Past

The aim of this two-day workshop is to bring together mathematicians, biologists and researchers from other disciplines whose work involves stochastic and multiscale phenomenon, to identify common methodologies to studying such systems, both from a numerical and analytical perspective.   Relevant topics include asymptotic methods for PDEs; multiscale analysis of stochastic dynamical systems; mean-field limits of collective dynamics.  Numerical methods, mathematical theory and applications (with a specific focus on biology) will all be discussed.  The workshop will take place on the 1st and 2nd of September, at the Mathematical Institute, Oxford University.   Please visithttps://sites.google.com/site/stochmultiscale2014/ for more information and to register.

The aim of this two-day workshop is to bring together mathematicians, biologists and researchers from other disciplines whose work involves stochastic and multiscale phenomenon, to identify common methodologies to studying such systems, both from a numerical and analytical perspective.   Relevant topics include asymptotic methods for PDEs; multiscale analysis of stochastic dynamical systems; mean-field limits of collective dynamics.  Numerical methods, mathematical theory and applications (with a specific focus on biology) will all be discussed.  The workshop will take place on the 1st and 2nd of September, at the Mathematical Institute, Oxford University.   Please visithttps://sites.google.com/site/stochmultiscale2014/ for more information and to register.

Economic dispatch is a critical part of electricity planning and 
operation. Enhancing the dispatch problem to improve its robustness 
in the face of equipment failures or other contingencies is standard 
practice, but extremely time intensive, leading to restrictions on 
the richness of scenarios considered. We model post-contingency 
corrective actions in the security-constrained economic dispatch 
and consider multiple stages of rescheduling to meet different 
security constraints. The resulting linear program is not solvable
by traditional LP methods due to its large size. We devise and 
implement a series of algorithmic enhancements based on the Benders'
decomposition method to ameliorate the computational difficulty.
In addition, we propose a set of online measures to diagnose
and correct infeasibility issues encountered in the solution process.

The overall solution approach is able to process the ``N-1'' 
contingency list in ten minutes for all large network cases
available for experiments. Extensions to the nonlinear setting will 
be discussed via a semidefinite relaxation.

July 5

9:30-10:30

Robert Langlands (IAS, Princeton)

Problems in the theory of automorphic forms: 45 years later 

11:00-12:00

Christopher Deninger (Univ. Münster)

Zeta functions and foliations     

 13:30-14:30          

Christophe Soulé (IHES, Bures-sur-Yvette)

 A singular arithmetic Riemann-Roch theorem           

14:40-15:40

Minhyong Kim (Univ. Oxford)

 Non-abelian reciprocity laws and Diophantine geometry

 16:10-17:10  

Constantin Teleman (Berkeley/Oxford)           

Categorical representations and Langlands duality 

July 6

 9:30-10:30

Ted Chinburg (Univ. Pennsylvania, Philadelphia)

 Higher Chern classes in Iwasawa theory

11:00-12:00          

Yuri Tschinkel (Courant Institute, New York)

Introduction to almost abelian anabelian geometry

13:30-14:30

Ralf Meyer (Univ. Göttingen)

Groupoids and higher groupoids

14:40-15:40

Dennis Gaitsgory (Harvard Univ., Boston)

Picard-Lefschetz oscillators for Drinfeld-Lafforgue compactifications

16:10-17:10

François Loeser (Univ. Paris 6-7)

Motivic integration and representation theory

July 7

9:00-10:00

Matthew Morrow (Univ. Bonn)                                                  

On the deformation theory of algebraic cycles

10:30-11:30

Fedor Bogomolov (Courant Institute, New York/Univ. Nottingham)

On the section conjecture in anabelian geometry                 

13:15-14:15                                                                      

Kevin Buzzard (ICL, London)

p-adic Langlands correspondences

14:45-15:45                                                                      

Masatoshi Suzuki (Tokyo Institute of Technology)

Translation invariant subspaces and GRH for zeta functions

16:00-17:00

Edward Frenkel (Univ. California Berkeley)

"Love and Math", the Langlands programme - Public presentation

July 8

9:15-10:15

Mikhail Kapranov (Kavli IMPU, Tokyo)

Lie algebras and E_n-algebras associated to secondary polytopes                                 

10:45-11:45          

Sergey Oblezin (Univ. Nottingham)

Whittaker functions, mirror symmetry and the Langlands correspondence

13:30-14:30                                                                      

Edward Frenkel (Univ. California Berkeley)

The Langlands programme and quantum dualities

14:40-15:40                                                                                              

Dominic Joyce (Univ. Oxford)

Derived symplectic geometry and categorification

16:10-17:10  

Urs Schreiber (Univ. Nijmegen, The Netherlands)

Correspondences of cohesive linear homotopy types and quantization

Universal fluctuations are shown to exist when well-known and widely used numerical algorithms are applied with random data. Similar universal behavior is shown in stochastic algorithms and algorithms that model neural computation. The question of whether universality is present in all, or nearly all, computation is raised. (Joint work with G.Menon, S.Olver and T. Trogdon.)

Seismology currently undergoes rapid and exciting advances fueled by a simultaneous surge in recorded data (in both quality and quantity), realistic wave-propagation algorithms, and supercomputing capabilities. This enables us to sample parameter spaces of relevance for imaging the Earth's interior 3D structure with fully numerical techniques. Seismic imaging is the prime approach to illuminate and understand global processes such as mantle convection, plate tectonics, geodynamo, the vigorous interior of the Sun, and delivers crucial constraints on our grasp of volcanism, the carbon cycle and seismicity. At local scales, seismic Earth models are inevitable for hydrocarbon exploration, monitoring of flow processes, and natural hazard assessment.

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With a slight focus on global-scale applications, I will present the underlying physical model of realistic wave propagation, its numerical discretization and link such forward modeling to updating Earth models by means of inverse modeling. The associated computational burden to solve high-resolution statistical inverse problems with precise numerical techniques is however entirely out of reach for decades to come. Consequently, seismologists need to take approximations in resolution, physics, data and/or inverse methodology. I will scan a number of such end-member approximations, and focus on our own approach to simultaneously treat wave physics realistically across the frequency band while maximizing data usage and allow for uncertainty quantification. This approach is motivated by decisive approximations on the model space for typical Earth structures and linearized inverse theory.

Peter Grindrod and Stephen Haben (UoOx)

We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass and angular momentum.

This is joint work with Angus Macintyre.

We prove that the first-order theory of a finite extension of the field of p-adic numbers is model-complete in the language of rings, for any prime p.

To prove this we prove universal definability of the valuation rings of such fields using work of Cluckers-Derakhshan-Leenknegt-Macintyre on existential

definability, quantifier elimination of Basarab-Kuhlmann for valued fields in a many-sorted language involving higher residue rings and groups,

a model completeness theorem for certain pre-ordered abelian groups which generalize Presburger arithmetic (we call finite-by-Presburger groups),

and an interpretation of higher residue rings of such fields in the higher residue groups.

Lie algebroids are geometric structures that interpolate between finite-dimensional Lie algebras and tangent bundles of manifolds. They give a useful language for describing geometric situations that have local symmetries. I will give an introduction to the basic theory of Lie algebroids, with examples drawn from foliations, principal bundles, group actions, Poisson brackets, and singular hypersurfaces.

This is a joint work with V. Lemaire

(LPMA-UPMC). We propose and analyze a Multilevel Richardson-Romberg

(MLRR) estimator which combines the higher order bias cancellation of

the Multistep Richardson-Romberg ($MSRR$) method introduced

in~[Pag\`es 07] and the variance control resulting from the

stratification in the Multilevel Monte Carlo (MLMC) method (see~$e.g.$

[Heinrich 01, M. Giles 08]). Thus we show that in standard frameworks

like discretization schemes of diffusion processes, an assigned

quadratic error $\varepsilon$ can be obtained with our (MLRR)

estimator with a global complexity of

$\log(1/\varepsilon)/\varepsilon^2$ instead of

$(\log(1/\varepsilon))^2/\varepsilon^2$ with the standard (MLMC)

method, at least when the weak error $\E Y_h-\EY_0}$ induced by the

biased implemented estimator $Y_h$ can be expanded at any order in

$h$. We analyze and compare these estimators on several numerical

problems: option pricing (vanilla or exotic) using $MC$ simulation and

the less classical Nested Monte Carlo simulation (see~[Gordy \& Juneja

2010]).

We are interested in large systems of agents collectively looking for a

consensus (about e.g. their direction of motion, like in bird flocks). In

spite of the local character of the interactions (only a few neighbours are

involved), these systems often exhibit large scale coordinated structures.

The understanding of how this self-organization emerges at the large scale

is still poorly understood and offer fascinating challenges to the modelling

science. We will discuss a few of these issues on a selection of specific

examples.

The accurate and efficient solution of linear systems Ax = b is very important in many engineering and technological applications, and systems of this form also arise as subproblems within other algorithms. In particular, this is true for interior point methods (IPM), where the Newton system must be solved to find the search direction at each iteration. Solving this system is a computational bottleneck of an IPM, and in this talk I will explain how preconditioning and deflation techniques can be used, to lessen this computational burden.

This is joint work with Jacek Gondzio.

I shall outline a general method for finding upper bounds on the diameters of finite groups, based on the Solovay-Kitaev procedure from quantum computation. This method may be fruitfully applied to groups arising as quotients of many familiar pro-p groups. Time permitting, I will indicate a connection with weak spectral gap, and give some applications.

Waldhausen defined higher K-groups for categories with certain extra structure. In this talk I will define categories with cofibrations and weak equivalences, outline Waldhausen's construction of the associated K-Theory space, mention a few important theorems and give some examples. If time permits I will discuss the infinite loop space structure on the K-Theory space.

 A continuum is a non-empty compact connected metric space. Given a continuum X let P(X) be the power set of X. We define the following set functions:
T:P(X) to P(X) given by, for each A in P(X), T(A) = X \ { x in X : there is a continuum W such that x is in Int(W) and W does not intersect A}
K:P(X) to P(X) given by, for each A in P(X), K(A) = Intersection{ W : W is a subcontinuum of X and A is in the interior of W}
S:P(X) to P(X) given by, for each A in P(X), S(A) = { x in T(A) : A intersects T(x)}
Some properties and relations between these functions are going to be presented.