Abstract

The existence and stability of localized patterns of criminal activity
are studied for the reaction-diffusion model of urban crime that was
derived by Short et.~al.~[Math. Models. Meth. Appl. Sci., {\bf 18},
  Suppl. (2008), pp.~1249--1267] as a continuum limit of an
agent-based model. Such patterns, characterized by the concentration
of criminal activity in localized spatial regions, are referred to as
hot-spot patterns and they occur in a parameter regime far from the
Turing point associated with the bifurcation of spatially uniform
solutions. Singular perturbation techniques are used to construct
steady-state hot-spot patterns in one and two-dimensional spatial
domains. Explicit stability thresholds for these patterns are obtained
by first deriving and then analyzing a new class of nonlocal
eigenvalue problems. In a certain parameter regime of this model,
analytical and numerical methods are used to analyze the phenomena of
hot-spot nucleation, whereby elevated regions of criminal activity
emerge ``spontaneously'' from a quiescent background. Finally, an
extended reaction-diffusion model that incorporates the effect of
police is analyzed, and optimal strategies to prevent the occurrence
of stable hot-spots are discussed.

Joint work with: Theodore Kolokolnikov (Dalhousie), Simon Tse (UBC), and
Juncheng Wei (CUHK, UBC).

Abstract

The equivariant embeddings of a split torus have been well-known since the 70s.  The isomorphism classes of such embeddings are classified by combinatorial objects called fans (after Demazure). In this talk, we address the classification of the embeddings of a non-necessary split torus and ask:  Are the isomorphisms classes of such embeddings classified by Galois-stable fans?  If time permits, we will discuss the analogous results in the setting of spherical homogeneous spaces.

Abstract

This talk concerns applications of ergodic theory to Euclidean Ramsey theory. We shall focus mainly on the problem of finding triangular configurations in thickened sets of positive density in the plane (due to Furestenberg, Katzenelson, and Weiss). The generalization for arbitrary configurations in dimension at least 2 (due to Ziegler) is also to be discussed.

Abstract

It is well known that an upper and lower bound on the Monge-Amp{\`e}re measure of a convex function u implies this function must actually be strictly convex. A lesser known result, also by Caffarelli, states that if the Monge-Amp{\`e}re of u has only a lower bound, the contact set between u and a supporting affine function must have affine dimension strictly less than n/2. By means of a careful geometric construction involving the subdifferential, we give an alternative proof of Caffarelli's result, and extend the result to optimal transportation problems with cost functions satisfying the weak Ma-Trudinger-Wang condition. This talk is based on a joint work with Young-Heon Kim.

Abstract

A closed geodesic on a surface can also be viewed as a closed orbit of the geodesic flow on the unit tangent bundle of the surface. In this talk I will discuss the main tool for studying the knot-properties of closed orbits of (three dimensional) flows. This tool is called a template, first defined and used by Birman and Williams to study the well known Lorenzbutterfly.

The theory of templates was first used for geodesic flows by Ghys. I will discuss questions related to his extraordinary result, that the closed geodesics on the modular surface equal the closed orbits on the Lorenz butterfly, and will discuss some generalization of his methods to other hyperbolic surfaces.

Abstract

Our goal will be to get a feeling for what Chow groups are and how do they "look like". We will discuss CH_0 of a smooth compact complex curve and see how the classical Abel-Jacobi theorem gives a very concrete description of it. We will then talk a little about the group of divisors on a smooth variety, for which the situation is not much worse. We will mention the group of zero-dimensional cycles on a surface, which is sometimes considerably more complicated. Finally we will see that while the Chow groups can be defined for non-smooth varieties, defining the intersection product is not a trivial matter at all.

Abstract

Suppose S: X --> X and T: Y --> Y are continuous selfmaps of metric spaces X,Y (especially, with X,Y compact). They are topologically conjugate if there is a homeomorphism h: X --> Y such that hS = Th . Various equivalence relations coarser than topological conjuacy have been considered for such maps S,T including the following: entropy conjugacy (as defined by Bowen); entropy conjugacy (as defined by Buzzi); Borel conjugacy (h is only required to be a Borel-bimeasurable bijection); Borel isomorphism after restriction to a set of measure one for every invariant Borel probability. I'll discuss theorems and questions around these ideas, especially with regard to countable state Markov shifts, for which another relation ("almost isomorphism", in the spirit of Adler-Marcus) is relevant. I'll include new results joint with Buzzi and Gomez, and also a new universality theorem of Mike Hochman.

Abstract

Abstract

Shift equivalence and strong shift equivalence of matrices over a semiring are matrix relations of great significance for symbolic dynamics; they  are also interesting simply for matrices. The classification of nonnegative integral matrices up to strong shift equivalence  is equivalent to the classification of shifts of finite type, the basic systems of symbolic dynamics. This classification problem has been open for 40 years.

I'll discuss the meaning of these relations and recent results (joint with Kim and Roush) for the case of positive matrices with entries from a dense subring of the reals. These results are obtained by extending the path methods approach developed by Kim and Roush in the early 1990s, which is based on a study of paths of conjugate positive real matrices.

Abstract

Extending the work of Jia and Sverak on self-similar solutions of the Navier-Stokes equations, we show the existence of large, forward, discrete self-similar solutions.

Abstract

We consider the contact process on finite trees. We assume that
the infection rate is larger than the critical rate for the one-dimensional
process. We show that, for any sequence of trees with increasing number of
vertices and degree bounded by a universal constant, the expected
extinction time of the process grows exponentially. Additionally, the
extinction time divided by its expectation converges in distribution to the
unitary exponential distribution. This is joint work with Thomas Mountford,
Jean-Christophe Mourrat and Qiang Yao.

Abstract

The talk will be an extended advertisement of a program to apply topological ideas to computation of essential dimension of groups.

Abstract

 We will continue the discussion from the previous talk on the basic ideas of Chow theory. We review the basic properties that make the Chow functor to be an oriented Borel-Moore homology theory, and then we will discuss some further topics including their equivariant version, their operational version and their universal version.

Abstract

Abstract

The Schubert subvarieties of a rational homogeneous variety X are distinguished by the fact that their homology classes form an additive basis of the integer homology of X. In general, the Schubert varieties are singular.

The cominuscule rational homogeneous varieties are those admitting the structure of a compact Hermitian symmetric space (eg. complex Grassmannians). In this case, type-dependent characterizations of the singular loci are known.

I will discuss a type-independent description, by representation theoretic data, of the singular loci. The result is based on a characterization (joint with D. The) of the Schubert varieties by an non-negative integer and a marked Dynkin diagram.

(If there is time left, I will discuss the project in which the integer-diagram characterization arose as a technical lemma. This work aims to determine whether or not the Schubert classes admit any algebraic representatives (other than the Schubert varieties). It is a remarkable consequence of Kostant's work that these algebraic representatives are solutions of a system of PDE; as a consequence, differential geometric techniques may be applied to this algebro-topological question.)

Abstract

Asphaltene deposition is one of the important problems of
oil production that requires an accurate predictive modeling.
Uncontrolled deposition of asphaltenes may lead to a significant
reduction of oil production, or even to a total pipe plugging. We
developed an asphaltene deposition model in a pipeline. The model is
based on data, which are obtained by experiments performed in a
Couette device, where the inner cylinder rotates, and deposition on
the outer wall is studied. A detailed theoretical analysis of an
applicability of a Couette device for imitation of the asphaltene
deposition in a pipe flow is presented. A hypothesis stating that the
asphaltene deposition is a particle size controlled process is
accepted. A population balance model is employed for modeling the
particle size evolution. A concept of the critical particle size is
introduced; only particles that are smaller than the critical size
can deposit. The model developed contains only three parameters that
are determined experimentally by using a Couette device. The model of
asphaltene deposition in a Couette device allows accurate describing
the deposit mass growth in time. Performance of the deposition model
for a pipeline with the coefficients obtained by a laboratory Couette
device is also demonstrated.

Abstract

Furstenberg-Katznelson's Theorem states that if "A" is a subset of Z^d with
positive upper density then for any finite set F of Z^d, A contains an affine image of F. We
wish to prove analogue theorem in prime tuples P^d where positive upper density is replaced by
relative upper density in P^d. This is partially done by Magyar and Cook in the case that no
two points in F have a common projection  on some axis. When we count such configurations P^d
behaves like a random subset of Z^d with density(1/log N)^d but this is not true in general
since P^d has direct product structure. In this talk, we will discuss how to use hypergraph
approach and Gowers' Transference Principle to deal with the case that F is a corner (i,e,.
simplex of the form {(x_1,..,x_d),(x_1+s,x_2,..,x_d),...,(x_1,...,x_d+s)},s \neq 0). We expect
that the same method should also work for any finite set F. This is a joint work with Akos
Magyar. 

Abstract

I will describe recent work that characterizes the Schubert varieties that arise as variations of Hodge structure (VHS).  I will also discuss the central role that these Schubert VHS play in our study of general VHS.  In particular: (i) infinitesimally their orbits under the isotropy action `span' the space of all VHS, yielding a complete description of the infinitesimal VHS; (ii) the cohomology classes dual to the Schubert VHS form an (integral) basis of the invariant characteristic cohomology associated to the system of PDE characterizing VHS.

Abstract

One of the biggest challenges in the field of random Schrödinger operators is to prove the existence of absolutely continuous spectrum for the Anderson model for small disorder in dimensions greater equal to 3. So far, the existence of absolutely continuous spectrum is only known for models on infinite-dimensional tree structures. The first proof, done by Abel Klein for a regular tree, dates back to 1994.
Recent developments considered trees of finite cone type and cross products of trees with finite graphs, so called tree-strips. I will present a proof for the existence of absolutely continuous spectrum for models on tree-strips of finite cone type. The proof uses a version of the Implicit Function Theorem in Banach spaces which are constructed by a supersymmetric formalism using Grassmann variables.

Abstract

In 1997 Elekes showed the following.  Suppose you have an $n\times n$ cartesian product $A \times B$ in the real plane.  If $cn$ lines each contain at least $cn$ points of the cartesian product then $c'n$ of the lines are parallel or $c'n$ of them have the same intersection point.  We show that this result can be extended by considering fewer lines.  Specifically, if we have $cn^{2/3+\beta/3}$ lines each containing $c'n$ points of the cartesian product then $c''n^{\beta} of the lines are parallel or have the same intersection point.  We used this to extend a theorem of Elekes and Ronyai regarding the structure of a surface containing many points of a cartesian product.  These results give a proof of a conjecture of Purdy saying that if two collinear sets in the plane give too few distinct distances then the lines are parallel or orthogonal.  This is joint work with Jozsef Solymosi and Frank de Zeeuw.

Details

Abstract

Let G be a finite group, we consider the homotopy colimit of classifying spaces of abelian subgroups of G. This space is a K(\pi,1) for certain finite groups, but there are examples when the space has non-vanishing higher homotopy groups. I will also talk about the complex K-theory of this space and give some examples.

Abstract

Planar maps are embeddings of graphs on compact orientable
surfaces. We shall consider a special class of maps: triangulations with an
infinite simple boundary. We shall consider measures supported on this
space satisfying two natural properties: translation invariance and domain
Markov property. We shall show these two characters completely characterize
them. The methods reveal some standard maps and some mysterious maps of
hyperbolic nature.

Abstract

 We will define the canonical dimension of a smooth projective variety and compute it in the case of quadratic surfaces.

Abstract

Abstract

One remarkable application of classical Schubert calculus on the cohomology of the Grassmannian is its close connection to the eigenvalue problem on sums of hermitian matrices.  The eigenvalue problem asks:  Given three sequences of real numbers, do there exist hermitian matrices A+B=C with eigenvalues given by the three sequences?  This problem has a generalization to eigenvalues of majorized sums of hermitian matrices where we replace "A+B=C" with "A+B>C".

In this talk, I discuss joint work with D. Anderson and A. Yong where we show that the eigenvalue problem on majorized sums is related to the Schubert calculus on the torus-equivariant cohomology of the Grassmannian in the same way that classical Schubert calculus is related to eigenvalue problem on usual sums of Hermitian matrices.  One consequence of this connection is a generalization of the celebrated saturation theorem to T-equivariant Schubert calculus.

Abstract

I will present an upper bound on the (d-1)-volume and covering numbers of a filtration of the non-differentiability locus of the distance function of a compact set in R^d. A consequence of this upper bound is that the projection function to a compact subset K depends in a Hoelder way on the compact set, in the L^1 sense. This in turn implies that Federer's curvature measure of a compact set with positive reach can be reliably estimated from a Hausdorff approximation of this set, regardless of any regularity assumption on the approximation.

Abstract

In the local Langlands program the (smooth) representation theory of p-adic reductive groups G in characteristic zero plays a key role. For any compact open subgroup K of G there is a so called Hecke algebra H(G,K). The representation theory of G is equivalent to the module theories over all these algebras H(G,K). Very important examples of such subgroups K are the Iwahori subgroup and the pro-p Iwahori subgroup. By a theorem of Bernstein the Hecke
algebras of these subgroups (and many others) have finite global dimension.

In recent years the same representation theory of G but over an algebraically closed field of characteristic p has become more and more important. But little is known yet. Again one can define analogous Hecke algebras. Their relation to the representation theory of G is still very
mysterious. Moreover they are no longer of finite global dimension. In joint work with R. Ollivier we prove that over any field the algebra H(G,K), for K the (pro-p) Iwahori subgroup, is Gorenstein. 

Abstract

Although directed random graph models are frequently used in modeling the electrical activity of networks of cortical neurons, experimental results consistently reveal that the actual network topology is complex, and tends to be clustered locally. This suggests that the random network assumption is unrealistic and that when analysing population dynamics in cortical networks, it is necessary to employ directed network models that incorporate clustering.

 

In this seminar I shall describe simulation results that demonstrate that replacing random connectivity with clustered connectivity can induce instability in subsets of neurons, in terms of significantly increased firing rates. Moreover, it is shown that one specific network topology gives rise to slow bistable switching between low and high states.

 

The aim of presenting this seminar is not to describe finished mathematical work, but rather to seek collaboration or assistance with finding mathematical explanations that predict that clustered connectivity can lead to the bistable or unstable states observed in simulations.

 

Abstract

UBC/UMC is the Undergraduate Mathematics Colloquium at UBC.

These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.

Title: Coding Theory and Practice

Abstract: In this talk, we will introduce the framework of coding theory, including some classical results, open problems and applications to data transmission and storage.

Details

In this thesis we present a reduction theory for the symmetrizable split maximal Kac-Moody groups. However there are many technical difficulties before one can even formulate a reduction theorem. Combining the two main approaches commonly seen in the literature we define a group, first over any field of characteristic zero and then on any commutative ring of characteristic zero. Then we prove a number of structural properties of the group such as representation in the highest weight modules, existence of a Tits system and an Iwasawa decomposition over $\RR$ and $\CC$. Finally we arrive at reduction theory which can only hold for part of the group.

Abstract

We cover some further definitions of canonical dimensions for smooth projective varieties.

Abstract

Abstract

The talk is based on a joint work with G. Alberti, P. Jones and D. Preiss.

Abstract

Higher stacks arise in many contexts in algebraic geometry and differential topology.  The simplest type are higher principal bundles, special cases of which include principal bundles and n-gerbes.  Locally, these objects are presentable by higher cocycles on a hypercover.  With ordinary principal bundles, we obtain a bundle from a cocycle by using the cocycle to construct a Lie groupoid over the trivial bundle on the cover, and then passing to its orbit space.  We establish the existence of an analogous construction for arbitrary higher principal bundles.  Unpacking this construction in examples, we recover the familiar definitions of principal bundles, bundle gerbes, multiplicative gerbes and their equivariant versions, now seen as instances of a single construction.  Applications beyond this include establishing a representability criterion for connected simplicial presheaves, and a Lie's 3rd theorem for finite dimensional L_oo-algebras.

Abstract

Abstract

Sparse signal recovery has been dominated by the basis pursuit denoise (BPDN) problem formulation for over a decade. In this paper, we propose an algorithm that outperforms BPDN in finding sparse solutions to underdetermined linear systems of equations at no additional computational cost. Our algorithm, called WSPGL1, is a modification of the spectral projected gradient for  minimization (SPGL1) algorithm in which the sequence of LASSO subproblems are replaced by a sequence of weighted LASSO subproblems with constant weights applied to a support estimate. The support estimate is derived from the data and is updated at every iteration. The algorithm also modifies the Pareto curve at every iteration to reflect the new weighted  minimization problem that is being solved. We demonstrate through extensive simulations that the sparse recovery performance of our algorithm is superior to that of  minimization and approaches the recovery performance of iterative re-weighted  (IRWL1) minimization of Candès, Wakin, and Boyd, although it does not match it in general. Moreover, our algorithm has the computational cost of a single BPDN problem.

Abstract

In this talk we will discuss the integer distances problem: can we find n points in a general position on the plane such that all the distances between the points are integers? So far this question has been resolved up to 7 points. We will present results related to the problem and our recent efforts to find 8 points with integer distances.

Abstract

The study of central simple algebras over a field is a venerable topic in ring theory. There is a generalization of central simple algebras to schemes in the étale topology (in fact to arbitrary ringed sites) due to Grothendieck. The group of equivalence classes of Azumaya algebras over X is known as the Brauer group of X. By comparing the étale topology on a smooth complex variety X with the classical topology, we are able to use results from classical obstruction theory in order to obstruct the existence of certain Azumaya algebras. After giving an introduction to Azumaya algebras and the Brauer group, we shall present one such result, which furnishes lower bounds on the ranks of Azumaya algebras on spaces of low cohomological dimension.