Graduate Blog

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Friday, June 21, 2013
11:00 a.m.
BA 6183, 40 St George St.

Ph.D. Candidate: Dana Bartosova

Ph.D. Advisor: Stevo Todorcevic

Thesis Title: Topological dynamics in the language of near ultrafilters and automorphism groups of $\omega$-homogeneous structures

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In this thesis, we present a new viewpoint of the universal minimal flow in the language of near ultrafilters. We apply this viewpoint to generalize results of Kechris, Pestov and Todorcevic about a connection between groups of automorphisms of structures and structural Ramsey theory from countable to uncountable structures. This allows us to provide new examples of explicit descriptions of universal minimal flows as well as of extremely amenable groups. We identify new classes of finite structures satisfying the Ramsey property and apply the result to the computation of the universal minimal flow of the group of automorphisms of $\cal P (\omega_1)/$fin as well as of certain closed subgroups of groups of homeomorphisms of Cantor cubes. We furthermore apply our theory to groups of isometries of metric spaces and the problem of unique amenability of topological groups.

The theory combines tools from set theory, model theory, Ramsey theory, topological dynamics and ergodic theory, and homogeneous structures.

A soft copy of the thesis can be obtained by contacting dana.bartosova@mail.utoronto.ca

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Wednesday, May 29, 2013
2:00 p.m.
BA 6183, 40 St George St.

Ph.D. Candidate: Patrick Walls

Ph.D. Advisor: Steve Kudla

Thesis Title: The Theta Correspondence and Periods of Automorphic Forms

http://www.math.toronto.edu/pjwalls/ThesisDraft/May8.pdf
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Thesis Abstract:

The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of $L$-functions attached to these modular forms.

In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if $G$  is a symplectic group and $H$  is an orthogonal group over a number field $k$ , these relations are identities equating Fourier coefficients of cuspidal automorphic forms on $G$  (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on $H$ over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger.  The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for $H$ is standard however the relative trace formula for $G$ is novel in that
it involves the trace of an operator built from theta functions.

The final part of this thesis explores some preliminary results on local height pairings of special cycles on the $p$-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Thursday, May 23, 2013
2:00 p.m.
BA 6183, 40 St George St.

Ph.D. Candidate: Travis Li

Ph.D. Advisors: Marco Gualtieri, Lisa Jeffrey

Thesis Title: Constructions of Lie Groupoids

http://www.math.toronto.edu/sli/Thesis.pdf

Thesis Abstract:

In this thesis, we develop two methods for constructing Lie groupoids.

The first method is a blow-up construction, corresponding to the elementary modification of a Lie algebroid along a subalgebroid over a hypersurface. This construction may be specialized to the Poisson groupoids and Lie bialgebroids. We then apply this method to the several cases. The first is the adjoint Lie groupoid integrating the Lie algebroid of vector fields tangent to a collection of normal crossing hypersurfaces. The second is the adjoint symplectic groupoid  of a log symplectic manifold. The third is the adjoint Lie groupoid integrating the tangent algebroid of a Riemann surface twisted by a divisor.

The second method is a gluing construction, whereby Lie groupoids defined on the open sets of an appropriate cover may be combined to obtain global integrations. This allows us to construct and classify the Lie groupoids integrating a certain Lie algebroid. We apply this method to the aforementioned cases, albeit with slight differences. The first case is the cateogry of integrations of the Lie algebroid of vector fields tangent to a single smooth hypersurface.The second case is the category of Hausdorff symplectic groupoids of a log symplectic manifold. The third case is the category of integrations of the tangent algebroid of a Riemann surface twisted by a divisor.

The September 2013 PhD Comprehensive Exam schedule is as follows:

Wednesday, September 4, 2013, 1-4 p.m., in BA 6183
Analysis (real and complex) PhD Comprehensive Exam
There is the option of writing only the real questions (2 hour exam)
or the complex questions only (1.5 hour exam).

Thursday, September 5, 2013, 1-4 p.m., in BA 6183
Algebra PhD Comprehensive Exam

Friday, September 6, 2013, 10:00 a.m.-1:00 p.m., in BA 6183
Topology PhD Comprehensive Exam

Ph.D. students must pass all their comprehensive exams within 12 months of
entering the program (i.e. by the September sitting of the second year).

Please inform Jemima (jmerisca@math) if you wish to write one or
more of the above exams.

Past comprehensive exam questions can be viewed at:
http://www.math.toronto.edu/graduate/pce/

The Statistics Department has scheduled their annual Probability
Comprehensive Exam on:

Friday, May 31, 2013, 12:00 to 5:00 p.m., in SS1088, 100 St. George St.

If you plan to take this exam as one of your three required PhD comprehensive exams,
please let me know and I will pass on your name to the Statistics graduate office.

Thanks,

Jemima

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Monday, May 13, 2013
11:00 a.m.
BA 6183, 40 St George St.

Ph.D. Candidate: Kathleen Smith

Ph.D. Advisors: Lisa Jeffrey, Yael Karshon

Thesis Title: Connectivity and Convexity Properties of the Momentum Map for Group Actions on Hilbert Manifolds

Thesis Abstract:

In the early 1980′s a landmark result was obtained by Atiyah and independently Guillemin and Sternberg:  the image of the momentum map for a torus action on a compact symplectic manifold is a convex polyhedron. Atiyah’s proof makes use of the fact that level sets of the momentum map are connected.  These proofs work in the setting of finite-dimensional compact symplectic manifolds.  One can ask how these results generalize.  A well-known example of an infinite-dimensional symplectic manifold with a finite-dimensional torus action is the based loop group.  Atiyah and Pressley proved convexity for this example, but not connectedness of level sets.  A proof of connectedness of level sets for the based loop group was provided by Harada, Holm, Jeffrey and Mare in 2006.

In this thesis we study Hilbert manifolds equipped with a strong symplectic structure and a finite-dimensional group action preserving the strong symplectic structure.  We prove connectedness of regular generic level sets of the momentum map.  We use this to prove convexity of the image of the momentum map.

A soft copy of the thesis can be obtained by contacting kndsmith@math.toronto.edu.

We are happy to announce that this year’s winners of the Daniel B. DeLury Teaching Awards for graduate students in mathematics are:

Iva Halacheva
Eric Hart
Patrick Walls

The selection committee consisted of Eckhard Meinrenken and Abe Igelfeld. Nominations were made by faculty members, course instructors, and undergraduate students.

The selection committee received many favourable comments about our TA’s. Fine work is being done by many of our teaching assistants, and we can take pride in their work.

Congratulations Iva, Eric and Patrick!

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Tuesday, April 30, 2013
BA 6183, 40 St. George Street
2:00 p.m.

PhD Candidate:  Brent Pym

PhD Advisor:  Marco Gualtieri

PhD Thesis Title: Poisson structures and Lie algebroids in complex geometry

http://www.math.toronto.edu/bpym/files/thesis/bpym-thesis-2013-04-26.pdf

Abstract:

This thesis is devoted to the study of holomorphic Poisson structures and Lie algebroids, and their relationship with differential equations, singularity theory and noncommutative algebra.

After reviewing and developing the basic theory of Lie algebroids in the framework of complex analytic and algebraic geometry, we focus on Lie algebroids over complex curves and their application to the study of meromorphic connections. We give concrete constructions of the corresponding Lie groupoids, using blowups and the Uniformization Theorem. These groupoids are complex surfaces that serve as the natural domains of definition for the fundamental solutions of ordinary differential equations with singularities. We explore the relationship between the convergent Taylor expansions of these fundamental solutions and the divergent asymptotic series that arise when one attempts to solve an ordinary differential equation at an irregular singular point.

We then turn our attention to Poisson geometry. After discussing the basic structure of Poisson brackets and Poisson modules on analytic spaces, we study the geometry of the degeneracy loci-where the dimension of the symplectic leaves drops. We explain that Poisson structures have natural residues along their degeneracy loci, analogous to the Poincaré residue of a meromorphic volume form. We discuss the local structure of degeneracy loci in small codimension, placing
a strong constraint on the singularities of the degeneracy hypersurfaces of generically symplectic Poisson structures. We use these results to give new evidence for a conjecture of Bondal.

Finally, we discuss the problem of quantization in noncommutative projective geometry. Using Cerveau and Lins Neto’s classification of degree-two foliations of projective space, we give normal forms for unimodular quadratic Poisson
structures in four dimensions, and describe the quantizations of these Poisson structures to noncommutative graded algebras.  As a result, we obtain a (conjecturally complete) list of families of quantum deformations of projective three-space. Among these algebras is an “exceptional” one, associated with a twisted cubic curve. This algebra has a number of remarkable properties: for example, it supports a family of bimodules that serve as quantum analogues of the classical Schwarzenberger bundles.

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Monday, April 29, 2013
BA 6183, 40 St. George Street
11:00 a.m.

PhD Candidate: Xiao Liu

PhD Advisor: Catherine Sulem

PhD Thesis Title: Analytical and numerical results for some classes of nonlinear Schrödinger equations

http://www.math.toronto.edu/liuxiao/index_files/xiao_thesis.pdf

Abstract:

This thesis is devoted to the study of nonlinear dispersive partial differential equations of Schrödinger type. The main questions we investigate are long-time behavior or occurrence of a finite time singularity, as well as stability properties of solitary wave solutions.

The derivative nonlinear Schrödinger (DNLS) equation is a nonlinear dispersive model that appears in the description of wave propagation in plasmas. The first part of this thesis concerns a DNLS equation with a generalized nonlinearity (gDNLS). We first investigate numerically the possible occurrence of singularities. We show that, in the L2-supercritical regime, singularities can occur. We obtain a precise description of the local structure of the solution in terms of the blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schrödinger equation (NLS) with supercritical power law nonlinearity. We also show that the gDNLS equation possesses a two-parameter family of solitary wave solutions and study their stability. We fully classify their orbital stability or orbital instability properties according to the strength of the nonlinearity and, in some instances, their velocity.

In linear quantum mechanical scattering theory, the phenomenon of resonant tunneling refers to the situation where incoming waves are fully transmitted through potential barriers at certain energies.  In the second part of this thesis, we
consider the one-dimensional cubic NLS equation with two classes of external potentials, namely the “box” potential and a repulsive 2-delta potential. We demonstrate numerically that resonant tunneling may occur in a nonlinear setting: Taking initial condition as a slightly perturbed, fast moving NLS soliton, we show that, under a certain resonant condition, the incoming soliton is almost fully transmitted. As the velocity of the incoming soliton increases, the transmitted mass of the soliton converges to the total mass.

The PIMS YRC is a conference for graduate students in math and stats. There have been over 80 participants at each of the three previous YRCs. The 2013 PIMS YRC is being held at the University of Alberta May 21 – May 24, 2013. Registration is now open. There is no registration fee and the cost of accommodations is covered by the conference. The registration deadline is Friday, April 26, 2013.

For additional information or to register see:

http://www.math.ualberta.ca/~game/pimsyrc13/registration.html