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. (A continuation of last week's talk)

Aim of the talk is to make a survey on some recent results concerning analysis over spaces with Ricci curvature bounded from below. I will show that the heat flow in such setting can be equivalently built either as gradient flow of the natural Dirichlet energy in L^2 or as gradient flow if the relative entropy in the Wasserstein space. I will also show how such identification can lead to interesting analytic and geometric insights on the structures of the spaces themselves. From a collaboration with L.Ambrosio and G.Savare'.

In Sujatha's talk in this seminar last semester, several equivalence relations (numerical equivalence, homological equivalence and algebraic equivalence)
appeared along with Jannsen's theorem that the category of motives is abelian semi-simple if and only if the equivalence relation is numerical equivalence. The proof of this theorem is beautiful, simple and short. (The main tool is the double-centralizer theorem.) I will try to do it justice in my lectures.

An edge-weighting vertex colouring of a (di)graph is an edge-weight
assignment such that the accumulated weights at the vertices yields
a proper vertex colouring. If such an assignment from a set S exists,
we say the graph is S-weight colourable. Using the Combinatorial
Nullstellensatz and a classical theorem of Schur, we show that every
digraph is S-weight colourable for any set S of size 2. It is
conjectured that every graph with no isolated edge is {1,2,3}-weight
colourable. We explore the problem of classifying those graphs which
are {1,2}-weight colourable.
This is joint work with Reza Naserasr, Mike Newman, Ben Seamone,
and Brett Stevens.

As part of Smale's program for smooth dynamics, David Ruelle gave a definition of a Smale space as (roughly) a topological dynamical system which has a local product structure of contracting and expanding directions for the dynamics. A special case is certain symbolic dynamical systems called shifts of finite type. In the late 1970's, Wolfgang Krieger, motivated by ideas from C*-algebra theory and K-theory, provided a beautiful algebraic invariant for shifts of finite type. The aim of this talk is to show how this invariant may be extended to the class of all Smale spaces as a kind of homology theory which provides a Lefschetz formula. Such a theory was conjectured by Bowen. (I will attempt to define and give examples of all the dynamical concepts: Smale space, shifts of finite type, etc.)

A type A_{n-1} root polytope is the convex hull in R^n of the origin and a subset of the points e_i-e_j, 1\leq i< j \leq n. A collection of triangulations of these polytopes can be described by reduced forms of monomials in an algebra generated by n^2 variables x_{ij}, for 1\leq i< j \leq n. In a closely related noncommutative algebra, the reduced forms of monomials are unique, and correspond to shellable triangulations whose simplices are indexed by noncrossing alternating trees. Using these triangulations we compute Ehrhart polynomials of a family of root polytopes. We extend the above results to more general families of polytopes and algebras of types C_n and D_n. Special cases of our results prove several conjectures of Kirillov regarding these algebras.

While in the theory of one-dimensional subshifts we have a fairly complete picture about when a factor map between two shifts of finite type exist. The situation for Z^d shifts of finite type is much more complicated and so far even the case of factoring onto the full Z^d shift has only a partial answer (which we will present in the talk).
Moreover we will introduce a class of Z^d matrix shifts whose entropy is constructed to be exactly log N and which come with a natural factor map onto the corresponding full N shift. We will give some examples of those equal-entropy extensions of Z^d full shifts realizing various mixing and periodic point conditions.
This will be a rather informal talk on some things I am working on and some related open questions.

Branched polymers are certain configurations of nonoverlapping disks in the plane. In 2003 Brydges and Imbie discovered some remarkable formulas for the volumes of configuration spaces of branched polymers. These formulas mysteriously involve combinatorial numbers like (n-1)!. We introduce branched polymers arising from any central hyperplane arrangement A and express the volume of the resulting configuration space through the characteristic polynomial of A. This is joint work with Alex Postnikov.

We describe Galois covers in the context of the étale site of a scheme, seen as a generalization of Galois field extensions. In particular, we use this idea to show that the étale cohomology of a field agrees with the standard galois cohomology.

The Hilbert scheme of projective space is a fundamental moduli space in algebraic geometry. The naive generalization of the Hilbert scheme can fail to exist for some spaces of interest, however. D. Rydh and I have generalized the Hilbert scheme, to the Hilbert stack, and have shown that the Hilbert stack is always algebraic. I will describe the Hilbert stack and some of the ideas behind the proof.

In Sujatha's talk in this seminar last semester, several equivalence relations (numerical equivalence, homological equivalence and algebraic equivalence) appeared along with Jannsen's theorem that the category of motives is abelian semi-simple if and only if the equivalence relation is numerical equivalence. The proof of this theorem is beautiful, simple and short. (The main tool is the double-centralizer theorem.) I will try to do it justice in my lectures.

The one-factorizations of the complete 3-uniform hypergraph with 9 vertices are classified by means of an exhaustive computer search. It is shown that the number of isomorphism classes of such one-factorizations is 103000.

We're pleased to have Bud Homsy back to give a UBC/UMC talk.

Title: Viscous Fingering in Porous Media: Modelling Unstable Growth Processes
Q: What do oil production, fractal aggregates, electrochemical deposition, and lightening have in common?

A: The underlying mathematics, of course! This talk will focus on modeling unstable flow in porous media, which important in enhanced oil production.It is also a good example of unstable growth processes and of free boundary problems in mathematics.I’ll start by reviewing the basic partial differential equations of fluid flow in porous materials and then discuss the stability analysis of the uniform flow state, with some emphasis on physical mechanisms.Then I’ll discuss the nonlinear regime of this instability which is only accessible through numerical simulation.This will help understand the connection between this instability and other unstable growth processes.

Determinantal (also called fermion) point processes are point processes (on discrete or continuous spaces) whose correlation functions are given by determinants of matrices coming from certain kernel functions. In this talk we start with a couple of examples where DPP's appear. Focusing on the discrete DPP's, we discuss the Gibbianness of them. We benefit from this property to construct some dynamics that leave a DPP invariant. We will typically construct Glauber and Kawasaki dynamics for DPP's.

The well understood phenomenon of Anderson localization says (in its dynamical formulation) that adding random fluctuations to the potential of a Schrodinger operator will lead to the absence of wave transport for the solution of the time-dependent Schrodinger equation. Several years ago it was argued by Burrell and Osborne that a corresponding phenomenon should hold in quantum spin systems. As an example they used the xy-spin chain to show on the physical level of rigor that the introduction of disorder will lead to zero-velocity Lieb-Robinson bounds. We will show
how recent results on Anderson localization can be used to make this result rigorous and, in fact, to improve on the conclusions reached by Burrell and Osborne.

There is a general principle for algebraic dynamical systems
that the growth rate of periodic points should be the entropy.
This has to be suitably interpreted, and I will formulate a
conjecture for which there are no known counterexamples. For
a single toral automorphism this is equivalent to a deep
result in diophantine analysis. For several commuting group
automorphisms the corresponding diophantine result is not
known, but I will describe recent work with Schmidt and
Verbitskiy using homoclinic points which provides an
alternative approach in many cases.

Let x be an algebraic number and let M(x) denote its Mahler measure. If x = x_{1}...x_{N} the t-metric
Mahler measure M_{t}(x) is a convenient way to study the smallest possible values of M(x_{n}) in
terms of x. In joint work with J. Jankauskas, we resolve an earlier conjecture
regarding M_{t}(x) for rational x. This result suggests a generalization to higher degree x,
which turns out, however, to be false. We provide an infinite family of quadratic
counterexamples and discuss how the conjecture should be modified.

Consider the negative Dirichlet or Neumann Laplacian on a square. Add a potential perturbation
which is supported on a small disk. How should the potential be placed in the square in order
to minimize the lowest eigenvalue of the resulting Schrodinger operator? The answer to this
question for the case of Neumann conditions is very different from the answer for the
Dirichlet case. In particular, for the Neumann case the answer is independent of the
sign of the potential. We will discuss how the solution of this problem ultimately
led to a proof of localization near the spectral boundary of the random displacement
model. The latter is a model for a random Schrodinger operator which is used to
model structural disorder in a crystal. A proof of localization for this model was long
considered challenging due to the non-monotone dependence of the operator on the random parameters.

We continue to describe sheaves of abelian groups on an arbitrary étale site. In particular, we prove that the category has enough injective; allowing us to define étale sheaf cohomology.

As in most cohomology theories over a topological space, calculations can be tricky. To solve this problem, we talk about Cech cohomology on an étale site, and show the equivalence of this cohomology to the étale cohomology whenever the base space satisfies some mild conditions.

I will talk about the sheaf associated to a group scheme, focusing on the case of the multiplicative group. I will discuss connections between its cohomology and the Picard and Brauer groups of a scheme.

This is the first of a series of talks in which, subject to time and interests, we will discuss some or all of the following:
-Definitions and basic results from the theory of Polynomial Identity (PI) rings.
-Major results from PI-theory: The Amitsur-Levitzki Theorem, Kaplansky’s Theorem, Posner’s Theorem.
-Construction of the ring of generic matrices and the universal division algebra.
-The centre of the universal division algebra.
-“Universal” properties of the universal division algebra: Galois subfields and rational specialization.
-The universal division algebra is an object of maximal essential dimension.

We continue on with the study of H^1(X_et, G) and its interpretation the set of principal homogenous spaces for G, when G is a sheaf of groups on X_et.

This is the second of a series of talks in which we will discuss some of the following:
-Major results from PI-theory: The Amitsur-Levitzki Theorem, Kaplansky’s Theorem, Posner’s Theorem.
-Construction of the ring of generic matrices and the universal division algebra.
-The centre of the universal division algebra.
-“Universal” properties of the universal division algebra: Galois subfields and rational specialization.
-The universal division algebra is an object of maximal essential dimension.

Let G be a compact connected Lie group act on a topological space X
in such a way that all isotropy subgroups are connected and of maximal
rank. In this talk we provide a criterion to determine when K_{G}^{*}(X) is
free over the representation ring R(G).

The Mayer expansion is a power series expansion that has a central
place in statistical mechanics. It is also full of combinatorial miracles that
relate it to graphs, forests and branched polymers. I will discuss the background,
the results, and some open problems.

Do your students (or other people you meet) say anything about math or learning math that makes you cringe?

Student perceptions of mathematics play a role in their motivation and approaches to learning in their math courses. Last term, we adapted an existing survey for Physics (*) and surveyed students at the beginning and end of a range of Math courses (including several first-year calculus courses). This has allowed us to assess student attitudes and perceptions, and to track how they shift over time.

In this Lunch Series, we will present our development of the survey and some of these initial results. We would also like to gather input from members of the department about the content of the survey: what perceptions or attitudes do mathematicians have about their own subject, and which would you hope students develop as they pursue their undergraduate degree? We hope to see you there for discussion and pizza.

Entropy" is a key notion in the study in of dynamical systems. This quantity reflects the "uncertainty", or "randomness" of a system. Subshifts are topological dynamical systems whose elements are sequences over a given finite alphabet. A translation-invariant measure on a subshift corresponds to a finite-valued stationary stochastic process. Measures obtaining maximal entropy are in some sense "most random" or "most uniform" among those with a given support. In this talk, I will present older and newer results of various authors regarding measures of maximal entropy.

Background:
The Canadian mathematics and statistics communities are engaged in a long range planning (LRP) exercise for 2012-2017/22. This exercise is funded by NSERC, and draws on the expertise of the societies, institutes and the communities of researchers in mathematics and statistics. The site longrangeplan.ca serves as the primary repository for information about the development of the plan.

Members of the steering committee are visiting universities to as part of their community consultations.

In this capacity, Rachel Kuske and Alejandro Adem will give a short presentation on the process, and the progress to date, which will be followed by general discussion.

A call for submission of written documents has also been issued, and we welcome input from interested parties. More details and guidelines for submission are provided on the home page of the website

The call for abstracts for Frontiers in Biophysics 2011 is now open! Please send your name, talk title, abstract, and affiliation (UBC, SFU, etc) to frontiersinbiophysics2011@gmail.com. This is a one-day conference highlighting interdisciplinary research within the areas of biophysics and computational and mathematical biology in the greater Vancouver area. Participants from undergraduates to emeritus are welcome and encouraged to attend and/or present a talk or poster. The fifth annual conference of its kind, Frontiers in Biophysics 2011 will be held on the UBC campus in Vancouver on Saturday February 26, 2011. The schedule will include student presentations, a poster session, and a keynote address by Professor Tim Elston from the University of North Carolina.

The structure of the zero set of a multivariate polynomial is a topic of
wide interest, in view of its ubiquity in problems of analysis, algebra,
partial differential equations, probability and geometry. The study of
such sets originated in the pioneering work of Jung, Abhyankar and
Hironaka and has seen substantial recent advances in an algebraic setting.

In this talk, I will mention a few situations in analysis where the study of
polynomial zero sets plays a critical role, and discuss prior work in this
analytical framework in two dimensions. Our main result (joint with
Tristan Collins and Allan Greenleaf) is a formulation of an algorithm for
resolving singularities of a real-analytic function in any dimension with
a view to applying it to a class of problems in harmonic analysis.

## Note for Attendees