
Wed 5 Dec 2012, 12:30pm
SPECIAL
Graduate Student Center, Room 200

Doctoral Exam

Graduate Student Center, Room 200
Wed 5 Dec 2012, 12:30pm3:00pm
Details
In this thesis which is a compendium of seven papers, we explore the behaviour of solutions of various semilinear elliptic equations and systems on both bounded and unbounded domains of dimension N.
On unbounded domains which is the main part of our work, our motivation is a celebrated conjecture of De Giorgi (1978) stating that bounded and monotone solutions of the AllenCahn equation, up to dimension 8, must be onedimensional. This conjecture is known to be true for N=<3 and with extra (natural) assumptions for 4=<N=<8.
Focusing on system of equations, we state a counterpart of the above conjecture for gradient systems introducing the concept of monotonicity for systems. Then, we prove this conjecture for dimensions up to three following ideas given for the scalar case but unfortunately for higher dimensions we are not able to give any (even partial) results. On the other hand, replacing the Laplacian operator by the divergence form operator for the AllenCahn equation, we ask under what conditions solutions of this new equation would be mdimensional. This leads us to define the concept of “mLiouville theorem” for PDEs. We say a PDE satisfies mLiouville theorem for 0<=m<N if all solutions of the PDE are at most mdimensional. The motivation to this definition is the Liouville theorem (or 0Liouville theorem) that we have seen in elementary analysis stating that bounded harmonic functions on the whole space must be constant (0dimensional). We present various 2 and higher Liouville theorems, however, we are not sure whether or not any of these results are optimal. 0Liouville theorem is at the heart of our work and this thesis includes various 0Liouville theorems for the HenonLaneEmden system, LaneEmden equation, Gelfand equation and gradient systems.
On bounded domains, following ideas observed for unbounded domains, we present regularity of solutions for gradient and twistedgradient systems. The novelty here is a stability inequality that gives us the chance to adjust the known techniques and ideas to systems.
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McGill

Wed 5 Dec 2012, 3:00pm
Probability Seminar
ESB 2012

Gaussian free field, random measure and KPZ on R^4

ESB 2012
Wed 5 Dec 2012, 3:00pm4:00pm
Abstract
A highlight in the study of quantum physics was the work of Knizhnik, Polyakov and Zamolodchikov (1988), in which they proposed a relation (KPZ relation) between the scaling dimension of a statistical physics
model in Euclidean geometry and its counterpart in the random geometry. Recently, Duplantier and Sheffield used the 2D Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and
gave the first mathematically rigorous formulation and proof of the KPZ relation in that setting. We have applied a similar approach to generalize part of their results to R^4 (as well as to R^(2n) for
n>=2). To be specific, we construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) given by the exponential of an instance of the 4D Gaussian free field.
We also establish the KPZ relation corresponding to this random measure. This is joint work with Dmitry Jakobson.
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UBC

Fri 7 Dec 2012, 11:00am
Algebraic Groups and Related Structures
MATX 1118

On canonical dimension of quadratic forms

MATX 1118
Fri 7 Dec 2012, 11:00am12:30pm
Abstract
We'll compute the canonical dimension of a quadratic form using the work of Karpenko and Merkujev.
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Department of Mathematics and Statistics, York University

Mon 10 Dec 2012, 12:00pm
Mathematical Education
MATH 202

Lunch Series on Teaching and Learning: An informal discussion of the "background tutorial" math remediation initiative for incoming firstyear science majors at York

MATH 202
Mon 10 Dec 2012, 12:00pm1:00pm
Abstract
At York, we have found firstyear science majors coming to us from the Ontario high school system in general rather poorly prepared for firstyear university mathematics. The result is very high dropplusfail rates in our firstyear math courses and a resulting high attrition rate in the early years of our degree programs. A major source of the problem appears to be the widespread use in the schools of an approach heavily emphasizing the memorization of solution problem templates, an approach which leaves a majority of our incoming science majors with deficiencies in very basic algebra, trigonometry, and, even more problematic, their intuitive understanding of the basic operations of arithmetic. In this discussion, I will outline an approach I have developed involving 4day, 4hourperday intensive remediation sessions focused on changing the way such students approach mathematics. The program was begun in 2005 and significantly expanded in 2009, now handling between 15 and 20% of the incoming class each year. I will present statistics outlining the significant impact we have seen on student performance. The aim is to keep the presentation very informal, leaving lots of time for discussion, feedback and suggestions on possibilities for further improving the initiative.
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University of Alberta

Tue 11 Dec 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127 (PIMS video conference room)

Multimarginal optimal transport and multiagent matching problems: uniqueness and structure of solutions

ESB 4127 (PIMS video conference room)
Tue 11 Dec 2012, 3:30pm4:30pm
Abstract
I will discuss uniqueness and Monge solution results for multimarginal optimal transportation problems with a certain class of cost functions; this class arises naturally in multiagent matching problems in economics. This result generalizes a seminal result of Gangbo and \'Swi\c{e}ch on multimarginal problems. I will also discuss some related observations about multimarginal optimal transport on Riemannian manifolds.
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Harvard

Wed 12 Dec 2012, 3:00pm
SPECIAL
Department Colloquium
MATH ANNEX 1100

Universality for beta ensembles

MATH ANNEX 1100
Wed 12 Dec 2012, 3:00pm4:00pm
Abstract
Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis concerns large but ﬁnite dimensional matrices. Spectacular progress was done in the past two decades to prove universality of random matrices with orthogonal, unitary or symplectic invariance. These models correspond to loggases with respective inverse temperature 1, 2 or 4.
I will first review the many occurrences of these statistics from the random matrices universality class. I will then report on a joint work with L. Erd{\H o}s and H.T. Yau, which yields universality for the loggases at arbitrary temperature at the microscopic scale.
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Harvard

Thu 13 Dec 2012, 2:00pm
SPECIAL
Probability Seminar
MATH 105

Mesoscopic analogies between random matrices and Lfunctions

MATH 105
Thu 13 Dec 2012, 2:00pm3:00pm
Abstract
Fluctuations of random matrix theory type have been known to occur in analytic number theory since Montgomery's calculation of the pair correlation of the zeta zeros, in the microscopic regime. At the mesoscopic scale, the analogy holds, through a limiting Gaussian field, which present an ultrametric structure similar to loggases. In particular we will consider an analogue of the strong Szeg{\H o} theorem for Lfunctions.
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Seminar Information Pages

Note for Attendees
Refreshments will be served in MATH 125 at 2:45 p.m.