Mathematics Dept.
  Events
PhD Candidate: Michael R Lindstrom
Mathematics, UBC
Thu 7 May 2015, 9:00am SPECIAL
Room 203, Graduate Student Centre, UBC
Exam: Investigation into the Feasibility and Operation of a Magnetized Target Fusion Reactor, and Qualitative Predictions of Magnetic Field Profile Perturbations Induced by Surface Roughness in Type II Superconductors: Insights from Mathematical Modelling
Room 203, Graduate Student Centre, UBC
Thu 7 May 2015, 9:00am-11:00am

Details

In this thesis we study two problems, one concerning fusion energy and another superconductivity.

Magnetized target fusion reactors are a modern idea to generate hydrogen fusion energy on earth. The design entails magnetically confining a plasma and crushing it in an imploding shell of molten metal. The design has many unresolved questions in its feasibility as a power source and its efficiency. We study the problem with two approaches. Firstly, we use a coordinate transformation and implement a novel flux-limited, split-step, finite volume scheme for nonlinear coupled conservation laws and do a parameter sensitivity analysis for the performance. Secondly, by a careful series of asymptotic arguments, we establish a leading order expression for the plasma compression. This expression is qualitatively consistent with numerical simulations, but it also gives new insights into the device operation. We then infer key design parameters for the success of magnetized target fusion.

The second problem involves computational modelling of superconductors. In type II superconductors where the coherence length ξ is small compared to the London penetration depth λ, the London equation predicts that magnetic fields decay exponentially in magnitude with the depth into the superconductor with length scale λ, provided the surface is flat. Various measurements of λ using low energy muon spin rotation on superconductors such as Yttrium-Barium-Copper-Oxide measure field profiles that differ from this prediction. There seems to be a dead layer, a distance δ over which the magnetic field magnitude does not decay. Speculation has been made that this may be due to surface roughness. Surface roughness has been studied for a simple sinusoidal model of surface roughness. We extend this work firstly by using Atomic Force Microscopy data of Yttrium-Barium-Copper-Oxide (a type II superconductor) crystals and predicting the field profiles the crystals could produce with the London model given their actual surface geometry; and secondly, we consider how roughness could affect experimental values for λ and δ. We find that dead layers are unlikely due to roughness alone, that the measurement of λ may be influenced by surface roughness, and that the field orientation may be perturbed, negligibly, within the superconductor.
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Tel Aviv University
Tue 12 May 2015, 3:00pm
Algebraic Groups and Related Structures
MATH 126
Real Galois cohomology of simply connected groups
MATH 126
Tue 12 May 2015, 3:00pm-4:00pm

Abstract

By the celebrated Hasse principle of Kneser, Harder and Chernousov, calculating Galois cohomology  H1(K,G) of a simply connected simple algebraic K-group G over a number field K reduces to calculating H^1(R,G) over the field of real numbers R. In some cases, in particular, for the split simply connected R-group G of type E_7, the first calculations of H^1(R,G) appeared only in 2013 and 2014, in preprints of Jeffrey Adams, Brian Conrad, and the speaker and Zachi Evenor. All of these calculations rely on the speaker's note of 1988.

In this talk, based on joint work with Dmitry Timashev (in progress), I will explain the method of Kac diagrams for calculating the Galois cohomology set H1(R,G)^for a simply connected simple algebraic R-group G. I will use groups of type E_7 as an example. No prior knowledge of Galois cohomology or of groups of type E_7 will be assumed.
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University of Utah
Wed 13 May 2015, 3:10pm
Diff. Geom, Math. Phys., PDE Seminar / Probability Seminar
MATH 203
Fluctuations for polymer models in intermediate disorder
MATH 203
Wed 13 May 2015, 3:10pm-4:00pm

Abstract

 Directed polymer models are finite-temperature versions of first- and last-passage percolation on the lattice. In 1+1 dimensions, the free-energy of the directed polymer is conjecturally in the Tracy-Widom universality class at all finite temperatures. However, this has only been proven for a small class of polymers - the so-called solvable models that include Seppalainen's gamma polymers and the O'Connell-Yor semi-discrete polymer - with special sets of shapes and edge-weight distributions. We present some new results towards the universality conjecture in the intermediate disorder scaling regime.

This is joint work with Jeremy Quastel.
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Université Paris 13
Wed 13 May 2015, 3:15pm
Topology and related seminars
ESB 4133
Automorphisms of p-completed classifying spaces of groups of Lie type
ESB 4133
Wed 13 May 2015, 3:15pm-4:15pm

Abstract

 
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Professor Vakhtang Putkaradze
University of Alberta
Thu 14 May 2015, 2:00pm SPECIAL
ESB 4127
Applied Math Seminar: Geometric theory of garden hose dynamics
ESB 4127
Thu 14 May 2015, 2:00pm-3:00pm

Details

Abstract: We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross-section available to the fluid motion during the dynamics, sometimes called collapsible tubes. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Using these methods, we derive the fully three dimensional equations of motion. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross-section. We also derive and analyze several analytical, fully nonlinear solutions of traveling wave type in two dimensions. Finally, we present results of preliminary experiments showing instability and re-stabilization elucidating the roles of rotation and boundary conditions. This research has been supported by NSERC and the University of Alberta.
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U.Western Ontario
Tue 19 May 2015, 3:15pm SPECIAL
Topology and related seminars
ESB 4127
Path categories and algorithms
ESB 4127
Tue 19 May 2015, 3:15pm-4:15pm

Abstract

Finite cubical complexes are abstract models for parallel processing systems. The vertices of a complex K are the states of the system, and the execution paths are morphisms of the corresponding path category P(K).

 

 

The theory of path categories and path 2-categories for finite oriented cubical and simplicial complexes will be reviewed. There is an algorithm for computing the path category P(K) of a finite complex K which is based on its path 2-category. This 2-category algorithm will be displayed, and complexity reduction methods for the algorithm will be discussed.

 

 

The 2-category algorithm works well only for toy examples. The size of the path category P(K) of a complex K can be an exponential function of the size of K. The algorithm has so far resisted parallelization.

One wants combinatorial local to global methods for addressing examples that are effectively infinite.

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George Bluman
UBC
Thu 21 May 2015, 2:00pm
Symmetries and Differential Equations Seminar
Math 125
Applications of Symmetry Methods to PDEs
Math 125
Thu 21 May 2015, 2:00pm-3:00pm

Abstract

This will be the introduction to a weekly sequence of seminars on modern developments in symmetry methods for PDE. Topics will include how to find the:conservation laws for any DE system--the extension of Noether's theorem to non-variational systems, local symmetries, higher-order symmetries, invertible and local mappings (including linearizations through symmetries and conservation law multipliers), nonlocally related PDE systems, nonlocal symmetries, nonlocal conservation laws, nonlocal mappings, and the nonclassical method to obtain solutions of PDEs. The main emphasis of these seminars will be on how to find systematically symmetries and conservation laws (local and nonlocal) of a given PDE system and how to use systematically symmetries and conservation laws for related applications.
        The first seminar will present an overview of  topics. 
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UCLA
Thu 21 May 2015, 3:30pm
Number Theory Seminar
room MATH 126
Analytic variation of Tate-Shafarevich groups
room MATH 126
Thu 21 May 2015, 3:30pm-4:30pm

Abstract

Analyzing known elementary relations between U(p) operators and Picard functoriality of the Jacobians of each tower of modular curves of p-power level, we get fairly exact control of the ordinary part of the limit Barsotti-Tate groups and the (p-adically completed) ind-limit Mordell-Weil groups with respect to the weight Iwasawa algebra. Computing Galois cohomology of these controlled Galois modules, we obtain good control of the (ordinary part of) limit Selmer groups and limit Tate-Shafarevich groups.
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Duke University
Mon 25 May 2015, 1:15pm SPECIAL
Department Colloquium
MATX 1100
Niven lecture: Surfing with wavelets
MATX 1100
Mon 25 May 2015, 1:15pm-2:15pm

Abstract

Wavelets provide a mathematical tool that emerged in the 1980s from a synthesis of ideas in mathematics, physics, computer science and engineering. They are now used in a wide range of mathematical applications, and provide a mathematical way to "zoom in" on details, without losing track of the large picture. The talk will describe some of the essential features of the approach, and illustrate with examples.

Note for Attendees

There will be the Grad Reception on Monday, May 25th at 11:30-1:00 (lunch and awards presentation).
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PhD Candidate: Iain R. Moyles
Mathematics, UBC
Wed 27 May 2015, 12:30pm SPECIAL
Room 203 of the Graduate Student Centre, 6371 Crescent Rd., UBC
Exam: Hybrid Asymptotic-Numerical Analysis of Pattern Formation Problems
Room 203 of the Graduate Student Centre, 6371 Crescent Rd., UBC
Wed 27 May 2015, 12:30pm-2:30pm

Details

ABSTRACT:  We present an analysis of the Gierer-Meinhardt model with saturation (GMS) on various curve geometries in two-dimensions. We derive a boundary fitted coordinate framework which translates an asymptotic two-component differential equation into a single component reaction diffusion equation with singular interface conditions. We create a numerical method that generalizes the solution of such a system to arbitrary two-dimensional curves and show how it extends to other models with singularity properties that are related to the Laplace operator. This numerical method is based on integrating logarithmic singularities which we handle by the method of product integration where logarithmic singularities are handled analytically with numerically interpolated densities.

In parallel with the numerical method, we present some analytical solutions to the GMS model on circular and slightly perturbed circular curve geometries. We see that for the regular circle, saturation leads to a hysteresis effect for two dynamically stable branches of equilibrium radii. For the near circle, we show that there are two distinct perturbations to the velocity profile, one which introduces angular dependence, and one which introduces a vertical shift caused by quadratic Fourier mode interactions. We perform a linear stability analysis to the true circle solution and show that there are two classes of eigenvalues leading to breakup or zigzag instabilities. For the breakup instabilities we show that the saturation parameter can completely stabilize perturbations that we show are always unstable without saturation and for the zigzag instabilities we show that the eigenvalues are given by the near-circle curve normal velocity. The breakup analysis is based on the reduction of an implicit non-local eigenvalue problem (NLEP) to a root finding problem. We derive conditions for which this eigenvalue problem can be made explicit and use it to analyze a stripe and ring geometry. This formulation allows us to classify certain technical properties of NLEPs such as instability bands and a Hopf bifurcation condition analytically.

The results for breakup and zigzag instabilities are verified with numerical simulations of the full model in both stripe and ring geometries. This includes confirmation of dominant breakup modes and demonstrating the stabilizing effect of saturation.
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George Bluman
UBC
Thu 28 May 2015, 2:00pm
Symmetries and Differential Equations Seminar
Math125
Direct construction of conservation laws and connections between symmetries and CLs. Part I
Math125
Thu 28 May 2015, 2:00pm-3:00pm

Abstract

In this first lecture on the construction of the conservation laws for a DE system, we present the Direct Method.  This leads to the direct construction of the CLs for essentially any DE system in a systematic framework.  The classical Noether's Theorem only works for variational systems and also requires the construction of the Lagrangian.  The Direct Method involves working directly with a given DE system.  The second lecture will show explicitly how the Direct Method generalizes Noether's Theorem and overcomes all of its limitations.
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