The first of a series of lectures on Severi-Brauer varieties. Starting from their definition and the geometry (line bundles, etc) they hold. And more importantly, the connection between Central Simple Algebra and Brauer-Severi varieties, such as how to relate the ground field, the Brauer group function field of the variety and the Picard group of the variety.

We show that the catenoid is the unique surface of least area (suitably understood) within a geometrically natural class of minimal surfaces. The proof relies on a techniques involving the Weierstrass representation used by Osserman and Schiffer to show the sharp isoperimetric inequality for minimal annuli. An alternate approach that avoids the Weierstrass representation will also be discussed. This latter approach depends on a conjectural sharp eigenvalue estimate for a geometric operater and has interesting connections with spectral theory. This is joint work with J. Bernstein

This is the weekly study seminar MATH 620A. If you are interested, please come to this meeting (or send an e-mail to Julia Gordon,
if you have not done so already). The future meetings of this seminar will not be announced on this page.

In 1960, Feit and Fine were interested in the question posed by the title and to answer it, they found a beautiful formula for the number of pairs of commuting n by n matrices with entries in the field F_q. Their method amounted to finding a stratification of the variety of commuting pairs of matrices into strata each of which is isomorphic to an affine space (of various dimensions). Consequently, their computation can be interpreted as giving a formula for the motivic class of the commuting variety, that is, its class in the Grothendieck group of varieties. We give a simple, new proof of their formula and we generalize it to various other settings. This is joint work with Andrew Morrison.

The second of a series of lectures on Severi-Brauer varieties. Starting from their definition and the geometry (line bundles, etc) they hold. And more importantly, the connection between Central Simple Algebra and Brauer-Severi varieties, such as how to relate the ground field, the Brauer group function field of the variety and the Picard group of the variety.

Consider percolation on the Hamming cube {0,1}^n at the critical probability p_c
(at which the expected cluster size is 2^{n/3}). It is known that if
p=p_c(1+O(2^{-n/3}), then the largest component is of size roughly 2^{2n/3} with
high probability and that this random variable is not-concentrated. We show that for any
sequence eps(n) such that eps(n)>>2^{-n/3} and eps(n)=o(1) percolation at
p_c(1+eps(n)) yields a largest cluster of size (2+o(1))eps(n)2^n.
This completes the description of the phase transition on the Hamming cube and settles a
conjecture of Borgs, Chayes, van der Hofstad, Slade and Spencer.

Our approach is to show that large percolation clusters have inherent randomness
causing them to clump together and form a giant cluster. The behavior
of the random walker on the Hamming cube plays a key role in the proofs of such statements.

Call an irreducible representation of SL_n(Z/qZ)$ "new" if it does not factor through SL_n(Z/r Z) for a proper divisor r of q. It is easy to obtain a lower bound on the dimension of a new representation which is linear in q (in the case of q prime and n=2 this is a classical theorem of Frobenius), and I will discuss the problem of getting a polynomial lower bound with larger exponent. This is part of joint work with Dubi Kelmer in which we obtain spectral gaps for congruence towers of arithmetic hyperbolic manifolds.

We study the probability that the origin is connected to the sphere of radius R in critical percolation in high dimensions, namely when the dimension d is a large fixed constant, or when d>6 and the lattice is sufficiently spread out. I will present highlights of the proof that this probability decays like R^{-2}. No knowledge of percolation will be assumed.

Professor George Bluman will be kicking off UBC/UMC this term.

Title: Dimensional Analysis, Modelling and Invariance

In a new field of research, for a given quantity of interest, called an unknown or "dependent variable", the first step of an expert is to determine the essential independent quantities that it can depend upon (including constants/parameters and variables ("independent variables") as well as the fundamental dimensions of all quantities. Often, the expert needs to conduct intelligent experiments to determine quantities and their dimensions. The only mathematics that can be used at this stage is "dimensional analysis".

The application of dimensional analysis reduces the number of essential independent quantities. For a real problem, any formula (equation, differential equation, etc.) relating quantities is invariant under any change of systems of units where each fundamental dimension is scaled by a positive factor. It follows that each formula can be made dimensionless. As a consequence, the expert can design proper (and safe) model experiments at more reasonable cost to predict the values of a quantity of interest in terms of relevant values of essential independent quantities in an actual situation.

The talk will give the mathematical background of dimensional analysis. Applications will include a proof of the Pythagoras Theorem and the deduction of the amount of energy released in the first atomic explosion of 1945 from motion picture data released in 1947 even though the amount of energy released was classified "top secret" until 1950! The talk will be accessible to all students and should serve as a motivation to learn different topics in linear algebra.

In this talk we present an algebraic context for knot theory. Knotted
trivalent graphs (KTGs) along with standard operations defined on them
form a finitely presented algebraic structure which includes knots, and in
which many topological knot properties are defineable using simple
formulas. Thus, a homomorphic invariant of KTGs provides an algebraic way
to study knots. We present a construction for such an invariant: the
starting point is extending the Kontsevich integral of knots to KTGs. This
was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki
in the late 90's using the theory of associators. We present an elementary
construction building on Kontsevich's original definition, and discuss the
homomorphic properties of the invariant, which, as it turns out,
intertwines all the standard KTG operations except for one, called the
edge unzip. We prove that in fact no universal finite type invariant of
KTGs can intertwine all the standard operations at once, and present an
alternative construction of the space of KTGs on which a homomorphic
universal finite type invariant exists. This space retains all the good
properties of the original KTGs: it is finitely presented, includes knots,
and is closely related to Drinfel'd associators. Partly joint work with
Dror Bar-Natan.

How do simple local interactions combine to produce complex large-scale structure and patterns? The abelian sandpile model provides a beautiful test case. I'll discuss a pair of conjectures about the scale invariance and dimensional reduction of the patterns formed. A new perspective on sandpiles views them as free boundary problems for the discrete Laplacian with an extra integrality condition. Joint work with Anne Fey and Yuval Peres.

Let $K$ be a field of characteristic $p>0$ and let $f(t_1,\ldots ,t_d)$ be a power series in $d$ variables with coefficients in $K$. We discuss a recent generalization of both Derksen's recent analogue of the Skolem-Mahler-Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices $(n_1,\ldots ,n_d)\in \mathbb{N}^d$ for which the coefficient of $t_1^{n_1}\cdots t_d^{n_d}$ in $f(t_1,\ldots ,t_d)$ is zero is generated by a finite-state automaton that accepts the base $p$ expansions of $d$-tuples of natural numbers as input. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to $S$-unit equations and more generally to the Mordell--Lang Theorem over fields of positive characteristic. (joint with Boris Adamczewski)

Note for Attendees

Refreshments will be served between the two talks.

Abstract: In a previous lecture I waved my hands at the subject of period integrals for automorphic forms on SL_2. If that lecture had a point, it was that the phenomenon of non-vanishing of quadratic twists of L-functions was somehow related to the non-vanishing of the symmetric square L-function. In this lecture, I will try to substantiate that claim by describing the relative trace formula of Jacquet, and by sketching the general shape of this formula for the group SL_2. In the end, I'll try to explain how one might get information about quadratic twists directly from the symmetric square if only one had a stable form of Jacquet's formula, and if one could explicitly evaluate the stable and kappa-stable portions.

A substantial part of extremal combinatorics studies relations existing between densities with which given combinatorial structures (fixed size ``templates'') may appear in unknown (and presumably very large) structures of the same type. Using basic tools and concepts from algebra, analysis and measure theory, we develop a general framework that allows to treat all problems of this sort in an uniform way and reveal mathematical structure that is common for most known arguments in the area. The backbone of this structure is made by commutative algebras defined in terms of finite models of the associated first-order theory.

In this talk I will give a general impression of how things work in this framework, and we will pay a special attention to concrete applications of our methods.

Start with n particles at the origin in the square grid Z2, and let each particle in turn perform simple random walk until reaching an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting random set of n occupied sites is close to a disk. We show that its fluctuations from circularity are, with high probability, at most logarithmic in the radius of the disk, answering a question posed by Lawler in 1995. These logarithmic fluctuations were predicted numerically by chemical physicists in the 1980's. Joint work with David Jerison and Scott Sheffield.

This will be a survey of what we do and don't know about what happens to potential Mathematics, Science, and Engineering majors as they make the transition from high school to university mathematics, highlighting where the need for more information is most pressing and how university programs are adapting to meet the needs of entering students.

This talk is presented by the CWSEI and the Mathematics Department.

A quantum cluster algebra is a subalgebra of an ambient skew field of rational functions in finitely many indeterminates. The quantum cluster algebra is generated by a (usually infinite) recursively defined collection called the cluster variables. Explicit expressions for the cluster variables are difficult to compute on their own as the recursion describing them involves division inside this skew field. In this talk I will describe the rank 2 cluster variables explicitly by relating them to varieties associated to valued representations of a quiver with 2 vertices. I will also indicate to what extent the theory. I present is applicable to higher rank quantum cluster algebras.

Informally, the essential dimension of a finite group is the minimal number of parameters required to describe any of its actions. It has connections to Galois cohomology and several open problems in algebra. I will discuss how one can use techniques from birational geometry to compute this invariant and indicate some of its applications to the Noether Problem, inverse Galois theory, and the simplification of polynomials.

Building on I. Dolgachev and V. Iskovskikh's recent work classifying finite subgroups of the plane Cremona group, I will classify all finite groups of essential dimension 2. In addition, I show that the symmetric group of degree 7 has essential dimension 4 using Yu. Prokhorov's classification of all finite simple groups with faithful actions on rationally connected threefolds.

We re-describe what a Severi-Brauer variety is by displaying a particularly good embedding into projective space. We also present a theorem by Chatelet, that uniquely describes the projective space among Severi-Brauer varieties.

Efforts to control the Mountain Pine Beetle infestation in British Columbia and Alberta include large-scale landscape manipulations such as clearcutting, and cost-intensive techniques such as green attack tree removal. Unfortunately, it is unclear just how effective these techniques are in practice. In order to determine and predict the effectiveness of various management strategies, we need to understand how MPB disperse through heterogeneous habitat, where heterogeneity is measured in terms of species composition and tree density on the landscape. In this talk I will present a spatially-explicit hybrid model for the Mountain Pine Beetle (MPB) dispersal and reproduction. The model is composed of reaction-diffusion-chemotaxis PDEs for the beetle flight period and discrete equations for the overwintering stage. Forest management activities are also included in the model. I will discuss the formation of beetle attack patterns and the impacts of management in the PDE model.

Introduced in 1963, the Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. I will survey a number of new results on the mixing of the dynamics at high, critical and low temperatures. At high temperatures the dynamics exhibits the cutoff phenomena, a sharp transition in the convergence to equilibrium. We show a polynomial upper bound on the mixing time at the critical temperature establishing the conjectured critical slowdown behaviour. Finally at low temperatures with all plus boundary conditions we prove a quasi-polynomial time upper bound on the mixing time.

Joint work with Eyal Lubetzky, Fabio Martinelli, and Fabio Toninelli.

In this talk I will introduce some of the combinatorics
behind affine permutations. These are a generalization of classical
permutations that, when viewed as a reflection group, amount to adding
an affine reflection to the set of generators (hence the name). We
now end up with an infinite group, however, many of the same results
from classical permutations still hold. Classifying families of
permutations in terms of the patterns they avoid has been studied for
a long time. However, applying pattern avoidance to affine
permutations is fairly new. I will discuss pattern avoidance, and in
particular, the role it plays in the geometry of the corresponding
affine Schubert varieties. Part of this talk is joint work with Sara
Billey.

The reconstruction problem on the tree concerns the propagation of information in Markov processes on trees and has been studied in probability, statistical physics, computational biology, information theory and theoretical computer science. I will discuss progress in establishing thresholds for the reconstruction problem and give an overview of its applications to phylogenetic reconstruction, mixing times of Markov chains, random constraint satisfaction problems and the computational complexity of counting problems.

I will first give a gentle introduction to LS category and
then review some of the progress that has been made over the last 10
years in connection with Iwase's counterexample to Ganea's conjecture.
This will include many problems that are still open. At the end I will
describe some recent work with Rodriguez.

This talk is directed to non-specialists and students. The Metropolis is a very famous algorithm which can used to simulate random variables. The Ising model is a model coming from statistical physics. After defining the notions needed, I will present the Metropolis algorithm and apply it to the simulation of the Ising model.

Abstract: The Langlands correspondence relates global Galois
representations with automorphic representations; the local
correspondence works at each prime. For any reductive group $G$ over
a local field $K$ we construct a complex reductive group $^LG$. For
any homomorphism from the Galois group of $K$ to $^LG$ (called a
Langlands parameter) we then construct a set of representations of
$G(K)$ (called an L-packet). I will make these constructions explicit
in the case that the Langlands parameter is discrete, tamely ramified
and regular and that $G$ is the unitary group associated to a tame
extension of
$K$.

We will prove that solutions to the defocusing energy-critical Schrodinger equations are global in the hyperbolic space H^3. The relevance of the energy-critical case is that in this case, one needs to understand how to take into account the scaling limits of the equation. In particular, one needs to see how to connect solutions to the corresponding equation on a Euclidian space to solutions of the original equation which concentrate as they evolve. To try and understand the influence of the geometry, we will also look at some results on other spaces in the other directions (the volume of balls grows slowly as the radius goes to infinity). This is a joint work with A. Ionescu and G. Staffilani.

Motivated by Hochman's notion of subdynamics of a Z^d subshift, we define and examine projective subdynamics of Z^d shifts of finite type (SFTs) where we restrict not only the action but also the phase space. In analogy with the notion of stable and unstable limit sets in cellular automata we distinguish between stable and unstable projective subdynamics.

First we review the classification of Z sofic shifts which can (not) appear as projective subdynamics of Z^2 (Z^d) SFTs both in the stable and unstable regime - these are results obtained jointly with Ronnie Pavlov.

In a second part of the talk we present results on the projective subdynamics of Z^d SFTs with some uniform mixing condition. In particular there is a compatibility condition assuring the projective subdynamics of Z^d SFTs has to be sofic and if time permits we explain a construction that allows to realize any mixing Z sofic as stable projective subdynamics of some strongly irreducible Z^2 (Z^d) SFT.

Dispersive equations are evolution equations in which different components move with different velocities. This way the solution is mixed and dispersed without being damped. Examples include the wave equation, Schrodinger equation, Beam equation...
Focusing on the fourth order case, we will show how one can understand and control the global dynamics of some nonlinear equations when there is a lot of volume to disperse, especially in the critical case, which corresponds to situations where there is no preferred scale.

I will explain an interpretation of Illusie's results on the deformation theory of commutative rings in terms of the cohomological classification of torsors and gerbes. Then I'll show how this point of view can be used to solve some other deformation problems. I'll also indicate some deformation problems that I don't know how to approach this way.

This will be an expository talk on some modern variants of the Hilbert transform. The main viewpoint will be from an L^p-theory perspective, but connections with other fields will also be discussed. I will focus on the double Hilbert transform along polynomial surfaces in \mathbb{R}^3, and outline the proof of a result by Carbery, Wainger and Wright that states such operators are bounded on all L^p, 1 < p < \infty, if and only if the polynomial surface is "well-behaved" enough with respect to its Newton diagram.

This is a first talk of a series about Hochschild (co)homology and cyclic (co)homology.
We will start with the basic definitions and later we will see some applications of these
theories to stringy topology

We define the Picard group of a variety using (Weil) divisors and show that the Picard group of projective space P^n is the integers. We also describe the two generators of the Picard group of P^n as line bundles.

Let K be a field extension of a field k and let sigma be a k-algebra automorphism of K. Then one can construct a skew field D=K(x;sigma), by creating a skew polynomial ring with relations ax=xsigma(a) for a in K, and then inverting all nonzero elements. Unlike commutative localization, however, noncommutative localization can be quite pathological and the division ring can sometimes have free subalgebras. In this talk, based on joint work with Daniel Rogalski, we show that dynamical properties of the automorphism sigma completely determine when this can occur.

The real trees form a class of metric spaces that extends the class of
trees with edge lengths by allowing behavior such as infinite total
edge length and vertices with infinite branching degree. We use
Dirichlet form methods to construct Brownian motion on a given locally
compact R-tree equipped with a Radon measure. We then characterize
recurrence versus transience.

This is joint work with Anita Winter and Michael Eckhoff.

Humans are currently jeopardizing the other species in life's fabric and potentially our own future due to our overuse of common resources. Over the last two decades, a large effort has focused on trying to persuade individuals to consume differently. These conservation efforts largely appeal to guilt and an individual's willingness to do the right thing. What about the role of shame in solving the tragedy of the commons? I will explore the differences between guilt and shame and then present results from a recent public goods experiment conducted with Christoph Hauert and others that tests the effects of shame on cooperation. I will also examine our findings in the context of shame's real world applications and concerns.

In modern language, Fermat's Descent Infini establishes that an elliptic curve has a Mordell-Weil group of rank 0. Since then, the method has been generalized to provide an upper bound on the rank of any elliptic curve and further work also allows the analysis of the Mordell-Weil group of Jacobians of many hyperelliptic curves. Reformulating work of Schaefer, we present a general framework, in principle applicable to any curve, which allows us, under certain technical conditions, to provide an upper bound on the rank of the Jacobian of any curve. In particular, we have been able to compute some rank bounds on Jacobians of smooth plane quartic curves. This is joint work with Bjorn Poonen and Michael Stoll.

Note for Attendees

Refreshments will be served between the two talks.

Piecewise isometries are a class of mappings of low complexity
and a test case for discontinuous dynamics. We analyze the dynamical
behaviour of a simple family of piecewise isometries and use methods from
uniform distribution to give a quantify the behaviour.

The Skyrme model is one of the important nonlinear sigma models in quantum field theory. In this talk I will report some recent progress on he dynamics of Skyrmions, focusing on the (3+1) space-time case.

The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. By using the dispersive Klein-Gordon effect, Guo first constructed a global smooth irrotational solution in the three dimensional case. It has been conjectured that same results should hold in the two-dimensional case. The main difficulty in 2D comes from the slow dispersion of the linear flow and certain nonlocal resonant obstructions in the nonlinearity. I will discuss a new method to overcome these difficulties and construct smooth global solutions for the 2D Euler-Poisson system.

We study the category of sheaves on the étale site of a scheme. In particular we show an equivalence between sheaves on (Spec(k))_ét and the category of discrete Gal(k_sep/k)-module.

I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to provide a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a result regarding topological obstructions for this problem. I will also address the question of whether a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.

I will present a time-discretization algorithm which allows to build Leray weak solutions of the Navier-Stokes equations. Such algorithm has a genuine variational structure and has been inspired by the study of gradient flows. From a collaboration with S.Mosconi.

## Note for Attendees

Refreshments will be served between the two talks.