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Abstract

Details

Abstract

The electromagnetic radio frequency spectrum is a scarce and valuable resource. Its utilization can be improved by allowing multi-layer communications, in which several signals are simultaneously transmitted and received in the same frequency band. In this talk I will describe some algorithms for the transmission and reception of multi-layer signals. These algorithms are compatible with commonly used transmitter and receiver equipment that was not designed for multi-layer communications. Such algorithms have great practical importance because they yield increased network capacity and at the same time allow telecom equipment manufacturers (including both handset and network vendors) and wireless operators to obtain additional returns on their multi-billion dollar investments.

Abstract

The Aharonov-Bohm effect is a quantum mechanical phenomenon where electrons passing through a region of vanishing magnetic field gets scattered due to topological effects. It turns
out that this phenomenon is closely related to the cohomology of forms with integer coefficients. We study this relationship from the point of view of the Calder´n problem and see that it can be captured in how Cauchy data of the connection laplacian determines uniquely the holonomy representation of the connection.

The work was partially supported by Finnish Academy of Science and by NSF Grant No.DMS-0807502.

Abstract

Schur functions were introduced early in the last century with respect to representation theory, and since then have become important functions in other areas such as combinatorics and algebraic geometry. They have a beautiful combinatorial description in terms of diagrams, which allows many of their properties to be determined.

These symmetric functions form a subalgebra of the algebra of quasisymmetric functions, which date from the 1980s. Despite this connection, the existence of a natural quasisymmetric refinement of Schur functions has been considered unlikely.

In this talk we introduce quasisymmetric Schur functions, which partition Schur functions in an intuitive way. Furthermore, we show how these quasisymmetric Schur functions refine many well-known Schur function properties with combinatorics that strongly reflect the classical case.  This is joint work with Christine Bessenrodt, Jim Haglund, Kurt Luoto and Sarah Mason.

The talk will require no prior knowledge of any of the above terms.

Abstract

In this talk I will give an introduction to Donaldson-Thomas invariants, and then their motivic incarnation.  I'll discuss motivic vanishing cycles and lambda rings, before moving to the main example of the talk - the one loop quiver with potential.  It turns out that the motivic DT invariants in this simple example have a neat presentation, and in a break with other worked out examples these invariants really involve the mondromy of the motivic vanishing cycle.

Abstract

Abstract

We show that any non-degenerate vector field u in L^{\infty}(\Omega, \R^N), where \Omega is a bounded domain in \R^N, can be written as {equation} \hbox{u(x)= \nabla_1 H(S(x), x) for a.e. x \in \Omega}, {equation} where S is a measure preserving point transformation on \Omega such that S^2=I a.e (an involution), and H: \R^N \times \R^N \to \R is a globally Lipschitz anti-symmetric convex-concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a self-dual version of Brenier's polar decomposition for the vector field u as u(x)=\nabla \phi (S(x)), where \phi is convex and S is a measure preserving transformation. We also describe how our polar decomposition can be reformulated as a self-dual mass transport problem.

Abstract

We continue our series on Central Simple Algebras with involution, introducing various group scheme associated with them.

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Abstract

We consider a certain class of non-symmetric Markov chains and obtain heat kernel bounds and parabolic Harnack inequalities. Using the heat kernel estimates, we establish a sufficient condition for the family of Markov chains to converge to non-symmetric diffusions. As an application, we approximate non-symmetric divergence forms with bounded coefficients by non-symmetric Markov chains. This extends the results by Stroock-Zheng to the non-symmetric divergence forms.

Joint work with Takashi Kumagai.

Abstract

This is the academic year's first UBC Undergraduate Mathematics Colloquium talk. UBC/UMC talks are meant for undergraduate students interested in mathematics beyond the curriculum. They are accessible at all levels. Our speakers are dynamic professional mathematicians with a reputation for interesting research and strong teaching.

Eric Cytrynbaum is our first speaker. Professor Cytrynbaum will talk about mathematical modelling. He will present two case studies from everyday life, including one describing the rise and fall of an air-breathing mammal attempting to maintain neutral buoyancy in the water.

Abstract

Given a polynomial f with complex coefficients, and a complex number z, we call the orbit of z under f the set of all images of z under the iterates of f. If the orbit of z under f is finite, we call the number z preperiodic for f. We study the following two basic questions regarding orbits of complex numbers under polynomials.

1) For two polynomials f and g, and for two complex numbers a and b, when does the orbit of a under f intersect the orbit of b under g in infinitely many points?

2) For two polynomials f and g, when there exists an infinite set of complex numbers z which are preperiodic both for f and for g?

Abstract

We investigate the probability that "Buffon's Needle" lands near a one-dimensional self-similar product set in the complex plane, where the similarity maps have rational centers and identical scalings. If the factors A and B are defined by at most 6 similarities, then the likelihood that the needle intersects an e^{-n}-neighborhood of such a set is at most Cn^{-p/\log\log n} for some p>0.

Abstract

Abstract

I will talk about the backward uniqueness of the heat equation in unbounded domains. It is known that a bounded solution of the heat equation in a half-space which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the half-space is replaced by cones with opening angle smaller than 90 degrees. In a joint work with Vladimir Sverak we show the result remains true for cones with opening angle larger than 110 degrees. Our proof covers heat equations having lower-order terms with bounded measurable coefficients.

Abstract

We define and discuss representable functors and groups schemes, as a first step towards describing the various group schemes associated to central simple algebras with involution.

Abstract

We prove that every closed, smooth manifold of at least dimension 3 admits a sequence of Riemannian metrics with pinched curvature, volume tending to infinity but whose first eigenvalue of the Laplacian remains bounded away from 0. As a consequence we construct sequences of hyperbolic knots whose complements have again volume tending to infinity and whose Cheeger constant is uniformly bounded away from 0. This is joint work with Marc Lackenby.

Abstract

Joint work with Thomas Mountford and Qiang Yao. We consider
the contact process on a random graph chosen with a fixed degree,
power law distribution, according to a model proposed by Newman,
Strogatz and Watts (2001). We follow the work of Chatterjee and
Durrett (2009) who showed that for arbitrarily small infection
parameter $\lambda > 0$, the limiting metastable density does not tend
to zero as the graph size becomes large.  We show three distinct
regimes for this density depending on the tail of the degree law.

References:
- Charterjee,S. and Durrett, R. (2009): Contact process on random
graphs with degree power law distribution have critical value zero.
Annals of Probability 37 (2009)
- Newman, M.E.J., Strogatz, S.H. and Watts, D.J. (2001): Random graphs
with arbitrary degree distributions and their applications. Physical
Review E. 64, paper 026118

Abstract

Abstract

 We continue to describe the foundation of group schemes theory to later discuss group schemes associated to central simple algebras with involution.

Abstract

Abstract

If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian L_b is an operator acting on the total  space of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian (when b -> 0) and the geodesic flow (when b -> \infty). Up to lower order terms, L_b is a weighted sum of the harmonic oscillator along the fibre TX and of the generator of the geodesic flow.  In the talk, we will explain the underlying algebraic, analytic and probabilistic aspects of its construction, and outline some of the applications obtained so far.

Other Information:Jean-Michel Bismut was born in 1948. He is a Professor of Mathematics at University Paris-Sud (Orsay), and a member of the Academie des Sciences, of the Academia Europaea, and of the Deutsche Akademie Leopoldina.  He received his 'Doctorat d'Etat' from Universite Paris VI in 1973 for his work in the control of stochastic processes.  His interests in probability theory led him to study refinements of the index theorem of Atiyah-Singer. Through his work on Quillen metrics, he participated to the proof of a Riemann-Roch theorem in arithmetic geometry.  He constructed an exotic Hodge theory, whose corresponding Laplacian is a hypoelliptic operator on the cotangent bundle of a Riemannian manifold. Recently, he used the hypoelliptic Laplacian to give a new approach to the evaluation of orbital integrals.  Jean-Michel Bismut was a plenary speaker at the International Congress of Mathematics in Berlin in 1998, and a vice-president of International Mathematical Union (I.M.U.) from 2002 to 2006. 

 

Abstract

Projectivized toric vector bundles are a large class of rational varieties that share some of the pleasant properties of toric varieties and other Mori dream spaces. Hering, Mustata and Payne proved that the Mori cones of these varieties are polyhedral and asked if their Cox rings are indeed finitely generated. We present the complete answer to this question. There are several proofs of a positive answer in the rank two case [Hausen-Suss, Gonzalez]. One of these proofs relies on the simple structure of the Okounkov body of these varieties with respect to a special flag of subvarieties. For higher ranks we study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces [Gonzalez-Hering-Payne-Suss]. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of M_{0,n}.

 

Abstract

I will explain the probabilistic interpretation of the hypoelliptic Laplacian L_b . To L_b, one can associate the diffusion on the manifold X that is a solution of the differential equation b^2 x'' = −x' + w'. For b = 0, we get x' = w', the equation of Brownian motion, and for b = +∞, we obtain the equation of geodesics x'' = 0. I will explain the rigorous results one can derive on the corresponding heat kernels via the Malliavin calculus. These will include uniform Gaussian decay of the hypoelliptic heat kernel over a symmetric space.

Abstract

Abstract

The Kompaneets equation describes the spectra of photons interacting with a  rarefied electron gas and includes three parameters. In 2010, Ibragimov obtained some time-dependent exact solutions for several restrictions of the parameters of this equation. In this talk, a tree of equivalent  nonlocally related PDE systems is constructed for the nonlinear Kompaneets (NLK) equation.  The tree includes some nonlocally related equivalent subsystems. For a two-parameter class of NLK equations, a point symmetry classification is given of these nonlocally related PDE systems and shown to yield previosuly unknown nontrivial nonlocal symmetries of the NLK equation.
        Invariant solutions arising from these nonlocal symmetries are shown to yield wider classes of time-dependent exact solutions for the NLK equation beyond those previously obtained by Ibragimov. In particular, for five classes of initial conditions, each involving two parameters, previously unknown explicit solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. Three of the classes exhibit quiescent behaviour, and the other two classes exhibit blow up behaviour in finite time. As a consequence, it is shown that the corresponding nontrivial stationary solutions, obtained by Dubinov in 2009, are unstable.  In particular, it is shown that only the stationary quiescent solution is stable.

Abstract

Third talk in the series. If G is a reductive Lie group with Lie algebra g, orbital integrals are key ingredient in Selberg’s trace formula. I will explain how one can think of the evaluation of orbital integrals as the computation of a Lefschetz trace. Using in particular the Dirac operator of Kostant, the standard Casimir operator of X = G/K is deformed to a hypoelliptic operator L_b acting on the total space of a canonically flat vector bundle on X, that contains TX as a subbundle. The symbol of this hypoelliptic operator is exactly the one described in the previous talks. When descending the situation to a locally symmetric space, the spectrum of the original Casimir remains rigidly embedded in the spectrum of the hypoelliptic deformation. Making b → +∞ gives an explicit evaluation of semisimple orbital integrals.

Abstract

A rational distance set is a subset of the real plane such that all
pairwise distances are rational numbers. It's not too hard to
construct an infinite rational distance set contained in a line or in
a circle, but if you do not allow 3 points on a line or 4 on a circle,
the current record is a set of 7 points, found a few years ago with a
computer. On the other hand, no one knows if a rational distance set,
no 3 points on a line or 4 on a circle, could be infinite. Erdős
conjectured that it would have to have a very special form, like an
algebraic curve.

In a paper with Jozsef Solymosi we showed that the only algebraic
curves that contain infinite rational distance sets are lines and
circles. In my talk I will explain the ideas involved and give an
outline of our proof.

Abstract

We develop path integral representations for Quantum Ising models. To
connect with classical Fortuin-Kasteleyn (FK) representation, we begin by
presenting the FK representation of the classical Curie-Weiss model (the
Ising model on complete graph) via the language of Poisson Point Processes.
We then show how to derive a general FK representation for Quantum Ising
model. This representation was originally developed by M. Campanino, A.
Klein, J.F. Perez (1991) and M. Aizenman, A. Klein, C.M. Newman (1993).
We apply the above to the quantum Curie-Weiss model in transversal field.
First, we present the full FK representation of this model. Examining the
form of the resulting measure and dropping the weight component from it
leads to the natural extension of the Erd\"os - Rényi random graphs.  Finally,
we consider the ground state of the quantum Curie-Weiss model via partial FK
representation. We prove the existence of a phase transition in the ground
state when the strength of the transversal field equals one.

Abstract

A central result in discrete geometry is the Szemeredi-Trotter theorem which gives a sharp bound on the number of point-line incidences in the Euclidean plane. The result has various generalizations and applications.

In this talk we prove an extension of the Szemeredi-Trotter theorem and we show some new applications. For example, using incidence bounds, we show that if M is an n-element set of k x k  matrices with real coefficients such that det(A - B) is not zero for any distinct A,B elements of M and V,W are n-element sets of k dimensional vectors, then |V +W| + |MW| >> n^{5/4}.

Part of the talk is based on joint work with Terry Tao.