achieving stable, accurate model solutions with large domain growth and arbitrary shape change, and some of the directions we are taking experimentally and analytically to further characterize the chemical control of plant shape.

Goresky and MacPherson observed that certain pairs of
algebraic varieties with torus actions have equivariant cohomology
rings that are "dual" in a sense that I will define. Examples of
such pairs come up naturally in both representation theory and
combinatorics. I will explain how this duality is in fact a shadow
of a much deeper relationship, in which certain categories of sheaves
on the varieties are Koszul dual to each other.

We present a new algorithm for the evaluation of the periodized Green's function for Helmholtz equation in two and three dimensions. A variety of classical algorithms (based on spatial and spectral representations, Ewald transformation, etc.) have been implemented in the past to evaluate such acoustic fields. As we show however, these methods become unstable and/or impractically expensive as the frequency of use of the sources increases. Here we introduce a new numerical scheme that overcomes some of these limitations allowing for simulations at unprecedented frequencies. The method is based on a new integral representation derived from the classic spatial form, and on suitable further manipulations of the relevant integrands to render the integrals amenable to efficient and accurate approximations through standard quadrature formulas. We include a variety of numerical results that demonstrate that our algorithm compares favorably with every classical method both for points close to the line where the poles are located and at high-frequencies while remaining competitive with them in every other instance.

We derive a general Hardy inequality and show most Hardy inequalities can be seen as special cases of this inequality. In addition we characterize the improvements of this inequality and (time permitting) we show an application of this inequality to the regularity of stable solutions to a nonvariational equation.

The representation dimension of a finite group G is the minimal dimension
of a faithful complex representation of G. I will discuss the maximal
representation dimension of groups of order p^n for fixed prime p and
integer n, and show that p-groups of maximal representation dimension can
be found (non-uniquely) in a certain class of groups of nilpotence class 2.

This talk is based on joint work with Zinovy Reichstein and Masoud
Kamgarpour.

Cluster expansions give convergent expressions for measures on infinite dimensional spaces, such as those that occur in statistical mechanics. One approach to these convergence questions is via fixed points for infinite dimensional increasing functions. Sometimes it is not so difficult to show that there is a fixed point, indeed, a unique least fixed point.

Abstract: Motivated by basic questions from robotics and molecular biology, we consider certain configuration spaces, and some associated maps to two dimensional Euclidian space. We are able to understand the singular sets for a natural subset of these maps.
For this subset, we apply variants of Bott-Morse theory to determine the structure of inverse images - in the configuration spaces - of curves and points in the plane.
In turn, these results answer key questions about the structure of spaces of length preserving immersions of polygons into Euclidean space and provide insights into the process of protein folding

The next talk for UBC/UMC, the undergraduate mathematics colloquium, will be given by David Kohler.

Title: The Missing Region Problem

Abstract:

In this talk, we will study the missing region problem and its solution. This will allow us to discuss various interesting mathematical ideas and concepts along the way, such as the strong law of small numbers and the Euler characteristic of a polyhedron ... some neat thinking outside the box, and we'll be back to Pascal's triangle in the end.

I will give a brief introduction to the development of soliton theory. Then I will present the constructive theory of nonlinear wave equations and its application to the (N + 1)-dimensional generalized nonlinear Klein-Gordon equation. Finally, the systematic method to compute the point symmetries of a PDE and its application will be introduced.

In geometry finite reflection groups (subgroups of the orthogonal group of an Euclidean space that are generated by reflections) appear, e.g., as groups of symmetries of certain regular polytopes. We will establish a correspondence between those groups and root systems, finite subsets of R^n that were first mentioned in Lie theory but have since then been observed in many different contexts like combinatorics or cluster algebras to name just a few. With help of those sets we will manage to give a nice description of finite reflection groups as well as a complete classification.

In the 1950's, reaction-diffusion equations were proposed as a model for the symmetry-breaking process that organisms undergo as they transition from single cells to embryos. There are plenty of other applications that involve them, most notably as a mechanism responsible for pattern formation in organisms. Perhaps the main reason for the popularity of the model is that linear stability analysis provides considerable insight into the types of patterns that the stationary solutions will have. The talk will serve as an introduction to reaction-diffusion equations and Turing analysis. For the people already familiar with the subject, we will also discuss the limitations that occur when considering domains in 2 or more dimensions.

A discussion of symmetry is good for the fluff section of any grant proposal. In this talk I will discuss finite symmetry groups of spheres. These questions have been part of topology from the beginning. Poincare showed that the binary icosahedral group acts freely on the 3-sphere and that the quotient is a homology 3-sphere. This led to his famous question about simply connected three manifolds.

Traditionally, three kinds of group actions on spheres have been studied, linear actions, smooth actions and continuous actions. The linear actions of a group are its orthogonal representations. A smooth action of a group is a homomorphism to Diff S^n and a topological actions is a homomorphism to Homeo S^n (the topological group of homeomorphisms from the sphere to itself). These correspond to three of the geometries of the sphere. Homotopy theory studies the most fundamental geometry, the one where only the toughest invariants are left.

There are several ways to define the homotopy actions of a group. In the easiest definition, a homotopy action of G on S^n is an action of G on a space X that is homotopy equivalent to S^n. But X can be complicated; it need not be a manifold or even finite dimensional.

The linear actions of G are the orthogonal representations of G. The smooth and topological actions are only completely understood when the group is acting freely, the so called spherical space form problem, and only a few finite groups can act freely on a sphere. Much to our surprise, Grodal-Smith have completely classified the homotopy actions of a finite group on a sphere. The surprise arises because the classification of homotopy actions is equivalent to computing

[BG,B Aut S^n]

where Aut S^n is the topological monoid of self equivalences of S^n; B Aut S^n and BG are the classifying spaces. Both are infinite dimensional and classical techniques are useless. The entry into computation is the modern technology for studying homotopy fixed points.

I will report on joint work with E. Macri on the space of stability
conditions for the derived category of the total space of the canonical
bundle on the projective plane. It is a 3–dimensional manifold, with
many chamber decompositions coming from the behaviour of moduli spaces
of stable objects under change of stability conditions.

I will explain how this space is related to classical results by Drezet
and Le Potier on stable vector bundles on the projective plane. Using
the space helps to determine the group of auto-equivalences, which
includes a subgroup isomorphic to \Gamma_1(3). Finally, via mirror symmetry,
it contains a universal cover of the moduli space of elliptic curves
with \Gamma_1(3)–level structure.

(Huang) We consolidate an unorganized point cloud with noise, outliers, non-uniformities, and in particular interference between close-by surface sheets as a preprocess to surface generation, focusing on reliable normal estimation. Our algorithm includes two new developments. First, a weighted locally optimal projection operator produces a set of denoised, outlier-free and evenly distributed particles over the original dense point cloud, so as to improve the reliability of local PCA for initial estimate of normals. Next, an iterative framework for robust normal estimation is introduced, where a priority-driven normal propagation scheme based on a new priority measure and an orientation-aware PCA work complementarily and iteratively to consolidate particle normals. The priority setting is reinforced with front stopping at thin surface features and normal flipping to enable robust handling of the close-by surface sheet problem.
We demonstrate how a point cloud that is well-consolidated by our method steers conventional surface generation schemes towards a proper interpretation of the input data.

(van den Berg) The use of l1 regularization in optimization problems to promote sparsity in the solution has increasingly become a standard technique. More recently, related types of regularization have been used in different contexts to obtain similar sparsifying effects. In this talk we present a framework that solves problem formulations in these new areas by applying root-finding on the Pareto curve.

The theory of optimal transport is concerned with phenomena arising when one matches two mass distributions in a most economic way, minimizing transportation cost of moving mass from one location to another. We consider an optimal transportation problem with costs satisfying certain type of degenerate curvature condition. This condition is a slightly stronger but still degenerate version of the Ma-Trudinger- Wang condition for regularity of optimal transport maps. We explain a continuity result of optimal maps with rough data on local and global domains. If time permits, we will also explain a connection to Principal- Agent problem in microeconomics. These reflect joint work in progress with Alessio Figalli and Robert McCann.

We study experimentally the effect of a mean flow, imposed on a buoyant exchange flow of two miscible fluids of equal viscosity, in a long tube oriented close to horizontal. We measure the evolution of the front velocity, Vf, as a function of the imposed velocity, V0. At low V0 an exchange flow dominated regime is found, as expected, and is characterized here by Kelvin-Helmholtz-like instabilities. With increasing V0 we unexpectedly observed that the flow becomes stable. Here also Vf increases linearly with V0, with slope > 1. At large V0 we find Vf ~ V0.

A simple random walk (SRW) on a graph is a Markov chain whose state space is the vertex set and the next state distribution is uniform among the neighbors of the current state. A graph is called recurrent if a SRW on it returns to the starting vertex with probability 1, and called transient otherwise. The path of a walk on a graph is simply the set of edges this walk has traversed. Our main result is that the path of a SRW on any graph is a recurrent graph. The proof uses the electrical network interpretation of random walks. We will give a sketch of the proof, including the necessary background, and discuss related questions and conjectures.

I will give a brief introduction to the development of soliton theory. Then I will present the constructive theory of nonlinear wave equations and its application to the (N + 1)-dimensional generalized nonlinear Klein-Gordon equation. Finally, the systematic method to compute the point symmetries of a PDE and its application will be introduced.

In this talk, through basic examples, I will try to illustrate how various classical problems of analysis arise when modeling the interactive motion of a fluid and a rigid structure: well-posedness, stability, asymptotic analysis, control and inverse problems...

We show how the method of Green and Tao can be used to obtain the correct order of magnitude for the number of k-term arithmetic progressions in the integers representable as the sum of two squares, with a similar Roth-like theorem for subsets of positive relative density. We'll discuss the possibility of sharpening this to an asymptotic formula.

This talk is based on a series of five lectures given by Doug Ulmer at the Park City Mathematics Institute in 2009. I will present Ulmer's proof that elliptic curves over function fields can have arbitrarily high Mordell-Weil rank.

Coin tosses are the very essence of probability theory. Yet, there are a number of simply stated problems that have resisted solution. These may involve Peter Winkler's clairvoyant demon, and they may sometimes be cast in the language of so-called dependent percolation. Three such problems and their extensions are discussed in this talk, together with some recent progress in collaboration with Ander Holroyd. The only prerequisite for the audience is an affinity for tossing a coin.

I will summarise how our quantitative description of the Universe as a whole has been dramatically improving, through advances in instrumentation which allow us to gather higher fidelity data which, combined with simple physical models and computationally-intensive statistical approach, enable us to determine a few cosmological parameters to high precision, leading (of course) to new questions.

I will present work-in-progress on the construction of compactifications of the moduli space of curves with A_k-singularities. These spaces conjecturally give moduli interpretations of certain log canonical models of the moduli space of curves. This is joint work with David Smyth and Fred van der Wyck.

## Note for Attendees

Cookies and tea will be served between the two talks.