Mathematics Dept.
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Mon 1 Jun 2015, 10:00am
Mon 1 Jun 2015, 10:00am-10:00am

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George Bluman
ubc
Fri 5 Jun 2015, 12:00pm
Symmetries and Differential Equations Seminar
Math 125
Direct construction of conservation laws and connections between symmetries and CLs. Part II
Math 125
Fri 5 Jun 2015, 12:00pm-1:00pm

Abstract

In the second lecture on the construction of the conservation laws for a DE system, we review the standard and extended formulations of the classical Noether's Theorem.   The classical Noether's Theorem only works for variational systems and requires the construction of the Lagrangian.  We show how the Direct Method generalizes Noether's Theorem and overcomes its serious limitations.  We also show how a symmetry maps a CL to another CL. Moreover, it will be shown that for a given PDE system, a solution pair consisting of a solution of its linearized system and a solution of the adjoint of its linearized system, directly yields a CL by a simple formula.
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George Bluman
UBC
Fri 12 Jun 2015, 12:00pm
Symmetries and Differential Equations Seminar
Math 125
Nonlocally related systems and nonlocal symmetries I
Math 125
Fri 12 Jun 2015, 12:00pm-1:00pm

Abstract

Often a given PDE system has no local symmetry and/or no local conservation law.  Moreover, its local symmetries may not be useful for a problem at hand. 

Aim: to extend existing methods for finding local symmetries and local CLs and their uses to PDE systems nonlocally related and equivalent to a given PDE system.

Two systematic and "natural" ways are presented for accomplishing this.  In particular, for any PDE system, (1) each local CL as well as (2) each Lie point symmetry systematically yields a nonlocally related system. The first lecture will present the conservation law based method.
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Ph.D. Candidate: Kyle D. Hambrook
Mathematics, UBC
Fri 19 Jun 2015, 2:00pm SPECIAL
Room 126, Mathematics Building
Doctoral Exam: Restriction Theorems and Salem Sets
Room 126, Mathematics Building
Fri 19 Jun 2015, 2:00pm-4:00pm

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ABSTRACT: In the first part of this thesis, we prove the sharpness of the exponent range in the L^2 Fourier restriction theorem due to Mockenhaupt and Mitsis (with endpoint estimate due to Bak and Seeger). The proof is based on a random Cantor-type construction of Salem sets due to Laba and Pramanik. The key new idea is to embed in the Salem set a small deterministic Cantor set that disrupts the restriction estimate for the natural measure on the Salem set but does not disrupt the measure’s Fourier decay.

In the second part of this thesis, we prove a lower bound on the Fourier dimension of certain sets on the real line arising from Diophantine approximation. This generalizes theorems of Kaufman and Bluhm. As a consequence, we obtain new explicit examples of Salem sets. We apply our result to metrical Diophantine approximation and compute the Hausdorff dimension of sets arising from Diophantine approximation in new cases. We also prove a higher-dimensional analog of our result.
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PhD Candidate: William F. Thompson
Mathematics, UBC
Mon 22 Jun 2015, 12:30pm SPECIAL
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Doctoral Exam: Parametrization and multiple time scale problems with non-Gaussian statistics related to climate dynamics
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Mon 22 Jun 2015, 12:30pm-2:30pm

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ABSTRACT: Many problems in climate modelling are characterized by their chaotic nature and multiple time scales. Stochastic parametrization methods can often simplify the analysis of such problems by using appropriate stochastic processes to account for degrees of freedom that are impractical to model explicitly, such that the statistical features of the reduced stochastic model are consistent with more complicated models and/or observational data. However, applying appropriate stochastic parametrizations is generally a non-trivial task. This is especially true when the statistics of the approximated processes exhibit non-Gaussian features, like a non-zero skewness or infinite variance. Such features are common in problems with nonlinear dynamics, anomalous diffusion processes, and multiple time scales. Two common topics in stochastic parameterization are model parameter estimation and the derivation of reduced stochastic models.

In this dissertation, we study both of these topics in the context of stochastic differential equation models, which are the preferred formalism for continuous-time modelling problems. The motivation for this analysis is given by problems in atmospheric or climate modelling. We estimate parameters of a dynamical model of sea surface vector winds using a method based on the properties of differential operators associated with the probabilistic evolution of the wind model. The parameter fields we obtain allow us to reproduce statistics of the vector wind data and inform us regarding the limitations of both the estimation method and the model itself. We also derive reduced stochastic models for a class of dynamical models with multiple time scales that are driven by α-stable stochastic forcing. The results of this project are then applied to derive a similar approximation for processes that are driven by a fast linear process experiencing additive and multiplicative Gaussian white noise forcing. The results of these chapters complement previous results for systems driven by Gaussian white noise and suggest a possible dynamical mechanism for the appearance of α-stable stochastic forcing in some climatic time series.
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Graz University of Technology
Tue 23 Jun 2015, 3:30pm
Number Theory Seminar
room MATH 126
Polynomial decomposition and Diophantine equations in two separated variables
room MATH 126
Tue 23 Jun 2015, 3:30pm-4:30pm

Abstract

Given a field K, a polynomial f(x) with coefficients in a field K is said to be indecomposable over K if it can not be represented as a functional composition of lower degree polynomials in K[x]. Any polynomial of degree greater than 1 can clearly be represented as a composition of indecomposable polynomials. Such a representation, called a complete decomposition, does not need to be unique. Ritt in the 1920's described the extent of the non-uniqueness for complex polynomials. In so doing, Ritt exhibited some invariants of complete decompositions of complex polynomials. Building on the methods developed by Ritt, Fried and others, and by shifting to the setting of maps between curves, we extend and generalize known results on invariants of complete decompositions. These results are obtained jointly with Michael Zieve. We further present some methods for showing indecomposability and discuss applications of such results to Diophantine equations. These methods are described in a joint survey paper with Robert F. Tichy.
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PhD Candidate: Alexandre Tomberg
Mathematics, UBC
Thu 25 Jun 2015, 9:00am SPECIAL
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Doctoral Exam: Renormalisation Group and Critical Correlation Functions in Dimension Four
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Thu 25 Jun 2015, 9:00am-11:00am

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Abstract: Critical phenomena and phase transitions are important subjects in statistical mechanics and probability theory. They are connected to the phenomenon of universality that makes the study of mathematically simple models physically relevant. Examples of such models include ferromagnetic spin systems such as the Ising, O(n) and n-component |\varphi |^4 models, but also the self-avoiding walk that has been observed to formally correspond to a "zero-component" spin model.

The subject of this thesis is the extension and application of a rigorous renormalisation group method developed by Brydges and Slade to study the critical behaviour of the continuous-time weakly self-avoiding walk and of the n-component |\varphi |^4 model on the 4-dimensional lattice \bbbmath Z ^4. Although a "zero-component" vector is mathematically undefined (at least naively), we are able to interpret the weakly selfavoiding walk in a mathematically rigorous manner as the n=0 case of the n-component |\varphi |^4 model, and provide a unified treatment of both models.

For the |\varphi |^4 model, we determine the asymptotic decay of the critical correlation functions including the logarithmic corrections to Gaussian scaling, for n\ge 1. This extends previously known results for n=1 to all n\ge 1, and also observes new phenomena for n>1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the "watermelon" network consisting of p weakly mutually and self-avoiding walks, for all p\ge 1, including the logarithmic corrections. This extends a previously known result for p=1, for which there is no logarithmic correction, to a much more general setting.
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Ph.D. Candidate: Wen Yang
Mathematics, UBC
Fri 26 Jun 2015, 12:30pm SPECIAL
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Doctoral Exam: Some new results on the SU(3) Toda system and Lin-Ni problem
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Fri 26 Jun 2015, 12:30pm-2:30pm

Details

ABSTRACT: In my dissertation, I mainly consider two problems. First, I study the SU(3) Toda system, which comes from the theoretic physics and differential geometry. For this system, by analyzing the bubbling solutions when the parameters tends to the critical parameters, I could compute the degree jump when the parameters cross the critical parameter. As a consequence, I can obtain a partial result on the degree counting formula for the system and some existence result provided the parameters are restricted in some range. In the second part of my dissertation, I study a nonlinear Neumann elliptic problem, which comes from biology. For this problem, I construct a nontrivial solution, i.e., non-constant solution, which disproves a conjecture during the past two decades.
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George Bluman
UBC
Fri 26 Jun 2015, 3:00pm
Symmetries and Differential Equations Seminar
Math 125
Nonlocally related systems and nonlocal symmetries II
PhD Candidate: Cindy M Blois
Mathematics, UBC
Mon 29 Jun 2015, 12:30pm SPECIAL
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Doctoral Exam: Functional Integral Representations for Quantum Many-Particle Systems
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Mon 29 Jun 2015, 12:30pm-2:30pm

Details

ABSTRACT: Formal functional integrals are commonly used as theoretical tools and as sources of intuition for predicting phase transitions of many-particle systems in Condensed Matter Physics. In this thesis, we derive rigorous versions of these functional integrals for two types of quantum many-particle systems.

We start with a brief review of quantum statistical mechanics and the formalism of coherent states, which form the basis for our analysis. For a mixed gas of bosons and/or fermions interacting on a finite lattice with a general Hamiltonian that preserves the total number of particles in each species, we rigorously derive a functional integral representation for the partition function, using a large-field cutoff for the boson fields. We then expand the resulting “action” in powers of the fields and find a recursion relation for the coefficients. In the case of a 2-body interaction (such as the Coulomb interaction), we also find bounds on the coefficients, which give a domain of analyticity for the action. This domain is large enough for use of the action in the functional integral, provided that the large-field cutoffs are taken to grow not too quickly. Next, we study a system of electrons and phonons interacting in a finite lattice, using the Holstein Hamiltonian. We again rigorously derive a coherent-state functional integral representation for the partition function of this system and then prove that the action in the functional integral is an entire-analytic function of the fields. However, since the Holstein Hamiltonian does not preserve the total number of bosons, our approach for the previous system requires some modification. In particular, we repeatedly use Duhamel expansions in powers of the interaction, rather than sums over particle numbers.
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Jean-luc Doumont
Tue 30 Jun 2015, 10:00am
PIMS Seminars and PDF Colloquiums
TRIUMF auditorium (on the UBC campus)
Structuring your research paper
TRIUMF auditorium (on the UBC campus)
Tue 30 Jun 2015, 10:00am-12:00pm

Abstract


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University of Alberta
Tue 30 Jun 2015, 3:00pm
Algebraic Groups and Related Structures
MATH 126
Upper-homogeneous varieties
MATH 126
Tue 30 Jun 2015, 3:00pm-4:00pm

Abstract

We introduce a new class of algebraic varieties which is closed under products and embraces, e.g., projective homogeneous varieties, generically split varieties, and standard norm varieties (associated to symbols in Galois cohomology groups). We prove for this class the characterization of canonical p-dimension in terms of algebraic cycles previously known for projective homogeneous and generically split varieties. As an application, we extend to this class the incompressibility criterion of products previously known only for certain type of projective homogeneous varieties.
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