Mathematics Dept.
  Events
Mon 2 Feb 2015, 1:00pm
Math Education Research Reading
MATX1118
"New Pedagogical Models for Instruction in Mathematics" by Greenberg and Williams
MATX1118
Mon 2 Feb 2015, 1:00pm-2:00pm

Abstract

 
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UBC
Mon 2 Feb 2015, 2:45pm
CRG Geometry and Physics Seminar
ESB 4127
Infinite root stacks of log schemes
ESB 4127
Mon 2 Feb 2015, 2:45pm-3:45pm

Abstract

 I will talk about the notion of "infinite root stack" of a logarithmic scheme, introduced by myself and Angelo Vistoli as part of my PhD thesis. It is a "limit" version of the generalization to log schemes of the stack of roots of a divisor on a variety, and we show, among other things, that its "bare" geometry closely reflects the "log" geometry of the base log scheme. After giving some motivation, I will briefly define log schemes and describe this infinite root construction. I will then state the results we get about it, and their relevance to log geometry, also in view of (hopefully) upcoming applications.

Note for Attendees

 Note the exceptional time!
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Los Alamos Lab
Mon 2 Feb 2015, 3:00pm SPECIAL
Institute of Applied Mathematics
LSK 460
Developing open-source tools for environmental applications
LSK 460
Mon 2 Feb 2015, 3:00pm-4:00pm

Abstract

Environmental applications, such as predicting climate impacts and feedbacks in critical watersheds, pose significant challenges for modeling and simulation. These applications are inherently multiscale, and uncertainty about models and model coupling is common. Thus, to manage this complexity, new interdisciplinary community-based approaches are needed.  Here, we highlight advances in open-source scientific libraries, frameworks, and software development methodologies that have led to a growing number of open-source analysis and simulation tools. We present results from our open-source simulator, Amanzi/ATS (Arctic Terrestrial Simulator), for subsurface contaminant transport at a representative waste site and coupled surface/subsurface flow in the Arctic.

Note for Attendees

 This is the annual IAM alumni lecture. 
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IBM TJ Watson Research Center
Mon 2 Feb 2015, 4:00pm
Department Colloquium
LSK 200
A conjugate IP approach for large scale non smooth programs
LSK 200
Mon 2 Feb 2015, 4:00pm-5:00pm

Abstract

Many scientific computing applications can be formulated as large-scale optimization problems, including inverse problems, medical and seismic imaging, classification in machine learning, data assimilation in weather prediction, and sparse difference graphs. While first-order methods have proven widely successful in recent years, recent developments suggest that matrix-free second-order methods, such as interior-point methods, can be competitive.

This talk has three parts. We first develop a modeling framework for a wide range of problems, and show how conjugate representations can be exploited to design a uniform interior point approach for this class. We then show a range of applications, focusing on modeling and special problem structure. Finally, we preview some recent work, which suggests that the conjugate representations admit very efficient matrix free methods in important special cases, and present some recent results for large scale extensions.   


Note for Attendees

Refreshments will be served at 3:40pm in the Math Lounge area, MATH 125 before the colloquium.
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Mathematical Modeling and Analysis, Los Alamos National Laboratory
Tue 3 Feb 2015, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133
Frameworks and Discretizations for Coupled Surface/Subsurface Flow
ESB 4133
Tue 3 Feb 2015, 12:30pm-2:00pm

Abstract

Modeling and simulation are playing an increasingly critical role in understanding and predicting climate impacts and feedbacks in terrestrial systems. Managing the complexity of these process-rich integrated hydrologic and biogeochemical models requires flexible software designs that enable exploration of model features and model coupling.  In addition, flexibility in meshing and robust discretization techniques are required to capture topographic features, such as hill slopes and rivers, and subsurface stratigraphy.

In this talk we highlight a flexible and extensible approach to multiphysics frameworks for these applications that specifies interfaces for coupled processes and automates weak and strong coupling strategies to manage this complexity. Process management is accomplished through a dual view of the model system: a high-level view ideal for model configuration represents the system of equations as a tree, where individual equations are associated with leaf nodes, and coupling strategies with internal nodes; and a low-level dynamically generated dependency graph that connects a variable to its dependencies, streamlining and automating model evaluation, easing model development, and ensuring models are modular and flexible.  We use this multiphysics framework, dubbed Arcos, to support both model and algorithm development for environmental applications in the open-source code Amanzi. For example, we have developed infrastructure for general unstructured polyhedral meshing with a flexible operator-based implementation of the Mimetic Finite Difference method, and used it to simulate coupled surface/subsurface flow.  Here we use a diffusive wave approximation for surface flow, and a Richards equation for subsurface flow.  Coupling is accomplished by ensuring continuity of both pressure and fluxes from the surface to the subsurface, and the system can be solved using either sequential or implicit coupling.  We show results for several benchmark problems, as well as physically relevant, large-scale simulation of rainfall on arctic tundra based upon LIDAR data from Barrow, Alaska. This demonstration shows the parallel performance of the code and its feasibility for use in watershed scale, high resolution simulations.

Note for Attendees

 Sushi lunch will be provided
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NYU
Tue 3 Feb 2015, 4:00pm
Department Colloquium
MATH 100
Multiscale modeling and simulation in active fluids
MATH 100
Tue 3 Feb 2015, 4:00pm-5:00pm

Abstract


Active fluids, the novel class of non-equilibrium materials made up of self-driven constituents, is attracting growing interest due to its impact on cell biology, condensed matter physics, and nanotechnology. Despite their differences in composition, active fluids orchestrate cooperative actions across various length and time scales, and accompany energy conversion from one form to another. In this talk, I focus on a recent study of nonlinear dynamics and pattern formation of microtubule/motor-protein assemblies using multiscale modeling and simulation. I explain how the local microtubule-motor and microtubule-microtubule interactions manifest themselves at macroscopic scales through hydrodynamic instability, as well as the connections between the coherent flow structures and topological defects observed in active nematics. I also briefly discuss the new physics and phenomena in other systems, and numerical and theoretical tool development for the quantitative study of general complex fluids.


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UT Austin
Thu 5 Feb 2015, 12:30pm
Department Colloquium
MATX 1100
Applications and Numerical Methods for Optimal Transportation
MATX 1100
Thu 5 Feb 2015, 12:30pm-1:30pm

Abstract

The problem of optimal transportation, which involves finding the most cost-efficient mapping between two measures, arises in many different applications. However, the numerical solution of this problem remains extremely challenging. After surveying several current applications, we describe a numerical method for the widely-studied case when the cost is quadratic. The solution is obtained by solving the Monge-Ampere equation, a fully nonlinear elliptic partial differential equation (PDE), coupled to anon-standard implicit boundary condition. Expressing this problem in terms of weak (viscosity) solutions enables us to construct a monotone finite difference approximation that provably converges to the correct solution. A range of challenging computational examples demonstrate the effectiveness of this method.

Note for Attendees

Sushi will be served.
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SFU
Thu 5 Feb 2015, 3:30pm
Number Theory Seminar
room ASB 10940 (SFU - IRMACS)
Some explicit Frey hyperelliptic curves
room ASB 10940 (SFU - IRMACS)
Thu 5 Feb 2015, 3:30pm-4:30pm

Abstract

Darmon outlined a program which is suited to potentially resolving one parameter families of generalized Fermat equations. He gave explicit descriptions of Frey representations and conductor calculations for Fermat equations of signature (p,p,r). Somewhat less explicit results are stated for signature (r,r,p), and even less for signature (q,r,p).
 
For the equation (r,r,p), there are at least three competing Frey curve constructions: superelliptic curves of hypergeometric type due to Darmon, hyperelliptic curves due to Kraus, and elliptic curves with models over totally real fields due to Freitas.
 
I will survey these Frey curve constructions and end by giving explicit Frey hyperelliptic curves for signatures (2,r,p) and (3,5,p).
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Los Alamos National Laboratory
Thu 5 Feb 2015, 4:00pm
Department Colloquium
ESB 2012 (note changed time and place)
Low Reynolds number flows through shaped and deformable conduits
ESB 2012 (note changed time and place)
Thu 5 Feb 2015, 4:00pm-5:00pm

Abstract

Unconventional fossil energy resources are revolutionizing the US energy market. While the techniques developed over the last 50 years lead to viable and profitable extraction of, e.g., trapped gas and hydrocarbons from almost-impermeable rock formations via hydraulic fracturing, the abysmal extraction rates (typically 15%) suggest the fluid mechanics of these processes is not well understood. In this talk, I will describe three basic theoretical fluid mechanics problems inspired by unconventional fossil fuel extraction. The first problem is flow in a deformable microchannel. Fluid-structure interaction couples the shape of the conduit to the flow through it, drastically altering the flow rate--pressure drop relation. Using perturbation methods, we show that the flow rate is a quartic polynomial of pressure drop for shallow channels, in contrast to the linear relation for rigid conduits. The second problem involves two-phase (gas-liquid) displacement in a horizontal Hele-Shaw cell with an elastic membrane as the top boundary. This problem arises at the pore-scale in enhanced oil recovery for large injection pressures. Once again, fluid-structure interaction alters the problem, leading to stabilization of the Saffman--Taylor (viscous fingering) instability below a critical flow rate. Using lubrication theory, we derive the stability threshold and show that it agrees well with recent experiments. The third problem involves the spread of a viscous liquid in a vertical Hele-Shaw cell with a variable thickness in the flow-wise direction, as a model for the spread of a plume of supercritical carbon dioxide through the non-uniform passages created by hydraulic fracturing. We show that the propagation regimes in such a shaped conduit are set by the direction of propagation. While the rate of spread in the direction of increasing gap thickness (and, hence, permeability) can be obtained by standard scaling techniques, the reverse scenario requires the construction of a so-called second-kind self-similar solution, leading to nontrivial exponents in the rate of spread.

Note for Attendees

Refreshments will be served at 3:45pm in the PIMS Lounge.
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Alex Bihlo
UBC
Thu 5 Feb 2015, 4:30pm
Symmetries and Differential Equations Seminar
Math 126
Invariant and conservative numerical schemes: Theory and applications
Math 126
Thu 5 Feb 2015, 4:30pm-5:30pm

Abstract

For centuries, geometric properties such as symmetries, conservation
laws and Hamiltonian forms play a central role in the study of
differential equations. Yet the importance of preserving geometric
properties also in the numerical solution of differential equations has
been pointed out only recently and is still not a sufficiently
acknowledged field in the numerical analysis of differential equations. In
this talk methods for finding invariant and conservative integrators
applicable to wide classes of ODEs and PDEs will be presented. Several
examples illustrating the practical relevance of these so-called geometric
integrators or mimetic schemes will be given.
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UBC
Tue 10 Feb 2015, 4:00pm
Discrete Math Seminar
ESB 4127
Ramsey Theory and Forbidden Configurations
ESB 4127
Tue 10 Feb 2015, 4:00pm-5:00am

Abstract

We say that a matrix F is a configuration in a matrix A if there is a submatrix of A which is a row and column permutation of F. We consider the following extremal function. Let G be a finite set of (0,1)-matrices. Let forb(m,G) denote the maximum number of columns in an m-rowed (0,1)-matrix A that has no repeated columns and no configuration F in the set G. It was already shown by Balogh and Bollobas that for any given k that if G consists of the three matrices I_k (identity matrix of order k),I_k^c (0-1 complement of I_k) and T_k (upper triangular (0,1)-matrix) then forb(m,G) is a constant where the constant is at least 2^{ck} for some constant c and at most 2^{2^k}. These three are unavoidable configurations in much the same way that a clique of size k and an independent set of size k are unavoidable in a large complete graph, you always get one or the other. We obtain a new argument (using Ramsey Theory) that brings the bound on forb(m,G) to 2^{ck^2}. This is joint work with Lincoln Lu.
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McGill University
Wed 11 Feb 2015, 3:00pm
Department Colloquium
ESB 2012
Some new numerical techniques to some old PDE problems
ESB 2012
Wed 11 Feb 2015, 3:00pm-4:00pm

Abstract

Problems involving complicated deforming time-dependent boundaries or interfaces (i.e. codim-1 surfaces) are ubiquitous in the modeling of physical systems. The resulting Partial Differential Equations (PDEs) often have irregular, and even discontinuous solutions along these surfaces. In turn, the solution of these PDEs couples back to flow and hence deform the surfaces in question. The numerical approximation of such systems isnotoriously challenging.

In a regular Cartesian grid setting, I propose to replace the original PDE by another PDE which better approximates, in the discrete setting, the exact solution to the original PDE. I will provide 3 examples of this approach.

First, in the active penalty method one does not enforce the PDE boundary conditions directly, but rather solves the PDE in a larger domain without boundary (e.g. a flat torus) and adds a carefully constructed source or penalty term that mimics boundary conditions. I will show how to systematically construct  penalty terms which improve the convergence rates of the penalized PDE, thereby allowing for higher-order finite-difference or Fourier-spectral numerical schemes to be applied to problems withnon-conforming boundaries.

Second, in the correction function method I tackle the problem of solving PDEs with jump discontinuities across a codim-1 surface, i.e. an interface. This is achieved by formulating an auxiliary local PDE which solution smoothly extends across the interface while enforcing the jump conditions. I will show that this approach is general, and can achieve arbitrary order of convergence while incurring (asymptotically) no additional computational cost.

Third, I will present methods for evolving in time arbitrary geometric objects such as boundaries or interfaces, but also general open/closed
surfaces with possibly no regularity (e.g. fractals). This is achieved by evolving in time the flow map and composing it with the initial conditions. This method fit naturally within the gradient-augmented level set framework and enables use of  a two-grid approach to achieve arbitrary order of convergence and optimal efficiency.

Throughout the talk I will illustrate these approaches with simulations of various physical systems including problems from fluid dynamics,
electromagnetism, solid mechanics for which these methods may need to be combined.

Note for Attendees

PIMS Afternoon Tea will be served at 2:45 p.m. in the PIMS Lounge (ESB 4133).
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Uriya First
UBC
Thu 12 Feb 2015, 3:30pm
Number Theory Seminar
room MATH 126
From systems of quadratic forms to single hermitian forms
room MATH 126
Thu 12 Feb 2015, 3:30pm-4:30pm

Abstract

I will describe a general technique to extend various results about non-degenerate quadratic and hermitian forms to systems of (possibly degenerate) hermitian forms. The idea is based on a certain categorical equivalence. Results that can be extended in this manner include: Witt’s cancellation theorem, Springer’s theorem, the weak Hasse principle (in certain cases), and statements about about finiteness of the genus.
(Joint work with Eva Bayer-Fluckiger and Daniel Moldovan)
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Anthony Wachs
IFP Energies nouvelles France
Thu 12 Feb 2015, 4:00pm
Department Colloquium
ESB 2012
Particulate flow across multi-scales: numerical strategies for momentum, heat and mass transfer
ESB 2012
Thu 12 Feb 2015, 4:00pm-5:00pm

Abstract

Particulate flows are ubiquitous in environmental, geophysical and engineering processes. The intricate dynamics of these two-phase flows is governed by the momentum transfer between the continuous fluid phase and the dispersed particulate phase. When significant temperature differences exist between the fluid and particles and/or chemical reactions take place at the fluid/particle interfaces, the phases also exchange heat and/or mass, respectively. While some multi-phase processes may be successfully modelled at the continuum scale through closure approximations, an increasing number of applications require resolution across scales, e.g. dense suspensions, fluidized beds. Within a multi-scale micro/meso/macro-framework, we develop robust numerical models at the micro and meso-scales, based on a Distributed Lagrange Multiplier/Fictitious Domain method and a two-way Euler/Lagrange method, respectively. Particles, assumed to be of finite size, potentially collide with each other and these collisions are modeled with a Discrete Element Method. We discuss the mathematical issues related to modeling this type of flows and present the main numerical and computational features of our simulation methods. We also illustrate what can be gained from massively parallel computations performed with our numerical code PeliGRIFF, in terms of physical insight into both fundamental questions and applications from the chemical engineering and process industry. Finally, we explain how knowledge gained at the micro scale can cascade upwards and contribute to the development of enhanced meso and macro-scale models.

 Speaker Biography: Dr Anthony Wachs received BS & MS degrees from University Louis Pasteur, Strasbourg and his PhD from the Institut National Polytechnique of Grenoble in 2000. He joined Institut Français du Pétrole in 2001 (now IFP Energies Nouvelles), passed his HDR in 2010 and is currently both scientific advisor and project manager.  He leads a team of researchers that develop both mathematical models and robust computational algorithms for the resolution of multi-phase flows (www.peligriff.com). His main research interests are non-Newtonian Flows, Multiphase Flows and High Performance Computing. He collaborates extensively with academic groups in Canada, Brazil, France and Germany.

 

Note for Attendees

Refreshments will be served in the PIMS Lounge (ESB 4133) at 3:15p.m. before this colloquium.
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Elsa Cardoso
UBC
Thu 12 Feb 2015, 4:30pm
Symmetries and Differential Equations Seminar
Math 125
Lie symmetry analysis of the primitive equations
Math 125
Thu 12 Feb 2015, 4:30pm-5:30pm

Abstract

The primitive equations are the main system of nonlinear partial differential equations on which modern weather and climate prediction models are based on. The Lie symmetries of the primitive equations are computed and the structure of the maximal Lie invariance algebra, which is infinite dimensional, is investigated. It is found that the maximal Lie invariance algebra for the case of a constant Coriolis force can be mapped to the case of vanishing Coriolis force. The same mapping allows one to transform the constantly rotating primitive equations to the equations in a resting reference frame. This mapping is used to obtain exact solutions for the rotating case from exact solutions from the nonrotating equations.
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Stanford U. and UC Irvine
Mon 16 Feb 2015, 3:30pm
Department Colloquium
ESB 2012 (PIMS)
Special PIMS colloquium: New results on the Bartnik mass
ESB 2012 (PIMS)
Mon 16 Feb 2015, 3:30pm-4:30pm

Abstract

We will introduce the Bartnik mass and survey the progress toward its understanding. We will then present a new upper bound which is sharp in certain cases. It is contained in a recent joint work with C. Mantoulidis and has relevance to a conjecture of G. Gibbons.
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Raouf Dridi
Thu 19 Feb 2015, 4:30pm
Symmetries and Differential Equations Seminar
Math 125
Algorithmic constructions for potential systems
Math 125
Thu 19 Feb 2015, 4:30pm-5:30pm

Abstract

I systemize a number of constructions appearing in the theory of potential systems
and nonlocal symmetries. I use the machinery of differential field extensions and differential Groebner basis.
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Bioengineering and Chemical and Biomolecular Engineering, University of Pennsylvania
Mon 23 Feb 2015, 3:00pm SPECIAL
Institute of Applied Mathematics
LSK 460
Adhesive Dynamics Simulations of Blood Cell Adhesion
LSK 460
Mon 23 Feb 2015, 3:00pm-4:00pm

Abstract

Adhesive Dynamics is a method to simulate the dynamics of cell adhesion to surfaces. Adhesion receptors are modeled as reactive mechanical entities with adhesive tips, and the formation and breakage of adhesion molecules with cognate ligands is simulated using random number sampling. Once the bonds form, the contact points they make with surfaces are tracked, and a force balance is used to calculate the motion of the cell. Specific rheological laws relate stress (or strain) to bond failure rates, and the parameters of these laws dictate the quantitative sensitivity of adhesion molecules to force, and ultimately affect the dynamics of cell adhesion as a whole. We summarize major findings of cell adhesion that have been enabled by Adhesive Dynamics - the development of state diagrams of adhesion, that link distinct states of adhesion to molecular identity, such as leukocyte rolling; specification of what is required for firm arrest of a leukocyte; a description of the shear threshold effect in which adhesion increases with shear rate; and understanding how two molecules can act in synergy to secure adhesion that cannot be secured by either molecule alone. Finally, we show how signal transduction networks can be integrated within adhesive dynamics to understand how cell activation can lead to changes in adhesion and arrest, and use predictions to understand how cells might behave when molecular components are altered or eliminated in knock-out mice, or in various diseases due to molecular defects.
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Northwestern University
Mon 23 Feb 2015, 4:00pm
Department Colloquium
LSK 200
Math Department Colloquium/Fluids Seminar: Sessile drop dynamics
LSK 200
Mon 23 Feb 2015, 4:00pm-5:00pm

Abstract

Oscillations of the sessile drop are of fundamental interest in a number of industrial applications, such as ink-jet printing and drop atomization. We generalize the stability analysis for the free inviscid drop (Rayleigh, 1879), focusing on the wetting properties of the solid substrate and mobility of the three-phase contact-line.  We report oscillation frequencies and modal structures for the `symmetry-broken’ Rayleigh drop that display spectral splitting/reordering and compare with experiments. To organize and explain the hierarchy of frequencies, we construct a corresponding `periodic table of mode shapes’ from the spectral data. In addition to the oscillatory spectrum, we report a new hydrodynamic instability that has fundamental implications for fluid transport.


Profile:  Dr Joshua Bostwick received bachelors degrees in Physics and Civil Engineering from University of Wisconsin-Milwaukee in 2005, and his PhD from Cornell University in 2011. He worked as a postdoctoral researcher at North Carolina State University and is currently Golovin Assistant Professor in the Department of Engineering Science and Applied Mathematics at Northwestern University. His research interests span surface tension, hydrodynamic instability, wetting and spreading, elastocapillarity, dynamical systems, constrained variational principles, symmetry methods.


Note for Attendees

Refreshments will be served at 3:40pm in the Math Lounge area, MATH 125 before the colloquium.
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Marie-Therese Wolfram
Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences
Tue 24 Feb 2015, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
Crowd motion: from modeling to simulations
ESB 4133 (PIMS Lounge)
Tue 24 Feb 2015, 12:30pm-1:30pm

Abstract

The dynamic motion of large human crowds is an ubiquitous phenomena in everyday life. First empirical studies on crowd motion started in the late 1950ties and spread into different fields like transportation research, psychology or urban and regional planning. Recently there has been a strong interest in crowd motion within the mathematical community, which initiated a lot of research on mathematical models, their analysis and simulations.
 
In this talk we focus on a fast exit scenario and consider a group, which wants to leave a room as quickly as possible. We present different modeling approaches, starting on the microscopic level and working our way up to the appropriate continuum limits. In particular we focus on Hughes model for pedestrian flow and give an interpretation from the mean field game perspective. Finally we discuss different challenges in the analysis and numerical simulations and illustrate the behaviour of the presented models with numerical simulations.
 

Note for Attendees

Lunch will be provided.
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IMAR (Bucharest) and Université Claude Bernard (Lyon)
Tue 24 Feb 2015, 2:00pm
CRG Geometry and Physics Seminar
ESB 4127
Serre's conjecture II : beyond the de Jong-He-Starr's theorem
ESB 4127
Tue 24 Feb 2015, 2:00pm-3:00pm

Abstract

 
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Ecole Polytechnique
Tue 24 Feb 2015, 4:00pm
Discrete Math Seminar
ESB 4127
Introduction to maps I: polygon gluings and classification of surfaces
ESB 4127
Tue 24 Feb 2015, 4:00pm-5:00pm

Abstract

Maps describe the way a surface can be obtained by gluing polygons together, and as such they provide a handy tool to prove that closed surfaces can be classified by a genus parameter  (both in the orientable and non-orientable case). I will review on this and on the equivalence between maps as polygon gluings and maps as graphs endowed with a rotation system. If time allows I will also review on a nice formula due to Harer and Zagier for the number of ways one can obtain the orientable surface of genus g by gluing pairwise the sides of a 2n-polygon.
 
(This is the first of a series of 3 talks on maps. The next two will occur later in March.)
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University of British Columbia
Wed 25 Feb 2015, 3:10pm
Probability Seminar
ESB 2012
A new proof of the sharpness of the percolation phase transition
ESB 2012
Wed 25 Feb 2015, 3:10pm-4:00pm

Abstract

The sharpness of the percolation phase transition, which is a crucial and much cited element of the theory, was first proved independently by Menshikov in 1986 and by Aizenman and Barsky in 1987. On February 11, 2015, a remarkable new proof was posted on arXiv by Duminil-Copin and Tassion.  I will describe their proof, which is completed in two pages, in its entirety.
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Stefan Erickson
Colorado College
Thu 26 Feb 2015, 3:30pm
Number Theory Seminar
room ASB 10940 (SFU - IRMACS)
Apollonian circle packings
room ASB 10940 (SFU - IRMACS)
Thu 26 Feb 2015, 3:30pm-4:30pm

Abstract

Apollonius's Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles ll all remaining spaces. A remarkable consequence of Descartes' Theorem is if the initial four tangent circles have integral curvatures, then all of the circles in an Apollonian circle packing will have integral curvatures. This process results a sequence of integers with fascinating arithmetic properties.

In this talk, we will investigate the arithmetic properties of Apollonian circle packings. We will describe the Apollonian group action on the set of Descartes quadruples. We will talk about modular restrictions and density conjectures and theorems. Finally, we will show a correspondence between the root quadruples and reduced binary quadratic forms and answer an open question about finding the root quadruple of a given Descartes quadruple.

 

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University of Melbourne
Thu 26 Feb 2015, 4:00pm
ESB 4133
Integrability, Solvability and Enumeration.
ESB 4133
Thu 26 Feb 2015, 4:00pm-5:00pm

Details

There are a number of seminal two-dimensional lattice models that are integrable, but have only been partially solved, in the sense that only some properties are fully known (e.g. the two-dimensional Ising model, where the free-energy is known, but not the susceptibility). Alternatively, critical properties are known for some lattices but not others. For
example, the critical point of the self-avoiding walk model is known rigorously for the honeycomb lattice, but not for other lattices. Similarly for the q-state Potts model and both bond and site percolation. The critical manifold of the former is known only for some lattices, likewise the percolation threshold is known only for some lattices.

A range of numerical procedures exist, based on exact enumeration, or other numerical work, such as calculating the eigenvalues of  transfer matrices, which, when combined with various structural invariants seem to give exact results in those cases that are known to be exact, but can be used to give increasingly precise estimates in those cases which are not exactly known. Reasons for this partial success are not well understood. In this talk I will describe four such procedures, and demonstrate their performance, and speculate on
their partial success.

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Kamran Fakhar
UBC and Kwantlen Polytechnic University
Thu 26 Feb 2015, 4:30pm
Symmetries and Differential Equations Seminar
Math 125
Lie symmetries and transformation of solutions for the Navier-Stokes equations
Math 125
Thu 26 Feb 2015, 4:30pm-5:30pm

Abstract

It is shown in this study that the Navier-Stokes equations allow an infinite-dimensional Lie group of point transformations, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group here depends on four arbitrary functions of time. Some new deformed solutions of Navier-Stokes equations in two and three dimensions are obtained by applying some of the elements of the symmetry group of these equations to their basic solutions. In order to explore the properties of deformed solutions, the analytic solutions are analyzed. It is noted that the corresponding deformed solutions behave as the basic solutions in the limiting sense for large time.
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Caltech
Fri 27 Feb 2015, 3:00pm
Department Colloquium
ESB 2012 (PIMS)
PIMS-UBC distinguished colloquium: Blowup or no blowup? The interplay between theory and computation in the study of 3D Euler equations.
ESB 2012 (PIMS)
Fri 27 Feb 2015, 3:00pm-4:00pm

Abstract

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities.  To resolve the nearly singular solution, we develop specially designed adaptive (moving) meshes with a maximum effective resolution of order 10^12 in each direction. A careful local analysis also suggests that the solution develops a highly anisotropic self-similar profile which is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. Very recently we prove rigorously that the 1D model develops finite time singularity.

This is a joint work of Prof. Guo Luo.

Note for Attendees

Coffee, tea and cookies served at 2:30pm in the PIMS Lounge, ESB 4133.
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