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 Events
University of Pennsylvania
Fri 30 Jan 2015, 3:00pm
Department Colloquium
LSK 200
The fractional Laplacian operator and its gradient perturbations
LSK 200
Fri 30 Jan 2015, 3:00pm-4:00pm

Abstract

The fractional Laplacian operator plays the same paradigmatic role in the theory of nonlocal operators that the Laplacian plays in the theory of local operators. We will present regularity results for solutions to problems defined by the fractional Laplacian operator with gradient perturbations. Our main results are the regularity of solutions in Sobolev spaces to the linear equation in the supercritical regime, when the operator is not elliptic, and the optimal regularity of solutions to the stationary obstacle problem in the supercritical regime.

This is joint work with Charles Epstein and Arshak Petrosyan.

Note for Attendees

Refreshments will be served at 2:40pm in the Math Lounge area, MATH 125 before the colloquium.
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Los Alamos Lab
Mon 2 Feb 2015, 2:00pm SPECIAL
Institute of Applied Mathematics
LSK 460
Developing open-source tools for environmental applications
LSK 460
Mon 2 Feb 2015, 2:00pm-10:00am

Abstract

 

Note for Attendees

 This is the annual IAM alumni lecture. 
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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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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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NYU
Tue 3 Feb 2015, 4:00pm
Department Colloquium
MATH 100
TBA
MATH 100
Tue 3 Feb 2015, 4:00pm-5:00pm

Abstract


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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.
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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 2:40pm in the Math Lounge area, MATH 125 before the colloquium.
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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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