In classical Lie theory a homomorphism of Lie groups f : H--> G, with H simply connected, is uniquely given by its effect on
the Lie algebras Lie(f) : Lie(H) --> Lie(G). When f : H --> G is a weak
morphism of Lie 2-groups, with H 2-connected (i.e., \pi_iH vanish
for i=0,1,2), we prove that f is uniquely given by Lie(f), where
Lie(f) : Lie(H) --> Lie(G) is the induced morphism in the derived
category of 2-terms diff. graded Lie algebras. We also exhibit a
functorial construction of the 2-connected cover H<2> of a Lie
2-group H.

The theory of geometric quantization is one way of producing a "quantum system" from a "classical system," and has been studied a great deal over the past several decades. It also has surprising ties to representation theory. However, despite this, there still does not exist a satisfactory theory of quantization for systems with singularities.

Geometric quantization requires the choice of a polarization; when using a real polarization to quantize a regular enough manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. However, there are many types of systems to which this result does not apply. One such type is the class of completely integrable systems, which are examples coming from mechanics that have many nice properties, but which are nontheless too singular for Sniatycki's theorem to apply.

In this talk we will explore one approach to the quantization of integrable systems, and show a Sniatycki-type relationship to Bohr-Sommerfeld fibres. However, some surprising features appear, including infinite-dimensional contributions and strong dependence on the polarization.

I will give at least a brief explanation of both geometric quantization and integrable systems, and hope to make the talk accessible to a general differential geometric audience.

When molten rock rises from deep in the earth, invades the earth's crust and cools, this forms new rock structures called igneous intrusions. These widespread features are often associated with the formation of mineral deposits, and there is hope that understanding their emplacement mechanisms will benefit engineering applications by giving insight into how rock behaves at large scale. Particular attention is given here to so-called large mafic sills and laccoliths, which can be proposed as natural analogues to hydraulic fractures that grow relatively close to a free surface.

The use of elastic plate theory to model the growth of shallow igneous intrusions has been debated for over 40 years. Investigation has typically resulted in the elastic plate model being heavily questioned or abandoned because it fails to predict the characteristic flat-topped, steep-sided thickness profiles of laccoliths or the strikingly uniform thickness of large mafic sills. However, upon coupling elastic plate theory with a fracture mechanics based propagation criterion and, crucially, the driving force due to the weight of the magma, the predicted thickness profiles and thickness to length relationships for both laccoliths and large mafic sills are consistent with an extensive collection of field data. Furthermore, analysis of the large time asymptotics predict that large mafic sills will attain a thickness that is not only spatially uniform, but also constant in time, depending only on physical properties of the magma and host rock. While a number of questions remain open, it is an exciting prospect that a single, basic model could provide a unifying framework to understanding what controls the first-order behaviour of the growth of laccoliths and large mafic sills.

This talk will present work that results from collaboration with Professor Alexander Cruden of the University of Toronto, Department of Geology.

Let G be a finite group. A G-gerbe over a space X may be
intuitively thought of as a fiber bundle over X with fibers being the
classifying space (stack) BG. In particular BG itself is the G-gerbe
over a point. A more interesting class of examples consist of G-gerbes
over BQ, which are equivalent to extensions of the finite group Q by G.
Considerations from physics have led to conjectures asserting that
the geometry of a G-gerbe Y over X is equivalent to certain "twisted"
geometry of a "dual" space Y'. A lot of progresses have be made recently
towards proving these conjectures in general. In this talk we'll try to
explain these conjectures in the elementary concrete examples of G-gerbes
over a point or BQ.

At the heart of Thurston's proof of Geometrization for Haken manifolds is a family of analytic functions between Teichmuller spaces called "skinning maps." These maps carry geometric information about their associated hyperbolic manifolds, and I'll discuss what is presently known
about their behavior. The ideas involved form a mix of geometry, algebra, and analysis.

In this talk I will describe the coupled integro-partial differential equations that model the evolution of a fluid-driven fracture propagating in a state of plain strain. I will discuss the use of the Mellin Transform and matched asymptotics to establish the asymptotic behavior of the solution in the vicinity of the fracture tip for a number of regimes of propagation. I also describe a novel cubic Hermite collocation scheme to solve these coupled equations. This algorithm involves special blended cubic Hermite-power law basis functions, with an arbitrary index 0<1, which are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. I also discuss the implementation of blended infinite elements to model semi-infinite crack problems. The cubic Hermite collocation algorithm is used to solve a number of different test problems and the results are compared to published similarity, asymptotic, and numerical solutions.

It is well know that for p = 2, the K(1)-localization of KO is EO_1,
and for p = 2; 3, the K(2)-localization of TMF is
EO_2. When does the K(n)-localization of TAF contain a factor of EO_n?
We will provide a complete answer. This is joint work with Mike
Hopkins.

The essential dimension of the projective linear group PGLn is a measure of
complexity of PGLn-torsors or, alternatively, central simple algebras. It was first raised by Procesi in the 1960 and the exact value is still
mostly unknown. We will discuss some recent developments which have led to both new lower
and upper bounds. These are obtained in part by studying classes of algebras with additional
structure, e.g. crossed-products or simple algebras split by a distinguished field extension.

Abstract: A Bers slice is a naturally embedded copy of the Teichmuller
space in the SL(2,C) character variety of a surface. We prove that Bers
slices are never algebraic. A corollary is that Thurston's skinning map
is never constant. The proof involves the theory of complex projective
structures and a little algebraic geometry.

We present the uniform distribution of roots of quadratic congruences on the unit circle. Several proofs have been rediscovered over the years. I will review some of the very different methods involved -- ergodic theory, exponential sums, automorphic forms. It constitutes an excellent introduction to the more delicate Linnik problems. I will then proceed to describe new generalizations. These have far reaching applications to arithmetic geometry and independence of Heegner points on rational elliptic curves.

In this talk, we discuss the behavior of heat kernel for symmetric jump-type process with jumping kernels comparable to radially symmetric function on the spaces. Parabolic Harnack principle and sharp two-sided heat kernel estimates for both small and large time will be discussed. This is a joint work with Zhen-Qing Chen and Takashi Kumagai.

This talk will be a friendly introduction to the basic concepts of game theory, starting with the Nash solution concept (more appropriately, the “Cournot-Nash” concept – you’ll see why). My goal is to highlight some of the central results, illustrate some interesting examples, and to give an idea why game theory has become so important in economic analysis (and other disciplines, too). Technicalities will be kept to a minimum, so it should be easy for everyone to walk away with a good idea of what game theory is all about.

This short talk aims at describing expander graphs and some of their fascinating applications to other fields of mathematics and computer science. Wether you are interested in coding theory, complexity theory, probability theory, number theory or group theory, (with or without a flavour of geometry and linear algebra on the side) there will be something for you. And if none of these really speak to you, at least you'll get a nice promenade in the mathematical landscape.

We discuss the question of quantitative bounds on the sup-norm of automorphic cusp forms. We present an improvement on a recent result by Blomer-Holowinski on Hecke-Maass forms on $X_0(N)$ with large level $N$. Analogous results are then established for all compact arithmetic surfaces by a geometric approach.

The stack of stable maps parameterizes maps from a complete curves having at worst nodal singularities into a smooth scheme. Generally this stack is not smooth, but we will explain how it can be made smooth by relaxing the condition that the source curves be complete. Although the resulting stack is not fibered in groupoids, and therefore may not be easily accessible to geometric intuition, it is a natural setting in which to construct the virtual fundamental class. We will discuss how this generalization can be used to prove a conjecture of Abramovich and Fantechi relating the virtual fundamental classes of two different moduli spaces parameterizing stable maps into mildly singular schemes.

The Langlands program relates complex representations of GL_n(Q_p) to Galois representations. For n = 1 this is explained by class field theory and for n = 2 this is closely related to the theory of modular forms. For general n, this is now understood by the work of Harris-Taylor and Henniart. In the last decade, a mod-p (as well as a p-adic) version of the Langlands program have been emerging, and they have already played an important role in some recent progress in number theory. But so far understanding has been limited to n = 1 and 2. We survey some of the known story in the classical and in the mod p case, and then discuss some recent progress on the classification of mod p representations of GL_n(Q_p), as time permits.

Kalman smoothing is an important topic in control theory, with a myriad of applications. We will discuss some of these applications, present the modeling framework amenable to solution by smoothing, and discuss two related approaches to making the smoother robust against errors in the measurement data. Specifically, we will consider two heavy tailed models for observation noise, discuss the merits of these models from a statistical point of view, show how in each case the statistical model gives rise to an optimization problem with special structure, and then solve these problems to find the a posteriori maximum likelihood (MAP) solution for each model. We will then compare the smoothers' performance on simulated data contaminated with different types of outliers and on real data in an underwater tracking experiment.

Let F be a finite extension of the p-adic numbers. We describe the classification of irreducible admissible smooth representations of GL_n(F) over an algebraically closed field of characteristic p, in terms of "supersingular" representations. This generalizes results of Barthel-Livne for n = 2. Our motivation is the hypothetical mod p Langlands correspondence for GL_n, which is supposed to relate smooth mod p representations to Galois representations.

Note for Attendees

This is a joint Number Theory/Algebraic Groups Seminar.

I will describe some background and recent results on singularity formation (and non-formation) for some simple, physical, and popular geometric PDE describing dynamics of maps into spheres -- the heat-flow, wave map, and Schroedinger map -- in the energy-critical 2D case. I'll try to keep it simple and accessible by illustrating the methods on a symmetric reduction of the heat-flow, leading to a single scalar PDE.

In recent work of Baas-Dundas-Richter-Rognes, the authors prove that the classifying space of 2-vector bundles, K(Vect) is equivalent to the algebraic K-theory of the connective K-theory spectrum ku. In this talk we will show that K(Vect) is the group completion of the classifying space of the 2-category of 2-vector spaces, which is a symmetric monoidal 2-category. We will explain how to use the symmetric monoidal structure to produce a $\Gamma$-2-category, which will give an infinite loop space structure on K(Vect). Then we will show that the equivalence of BDRR is a map of infinite loop spaces.

The first talk this term for UBC/UMC, the undergraduate mathematics colloquium, will be given by Mike Bennett.

Title: Diophantine equations for fun (and profit?)

Diophantine equations are one of the oldest, frequently celebrated and most abstract objects in mathematics. In this talk, I'll attempt to show some of the roles these equations play in modern mathematics and maybe even reveal how they can be used to make a (not particularly) fast buck.

This is an expository talk about the distributive law of algebra and its role in combinatorics, probability, and physics. The main idea is that one can run the distributive law either way, a process sometimes called re-summation. This extraordinarily simple idea underlies a fair amount of current research in probability and mathematical physics.

Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$. Dubickas and Smyth constructed a modified version $M_1$ of $M$ having the triangle inequality. $M_1$ is called the metric Mahler measure. We produce an entire parametrized family $\{M_t\}$ of metric Mahler measures which gives rise to a new reformulation of Lehmer's problem. We further examine the functions $t\mapsto M_t(\alpha)$, for fixed $\alpha$, showing that they are constructed piecewise from certain simpler functions.

Note for Attendees

Cookies and tea will be served between the two talks.

This is the first talk in the teaching seminar associated with the TA Accreditation Program. (All are welcome!)

Title: Engaging students in the classroom: what is all this 'clicker' nonsense?

We will explore ways to engage students to think actively about course material in class using "clickers." We will make use of iClickers throughout this session.

Starting from a classical result of Skolem, Mahler and Lech for linear recurrence sequences, we present an algebraic geometric generalization of it. Then we interpret our result from the point of view of dynamics, linking it with the Mordell-Lang conjecture from Diophantine Geometry. We conclude by studying another dynamical question which generalizes the Manin-Mumford conjecture.

Motivated by the classical Mordell-Lang problem we formulate a dynamical generalization, which we show that it doesn't always hold. Then we discuss the cyclic case of our question, which we call the Dynamical Mordell-Lang Conjecture. We present several positive results which support our conjecture, and discuss the difficulties one has for proving the full conjecture. In particular, our work answers a basic question from complex dynamics.

In the first half of the talk, I will explain the notion of PT stability, as defined by Bayer. I will also explain how it is related to classical stability conditions on sheaves, and other Bridgeland-type stability conditions. In the second half of the talk, I will discuss results on the moduli space of PT-stable objects from my thesis. In particular, I will explain how to use semistable reduction to obtain the valuative criterion of completeness for PT-stable objects.

Temporal cycles in natural populations have long been observed. More recent field studies have shown that in some situations, the population cycles have different phases in different locations, consistent with the existence of a travelling wavetrain of population density, passing through the region. Numerical simulations of predator-prey models with spatial dependence show that travelling wavetrains and other patterns can form behind invading fronts of predators. For reaction-diffusion and similar models, mathematical analysis can be used to predict in some cases what spatiotemporal patterns will form behind an invading front. In this talk I will describe some of the analysis that is possible using dynamical systems methods, especially for predicting the properties of wavetrains that form behind an invading front.

The problem of determining the electrical conductivity of a body by making voltage and current measurements on the object's surface has various applications in fields such as oil exploration and early detection of malignant breast tumour. This classical problem posed by Calder\'on remained open until the late '80s when it was finally solved in a breakthrough paper by Sylvester-Uhlmann. In the recent years, geometry has played an important role in this problem. We will look at the connection between this analysis problem with seemingly unrelated fields such as symplectic geometry and differential topology as well as geometric scattering theory.

The speaker is partially supported by NSF Grant No. DMS-0807502}

This SCAIM lecture is more of a tutorial than a talk. I will present a simple Finite Element Method (FEM) code as a model for the more complex codes in common modern use. Code components include: mesh generation, matrix assembly, a linear solver, and a post-processor. I will discuss these components with reference to their object-oriented (C++) implementations. This tutorial is aimed at those needing a framework for modifying and using existing codes. As a model, the example computes a piecewise-linear approximation of the solution to the Poisson problem on a circular domain with Neumann or Dirichlet boundary conditions. Focus is on implementation issues rather than on issues of accuracy and convergence. The example C++ code will be provided on-line following the talk for use as a teaching and reference tool.

We show that on a smooth compact Riemann surface with boundary (M_0, g) the Dirichlet-to-Neumann map of the Schr\"odinger operator \Delta_g + V determines uniquely the potential V. This seemingly analytical problem turns out to have connections with ideas in symplectic geometry and differential topology. We will discuss how these geometrical features arise and the techniques we use to treat them.

This is joint work with Colin Guillarmou of CNRS Nice.

The speaker is partially supported by NSF Grant No. DMS-0807502 during this work.

We wish to understand the boundary between forbidden configurations on
4 rows that yield a quadratic bound and those that have cubic
constructions. The result is joint with my supervisor and Attila Sali.
The bounds we are concerned with are the following: For a (0,1)-matrix
F, we define forb(m,F) to be the maximum number of columns in an
m-rowed (0,1)-matrix which has no repeated columns and has no
submatrix which is a row and column permutation of F. The asymptotics
of forb(m,F) for arbitrary F have been conjectured by Anstee and Sali.

Abstract
The "colored Tverberg problem" asks for a smallest size of the color
classes in a (d+1)-colored point set C in R^d that forces
the existence of an intersecting family of r "rainbow" simplices with
disjoint, multicolored vertex sets from C. Using equivariant topology
applied to a modified problem, we prove the optimal lower bound
conjectured by Barany and Larman (1992) for the case of partition into
r parts, if r+1 is a prime.
The modified problem has a "unifying" Tverberg-Vrecica type
generalization, which implies Tverberg's theorem as well as the ham
sandwich theorem.
This is joint work with Pavle V. Blagojevic and Gunter M. Ziegler.

In the early 60s Kesten showed that self-avoiding walk in the upper half plane has a decomposition into an i.i.d. sequence of "irreducible bridges". Loosely defined, a bridge is a self-avoiding path that achieves its minimum and maximum heights at the start and end of the path (respectively), and it is irreducible if it contains no smaller bridges. Considering only the 2-dimensional case, one can ask if the (likely) scaling limit of self-avoiding walk, the SLE(8/3) process, also has such a decomposition. I will talk about recent work with Hugo Duminil from Ecole Normale Superieure that provides a positive answer, using only the restriction property of SLE(8/3). In the end we are able to decompose the SLE(8/3) path as a Poisson Point Process on the space of irreducible bridges, in a way that is similar to Ito's excursion decomposition of a Brownian motion according to its zeros. Our decomposition can actually be generalized beyond SLE(8/3) and applied to an entire family of "restriction measures", hence the title of the talk. If time permits I will also talk about the natural time parameterization for SLE(8/3), which has immediate applications towards the bridge decomposition.

A vector bundle is a continuously varying family of vector spaces; for example, the set of lines that are tangent to a smooth curve is a vector bundle. In this talk, we will draw pictures, give examples, state applications, and (time permitting) learn how vector bundles can mend a heart broken by Liouville's theorem. Our emphasis will be on intuition, and all technical details will be suppressed.

In statistical physics we study models for interacting particles. One of the main open questions is to find models which exhibit a phase transition (think about water turning into gas at 100 Celcius). First I will introduce all the big words people use in statistical physics. Then I will show a result of Lebowitz and Penrose on the existence of a phase transition for a certain class of models. If there is some time left I can talk about some extensions of the Lebowitz and Penrose result.

Dissipative nonlinear systems such as fluid dynamical systems can reach a chaotic state when the parameter measuring the nonlinearity is large. For instance, parallel shear flows of Newtonian fluids are turbulent when the ratio of the nonlinear inertial term and the viscous dissipation term, defined by the Reynolds number is sufficiently important. In non-Newtonian fluid flows, an additional nonlinearity is introduced via the constitutive equation. For viscoelastic fluids, this nonlinearity can give rise to turbulent flow at low Reynolds number (Larson Nature 2000 and Groisman and Steinberg Nature 2000). The degree of nonlinearity is expressed by the Weissenberg number which is a product of a characteristic rate of deformation and the relaxation time of the polymer. The shear-thinning behaviour, non linear decrease of the effective viscosity with the shear rate, is the most common property of non Newtonian fluids. It is reasonable to inquire, whether an interplay between this nonlinearity and inertia can lead to a chaotic flow. This point has been addressed in (Ashrafi and Khayat PRE 2000) using low order dynamical system (generalized Lorenz system) in the Taylor-Couette flow of weakly-thinning fluid. It is shown that the additional nonlinearity gives rise to a Hopf bifurcation otherwise non existent for Newtonian fluid. In the previous talk dealing with the transition to turbulence for a yield-stress shear-thinning fluid in a pipe, a new state with a robust coherent structure characterized by two weakly modulated counter-rotating longitudinal vortices was described. In this nonlinear asyrnmetric state, time-averaged axial velocity profiles exhibited increasing asymmetry with increasing Reynolds number. In the present talk, velocity fluctuations are analysed, and it is shown that this state displays the salient feature of chaos, namely, randomly fluctuating motion excited in a broad range of spatial and temporal scales. Beside the experimental part, a spectral Petrov-Galerkin method is used to study the nonlinear stability of Hagen-Poiseuille flow of shear-thinning fluid. In the first step and as suggest the experimental results, the perturbation is assumed homogeneous in the axial direction. In this situation, the numerical results show that travelling waves with an azimuthal wave number m=1, are not sustained.

Traveling fronts are special solutions of reaction-diffusion equations which model phenomena such as propagation of species in an environment or spreading of flames in combustible media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in general inhomogeneous media. We will show that in certain circumstances a unique front exists and it is a global attractor of the corresponding parabolic evolution, thus describing long time dynamics for very general solutions of the PDE. In contrast to this, we will also present examples of media where no traveling front solutions exist.

Reaction-diffusion equations are parabolic partial differential equations used in the modeling of phenomena such as propagation of species in an environment or spreading of flames in combustible media. Their general solutions exhibit two basic behaviors, extinction (quenching) and spreading. In this talk we will review recent progress in our understanding of how the motion of the underlying medium, modelled by a fluid flow, affects both the occurence of quenching and the speed of spreading of reaction. The problem turns out to have fruitful connections to questions about mixing effixiency of flows and homogenization of advection-diffusion operators.

## Note for Attendees

Cookies and tea will be served after the talk.