Past

Would you like to come see some toys?

'Toys' here have a special sense: objects of daily life which you can find or make in minutes, yet which, if played with imaginatively reveal surprises that keep scientists puzzling for a while. We will see table-top demos of many such toys and visit some of the science that they open up. The common theme is singularity.

Tadashi Tokieda is the Director of Studies in Mathematics at Trinity Hall, Cambridge and the Poincaré Professor in the Department of Mathematics, Stanford.

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Low-rank compression of matrices and tensors is a huge and growing business.  Closely related is low-rank compression of multivariate functions, a technique used in Chebfun2 and Chebfun3.  Not all functions can be compressed, so the question becomes, which ones?  Here we focus on two kinds of functions for which compression is effective: those with some alignment with the coordinate axes, and those dominated by small regions of localized complexity.

The class of multiserial algebras contains many well-studied examples of algebras such as the intensely-studied biserial and special biserial algebras. These, in turn, contain many of the tame algebras arising in the modular representation theory of finite groups such as tame blocks of finite groups and all tame blocks of Hecke algebras. However, unlike  biserial algebras which are of tame representation type, multiserial algebras are generally of wild representation type. We will show that despite this fact, we retain some control over their representation theory.

An important observation in compressed sensing is the exact recovery of an l0 minimiser to an underdetermined linear system via the l1 minimiser, given the knowledge that a sparse solution vector exists. Here, we develop a continuous analogue of this observation and show that the best L1 and L0 polynomial approximants of a corrupted function (continuous analogue of sparse vectors) are equivalent. We use this to construct best L1 polynomial approximants of corrupted functions via linear programming. We also present a numerical algorithm for computing best L1 polynomial approximants to general continuous functions, and observe that compared with best L-infinity and L2 polynomial approximants, the best L1 approximants tend to have error functions that are more localized.

Joint work with Alex Townsend (MIT).

I'll explain the formalism of extended QFT, while
focusing on the cases of two dimensional conformal field theories,
and three dimensional topological field theories.

It has been shown that there are global solutions to 
Maxwell-Klein-Gordon equations in Minkowski space with finite energy 
data. However, very little is known about the asymptotic behavior of the 
solution. In this talk, I will present recent progress on the decay 
properties of the solutions. We show the quantitative energy flux decay 
of the solutions with data merely bounded in some weighted energy space. 
The results in particular hold in the presence of large total charge. 
This is the first result that gives a complete and precise description 
of the global behavior of large nonlinear fields.
 

Finiteness properties of groups come in many flavours, I will discuss topological finiteness properties. These relate to the finiteness of skelata in a classifying space. Groups with interesting finiteness properties have been found in many ways, however all such examples contains free abelian subgroups of high rank. I will discuss some constructions of groups discussing the various ways we can reduce the rank of a free abelian subgroup. 

Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions, 
with K-contact manifolds corresponding to symplectic manifolds. It is an interesting problem to find
obstructions for a closed manifold to admit such types of structures and in particular, to construct
K-contact manifolds which do not admit Sasakian structures. In the simply-connected case, the
hardest dimension is 5, where Kollar has found subtle obstructions to the existence of Sasakian 
structures, associated to the theory of algebraic surfaces.
In this talk, we develop methods to distinguish K-contact manifolds from Sasakian ones in 
dimension 5. In particular, we find the first example of a closed 5-manifold with first Betti number 0 which is K-contact but which carries no semi-regular Sasakian structure.

 (Joint work with J.A. Rojo and A. Tralle).

We establish a correspondence between the ABC Conjecture and N=4 super-Yang-Mills theory. This is achieved by combining three ingredients:

(i) Elkies' method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings;

(ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and

(iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d'enfant in the sense of Grothendieck. 
 

We explore this correspondence for the highest quality ABC-triples as well as large samples of random triples. The Conjecture itself is mapped to a statement about the fundamental domain of the toroidal compactification of the string realization of N=4 SYM.

What is the purpose of journals?  How should you choose what journal to submit a paper to?  Should it be open access?  And how would you like your work to be evaluated?

Given some class of "geometric spaces", we can make a ring as follows. Additive structure: when U is an open subset a space X,  [X] = [U] + [X - U]. Multiplicative structure:  [X][Y] = [XxY]. In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology.  I will discuss some remarkable
statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural).  A motivating example will be polynomials in one variable. This is joint work with Melanie Matchett Wood.

In this talk, a concrete realization of the Bern-Carrasco-Johansson (BCJ) duality between color and kinematics in non-abelian gauge theories is presented. The method of Berends-Giele to package Feynman diagrams into currents is shown to yield classical solutions to the non-linear Yang-Mills equations. We describe a non-linear gauge transformation of these perturbiner solutions which reorganize the cubic-diagram content such that the kinematic dependence obeys the same Jacobi identities as the accompanying color factors. The resulting tree-level subdiagrams are assembled to kinematic numerators of tree-level and one-loop amplitudes which satisfy the BCJ duality.

We describe the representation theory of loop groups in
terms of K-theory and noncommutative geometry. For any simply
connected compact Lie group G and any positive integer level l we
consider a natural noncommutative universal algebra whose 0th K-group
can be identified with abelian group generated by the level l
positive-energy representations of the loop group LG.
Moreover, for any of these representations, we define a spectral
triple in the sense of A. Connes and compute the corresponding index
pairing with K-theory. As a result, these spectral triples give rise
to a complete noncommutative geometric invariant for the
representation theory of LG at fixed level l. The construction is
based on the supersymmetric conformal field theory models associated
with LG and it can be generalized, in the setting of conformal nets,
to many other rational chiral conformal field theory models including
loop groups model associated to non-simply connected compact Lie
groups, coset models and the moonshine conformal field theory. (Based
on a joint work with Robin Hillier)

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Fluids and solids leave our bodies everyday.  How do animals do it, from mice to elephants?  In this talk, I will show how the shape of urinary and digestive organs enable them to function, regardless of the size of the animal.  Such ideas may teach us how to more efficiently transport materials.  I will show how the pee-pee pipe enables animals to urinate in constant time, how slippery mucus is critical for defecation, and how the motion of the gut is related to the density of its contents, and in turn to the gut’s natural frequency. 

More info is in the BBC news here: http://www.bbc.com/news/science-environment-34278595

We hope to bring together all Oxford researchers interested in Cryptography, in Quantum Computing and in the interactions between the two.

Please register at:  http://oxford-cryptography-day.eventbrite.co.uk

The accumulation of surface meltwater on ice shelves can lead to the formation of melt lakes. These structures have been implicated in crevasse propagation and ice-shelf collapse; the Larsen B ice shelf was observed to have a large amount of melt lakes present on its surface just before its collapse in 2002. Through modelling the transport of heat through the surface of the Larsen C ice shelf, where melt lakes have also been observed, this work aims to provide new insights into the ways in which melt lakes are forming and the effect that meltwater filling crevasses on the ice shelf will have. This will enable an assessment of the role of meltwater in triggering ice-shelf collapse. The Antarctic Peninsula, where Larsen C is situated, has warmed several times the global average over the last century and this ice shelf has been suggested as a candidate for becoming fully saturated with meltwater by the end of the current century. Here we present results of a 1-D mathematical model of heat transfer through an idealized ice shelf. When forced with automatic weather station data from Larsen C, surface melting and the subsequent meltwater accumulation, melt lake development and refreezing are demonstrated through the modelled results. Furthermore, the effect of lateral meltwater transport upon melt lakes and the effect of the lakes upon the surface energy balance are examined. Investigating the role of meltwater in ice-shelf stability is key as collapse can affect ocean circulation and temperature, and cause a loss of habitat. Additionally, it can cause a loss of the buttressing effect that ice shelves can have on their tributary glaciers, thus allowing the glaciers to accelerate, contributing to sea-level rise.

We give necessary and sufficient conditions for the variance of the partial sums of stationary processes to be regularly varying in terms of the spectral measure associated with the shift operator. In the case of reversible Markov chains, or with normal transition operator we also give necessary and sufficient conditions in terms of the spectral measure of the transition operator.  

The two spectral measures are then linked through the use of harmonic measure.



This is joint work with S. Utev(University of Leicester, UK) and M. Peligrad (University of Cincinnati, USA).

Fang, Lu and Yoshikawa conjectured a few years ago that a certain string-theoretic invariant (originally introduced by the physicists M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa) of Calabi-Yau threefolds is a birational invariant. This conjecture can be viewed as a "secondary" analog (in dimension three) of the birational invariance of Hodge numbers of Calabi-Yau varieties established by Batyrev and Kontsevich. Using the arithmetic Riemann-Roch theorem, we prove a weak form of this conjecture. 

In this talk I will give several perspectives on the role of
quasi-abelian categories in analytic geometry. In particular, I will 
explain why a certain completion of the category of Banach spaces is a
convenient setting for studying sheaves of topological vector spaces on
complex manifolds. Time permitting, I will also argue why this category
may be a good candidate for a functor of points approach to (derived)
analytic geometry.

We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.