Motivated by the classical Mordell-Lang conjecture, we formulate a natural dynamical analogue in the context of semiabelian varieties, which we prove under a stronger hypothesis. We also present counterexamples which
occur in the Dynamical Mordell-Lang problem once we relax our hypothesis.

Self-avoiding polygons are simple closed curves embedded in a regular lattice. They belong to a family of objects, known collectively as lattice animals, that lie at the heart of lattice models of polymers, magnets and other phenomena. This large family of discrete structures has been a source of combinatorial problems for over 50 years and many
basic questions remain stubbornly unsolved. For example, the simple enumerative question - "How many self-avoiding polygons are there of length n?" is currently best answered by an algorithm that requires exponential time and memory.

The question of counting polygons on the cubic lattice is enriched by topology - self-avoiding polygons have well defined knot types. The exact computation of the number of polygons of length n and fixed knot type K is extremely difficult - indeed the current best algorithms can barely touch the first knotted polygons. In this talk I will discuss a
different approach to the problem. Instead of of exact methods, we have used an approximate enumeration method - which we call the GAS algorithm. This is a generalisation of the famous Rosenbluth method for sampling self-avoiding walks.

Using this algorithm we have estimated the number of polygons of different lengths and knot types on three different cubic lattices. These give direct evidence that the asymptotic growth of the number of polygons of a fixed knot type K is simply related to the growth of the number of unknotted polygons and the number of prime components in K. We have also studied the relative frequencies of different knots - for example, the ratio of the number of trefoils to the number of figure-eights. We see strong evidence that these ratios are universal suggesting that a very long closed curve is about 27 times more likely to be a trefoil than a figure-eight.

Markley & Paul, Engel, Calkin-Wilf and Friedland show a sequence of computable and very good lower and upper bounds

on the entropy of 2-D symmetric SFT's. I'll describe the method and show Engel's observation that these bounds monotonically converge to the entropy. I'll also discuss some of Engel's related conjectures and disprove some of them."

Motivated by applications to Brill-Noether theory and higher-rank Brill-Noether theory, we discuss several variations on Grassmannians. These include "doubly symplectic Grassmannians", which parametrize subspaces which are simultaneously isotropic for a pair of symplectic forms, "linked Grassmannians", which parametrize tuples of subspaces of a chain of vector spaces linked via linear maps, and "symplectic linked Grassmannians", which is an amalgamation of the linked Grassmannian and symplectic Grassmannian.

We introduce cohomological dimension following Serre's Galois Cohomology book. With the goal of proving some theorems on fields of cohomological dimension less or equal to 1.

I will describe new and old conjectures and some recent results
concerning typical maximal words in the group of signed permutations,
called signed sorting networks. These mirror similar results and
conjectures, concerning sorting networks, and suggest that these
models are in a common universality class.

The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers, particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.

The recent solution of M. Hill, M. Hopkins and D. Ravanel of the Kervaire invariant problem is likely to lead to a new area in the study of homotopy groups of spheres. In this talk I will explain that problem, and indicate what the above three authors have achieved. I’ll discuss the impact of their result on the study of vector fields and mention some problems which now open up in this “Post Kervaire-Milnor” era

The next talk in the TAAP Seminar Series will be by David Kohler and Desiree Mou. David Kohler is a senior graduate student in our department, and a recent winner of the UBC Graduate Teaching Award. Desiree Mou is a facilitator and instructional developer.

Title: How to assess your teaching skills
Abstract: This session will discuss, in a very honest and down-to-earth fashion, how to improve your ability to assess your own teaching skills. The focus will be on two specific items: getting better feedback and using goal-setting techniques.

I'll describe a joint result with Michal Hochman characterizing entropy
numbers of multidimensional subshifts of finite type (SFTs) in
recursive-theoretic terms. This result relates to Berger's theorem which
is the solution to Wang's problem: a non-empty 2 dimensional SFT may not
have periodic points, and (thus) the emptiness problem is undecidable.
The first part of this talk will be devoted to background on Berger's
theorem, apriodic SFTs, undecidability, and substitution systems.

The period of a principal homogeneous space (a torsor) of an algebraic torus S over a field F is the order of its class in the group H^1(F,S). We compute the period of a generic torsor of S in terms of the character lattice of the torus S.

MATH ANNEX 1100 (PIMS/UBC distinguished colloquium)

Fri 8 Oct 2010, 3:00pm-4:00pm

Abstract

Essential dimension of an algebraic object is the smallest number of algebraically independent parameters required to define the object. This notion was introduced by Buhler and Reichstein. Relations to algebraic geometry and representation theory will be discussed.

Just as Cheeger's inequality relating spectral gap and
isoperimetric constants for manifolds has an analogue for graphs, so it has
an analogue for an action of a group preserving a probability measure. In
the case of Bernoulli actions, one can determine the spectral gap. This
allows one to analyze when certain combinatorial constructions are possible
as factors of IID random variables. (Joint work with Fedja Nazarov.)

We study the space M_g of isometry classes (or conformal equivalence classes) of smooth manifolds, diffeomorphic to #^g(S^d \times S^d), the connected sum of g copies of S^d \times S^d. For 2d=2, this is essentially the moduli space of Riemann surfaces. There is a variant M_{g,1} where we consider moduli of manifolds with an embedded D^{2d}; connected sum with S^d \times S^d gives a map M_{g,1} \to M_{g+1,1}, and we can form the direct limit M_{\infty,1}. The work of Madsen and Weiss on Mumford's conjecture determines the homology of M_{\infty,1} in the case 2d=2. We give a similar description of the homology of M_{\infty,1} in higher dimensions (2d \geq 6). This is joint work with Oscar Randal-Williams.

Speaker: Tyler Helmuth Title: What is Percolation?

Percolation, in the mathematical sense, is an abstraction of the physical phenomena of percolation: the movement of fluids through porous materials. I'll introduce the most classical percolation models, along with some basic questions and results. I'll also describe some more sophisticated questions that are related to the Fields Medal winning work of Werner (2006) and Smirnov (2010).

Speaker: Maxim Stykow Title: What is Cohomology?

We've all heard this "too-abstract-for-it's-own-good" word float around the math department but what actually does it mean? I will describe the motivation and intuitive ideas behind cohomology and provide insight into different flavors of cohomology and their applications. Finally, I will discuss the surprising fact that homotopy theory is the mother of all cohomology theories - great and small.

The Rallis inner product formula provides a precise relationship between the inner product of a theta lift and the special value of a $L$-function. We hope to give a flavor of the ingredients involved and a rough description of the recipe for cooking up this formula. We also hope to discuss/speculate on possible applications.

Location Changed to IAM Lounge - Room 306 Leonard S. Klinck Bldg.

Thu 14 Oct 2010, 2:00pm-3:00pm

Abstract

Animal cells crawl on surfaces using the lamellipod, a flat dynamic network of actin polymers enveloped by the cell membrane. Recent experiments showed that the cell geometry is correlated with speed and with actin dynamics. I will present mathematical models of actin network self-organization and viscoelastic flow explaining these observations. According to this model, a force balance between membrane tension, pushing actin network and centripetal myosin-powered contraction of this network can explain the cell shape and motility. In addition, I will discuss Darci flow of cytoplasm and its role in the cell movements.

After an introduction to the classical category of motives, we shall explain the construction of the category of birational motives which was carried out in joint work with B. Kahn, along with some applications.

Microtubules confined to the two-dimensional cortex of elongating plant cells must form a parallel yet dispersed array transverse to the elongation axis for proper cell wall expansion. Collisions between microtubules, which migrate via hybrid treadmilling, can result in plus-end entrainment (“zippering”) or catastrophe. Here, I present (1) a cell-scale computational model of cortical microtubule organization and (2) a molecular-scale model for microtubule-cortex anchoring and collision-based interactions between microtubules. The first model treats interactions phenomenologically while the second addresses interactions by considering energetic competition between crosslinker binding, microtubule bending and microtubule polymerization. From the cell-scale model, we find that plus-end entrainment leads to self-organization of microtubules into parallel arrays, while collision-induced catastrophe does not. Catastrophe-inducing boundaries can tune the dominant orientation. Changes in dynamic-instability parameters, such as in mor1-1 mutants in Arabidopsis thaliana, can impede self-organization, in agreement with experiment. Increased entrainment, as seen in clasp-1 mutants, conserves self-organization, but delays its onset. Modulating the ability of cell edges to induce catastrophe, as the CLASP protein may do, can tune the dominant direction and regulate organization. The molecular-scale model predicts a higher probability of entrainment at lower collision angles and at longer unanchored lengths of plus-ends. The models lead to several testable predictions, including the effects of reduced microtubule severing in katanin mutants and variable membrane-anchor densities in different plants, including Arabidopsis cells and Tobacco cells.

In finite fields and Euclidean space, we shall discuss criteria under which the set $A\cdot A+A \cdot A+\dots+A \cdot A$ is substantially larger than the set $A$ itself. We shall also discuss connections between this problem and the question of distribution of simplexes in vector spaces over finite fields and Euclidean space.

Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big divisor D, Okounkov showed how to construct a convex body in R^d, and this construction has recently been developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata. In general, this Okounkov body is quite hard to understand, but when X is a toric variety, it is just the polytope associated to D via the standard yoga of toric geometry. I'll describe a more general situation where the Okounkov body is still a polytope, and show that in this case X admits a flat degeneration to the corresponding toric variety. This project was motivated by examples, and as an application, I'll describe some toric degenerations of flag varieties and Schubert varieties. There will be pictures of polytopes.

We prove the unique existence of solutions of the 3D incompressible Navier-Stokes equations in an exterior domain with small non-decaying boundary data, for all $t \in R$ or $t \in (0,\infty)$. In the case $t \in (0,\infty)$ it is coupled with a small initial data in weak $L^{3}$. If the boundary data is time-periodic, the spatial asymptotics of the time-entire solution is given by a Landau solution which is the same for all time. If the boundary data is time-periodic and the initial data is asymptotically discretely self-similar, the solution is asymptotically the sum of a time-periodic vector field and a forward discretely self-similar vector field as time goes to infinity. This is a joint work with Kyungkuen Kang and Hideyuki Miura.

Let R be an associative algebra with linear basis {x_w}. The problem of
determining the multiplicative structure coefficients of R with respect to
the basis {x_w} has been an interesting problem in combinatorics, algebra
and geometry. For example, the algebra of Symmetric polynomials has a
basis of Schur functions whose structure coefficients are the classical
Littlewood-Richardson numbers.

In this talk, I will discuss a new formula for computing
Littlewood-Richardson type coefficients for a large class of algebras
including cohomology rings of flag varieties of a Kac-Moody group and
coinvariant algebras of finite Coxeter groups. This formula relies only on
the data given in a Cartan matrix. This talk will be similar to the one I
gave in the AG seminar a few weeks ago, but with an emphasis on the
combinatorics involved.

There is a general cohomology defined by Sweedler for
co-commutative Hopf algebras, generalizing the usual cohomology of a
group or a Lie algebra. Recently it was discovered that
low-dimensional groups could be defined without the co-commutativity
requirement. In joint work with Christian Kassel, we have given the
first few examples of computations with these, in the case of algebras
of functions on groups. These turn out to be related to torsors in
algebraic geometry, and Drinfeld twists in quantum groups theory

I will review some recent (and also not-so-recent) progress
on the understanding of the behavior of the random walk on the
hypercubic lattice among bounded i.i.d. random, nearest neighbor
conductances that have a heavy lower tail at zero. The center of
focus of my talk will be the large-$n$ behavior of the probability
that the random walk in this random medium returns back to the
starting point after $2n$ steps. First I will show that there are
universal upper bounds on this probability, which enforce the
standard (diffusive) decay in spatial dimensions $d=2,3$ but permit
subdiffusive decay in higher dimensions. Then I will show how one
constructs examples for which the decay is actually provably
subdiffusive in all $d\ge4$. In $d=4$ this is particularly subtle
because there one has to control the walk over a whole range of
spatial scales. Based on various joint papers with N. Berger, O.
Boukhadra, C. Hoffman , G. Kozma and T. Prescott.

The mean ergodic theorem says that when T is a measure preserving transformation of a probability space (X,m), averages of foT^n converge to a limit L when the parameter n is averaged over longer and longer intervals, assuming f is in L^{2}(m). We consider averages the form foT^(a(n)), where a(n) is some sequence growing polynomially. For a large class of sequences a(n), the averages of foT^(a(n)) still converge to the same limit L as above.

We will introduce some of the relavent background, prove some special cases of these results, and survey the situation in general. A subsequent talk will apply these results to additive combinatorics.

I will describe work with Dubi Kelmer on the first Laplace eigenvalue in towers of manifolds covered by real or complex hyperbolic space. All congruence quotients in a given dimension have a uniform spectral gap; we show how to deduce from this a uniform spectral gap for the family of congruence covers of a fixed arithmetic (non-congruence) manifold.

Abstract:
The exclusion process is one of the basic particle processes. In this process, particles move randomly, with the constraint that two particles may not occupy the same position. Of particular interest -- both mathematical and as a model for flow along a path -- is the process where the individual motions are along a line, with a drift in one direction.

Certain instances of this process are closely related to many other seemingly different models. These include a random growth model for a pile, and a certain card shuffling process. These connections allow us to prove certain properties of the exclusion processes. It is commonly believed that many large scale properties of such models, do not depend on the microscopic parameters of the model. This is referred to as universality and verifying it is a major open problem.

I will describe several aspects of the exclusion process, including its so called hydrodynamic limit, its stationary distributions and the asymptotic speed of particles. I will discuss some progress on extending these results to less specific exclusion processes.

Note for Attendees

The speaker aims to keep the talk accessible to a general mathematical audience:

Familiarity with continuous time Markov chains and stationary distributions may be useful, but is NOT crucial, since very concrete examples will be discussed. Some of the finer points alluded to depend on other notions from an undergraduate probability course.

In this talk we use a combination of the methods developed by Gowers and
refined by Green and Tao in the proof of Szemeredi's Theorem on long
arithmetic progressions in sets and the Hardy-Littlewood circle method to
obtain a bound for the upper density of a subset A of the integers Z such
that if d>=7, then the set A^d contains no triangles which are similar to a
given triangle. The bound is independent of the given triangle.

Yau conjectured that the first eigenvalue of the Laplacian on compact embedded minimal surfaces in $\mathbb S^3$ should be equal to 2. We prove that Yau's conjecture is true for all minimal surfaces that are known to exist so far: the minimal surfaces constructed by Lawson, by Karcher-Pinkall-Sterling, and by Kapouleas-Yang. (Joint work with M. Soret)

We are interested here in interacting particle systems. Such systems
are used to approximate probability measures. The two examples I will talk about
are the genetic particle systems and Bird/Nanbu-like particle systems.
Propagation of chaos is a property shared by these systems, it is an asymptotic
property in term of the number of particles. I will show how one can do precise
computation concerning this propagation of chaos and deduce convergence results
for these systems.

I am going to explain how representations of the braid
groups endowed with a compatible symplectic form give rise to link
invariants with values in the Witt ring of the field considered. The
construction makes use of Maslov indices. In the end, using the Burau
representation, we get one invariant which "contains" many others:
signatures, Jones metaplectic invariants, and a polynomial which is
almost the one by Alexander-Conway.

A wide range of questions in discrete mathematics begin with ``How many...''. The main focus of this talk will be to discuss the use of generating functions as a means of constructing, counting, and encoding information about combinatorial classes. In particular, the question of how to construct bigger combinatorial classes using smaller ones will be discussed. Examples of classes that we can count and those where counting is not so easy will be used throughout to illustrate the concepts and ideas presented.

The theory of complex multiplication provides algorithms for obtaining elliptic curves over finite fields with a number of points known in advance, which finds applications in cryptography and primality proving. The main ingredient is the construction of Hilbert class fields of imaginary-quadratic number fields. While these are of exponential size with respect to the input, several approaches have been described that are quasi-linear in the output. I will give a self-contained overview of the algorithms and the latest record computations.

Note for Attendees

Refreshments will be served between the two talks.

In applications ranging from energy harvesting to toxin detection, nanocrystalline materials promise to yield revolutionary new technologies. The ability to produce nanomaterials with controlled morphologies remains a difficulty, however. Producing nanomaterials using self-assembly promises to be a low-cost, high-yield approach. However a fundamental understanding of growth instabilities that occur and lead to the natural formation of nanostructures and patterns is still needed. In this talk, we will discuss mathematical models and simulation methods for such problems that account for the influence of strongly anisotropic surface tension forces and elastic stresses that arise during growth of nanocrystalline materials. In the context of epitaxial growth of thin films, we demonstrate the dependence of nanostructure dynamics on the growth conditions, such as the deposition rate and the temperature.

We introduce a collection of polynomials G_N in Z[z] having the following property: the Nth cyclotomic polynomial divides G_N if and only if N cannot be represented as a sum of two odd primes. Numerical evidence suggests that, in fact, G_N is irreducible and has no roots on the unit circle. We proceed to discuss some basic properties of G_N, including giving asymptotic estimates on the size of their coefficients. At various stages, this work is joint with P. Borwein, K.K. Choi, and G. Martin.

Most tissues are hierarchically organized into lineages, which are sets of progenitor-progeny relationships where the cells differ progressively in their character due to differentiation. It is increasingly recognized that lineage progression occurs in solid tumors. In this talk, we develop a multispecies continuum model to simulate the dynamics of cell lineages in solid tumors. The model accounts for spatiotemporally varying cell proliferation and death mediated by the heterogeneous distribution of oxygen and soluble chemical factors. Together, these regulate the rates of self-renewal and differentiation of the different cells within the lineages and lead to the development of heterogenous cell distributions and formation of niche-like environments for stem cells. As demonstrated in the talk, the feedback processes are found to play a critical role in tumor progression, the development of morphological instability, and response to treatment.

The multiplicative group of units in the ring Z/nZ is one of the first groups an undergraduate student of algebra encounters. As n varies over the positive integers, these multiplicative groups form a naturally occurring family of finite abelian groups, whose structure encodes various interesting arithmetic invariants of the modulus n (for example, the number of prime factors of n). Their algebraic structure is already somewhat complicated, but the real fun begins when we want to understand the statistical distribution of these invariants. What distribution do we get, for example, if we choose positive integers "at random" and record how many prime factors they have? The answer, in a sense that analytic number theory has made rigorous, is surprisingly a Gaussian distribution.

This combination of an algebraic setting, analytic number theory questions, and probabilistic answers provides a thriving ecosystem in which to do field research (well, group research, anyway). Many of the speaker's current projects involve dissecting multiplicative groups and seeing what structures there are to discover. In this talk we take a tour of several interesting arithmetic invariants connected to the multiplicative group (mod n) and the investigation of their distribution.

Note for Attendees

Background knowledge that will help:

The quotient ring Z/nZ and its multiplicative group of units (Z/nZ)^x

The Euler phi function

The structure theorem for finite abelian groups

The Gaussian (normal) distribution
Refreshments will be served in MATH 125 at 2:45 p.m.

## Note for Attendees

Refreshments will be served between the two talks.