Past

Ousman Kodio

Lubricated wrinkles: imposed constraints affect the dynamics of wrinkle coarsening

We investigate the problem of an elastic beam above a thin viscous layer. The beam is subjected to
a fixed end-to-end displacement, which will ultimately cause it to adopt the Euler-buckled
state. However, additional liquid must be drawn in to allow this buckling. In the interim, the beam
forms a wrinkled state with wrinkles coarsening over time. This problem has been studied
experimentally by Vandeparre \textit{et al.~Soft Matter} (2010), who provides a scaling argument
suggesting that the wavelength, $\lambda$, of the wrinkles grows according to $\lambda\sim t^{1/6}$.
However, a more detailed theoretical analysis shows that, in fact, $\lambda\sim(t/\log t)^{1/6}$.
We present numerical results to confirm this and show that this result provides a better account of
previous experiments.

Edward Rolls

Multiscale modelling of polymer dynamics: applications to DNA

We are interested in generalising existing polymer dynamics models which are applicable to DNA into multiscale models. We do this by simulating localized regions of a polymer chain with high spatial and temporal resolution, while using a coarser modelling approach to describe the rest of the polymer chain in order to increase computational speeds. The simulation maintains key macroscale properties for the entire polymer. We study the Rouse model, which describes a polymer chain of beads connected by springs by developing a numerical scheme which considers the a filament with varying spring constants as well as different timesteps to advance the positions of different beads, in order to extend the Rouse model to a multiscale model. This is applied directly to a binding model of a protein to a DNA filament. We will also discuss other polymer models and how it might be possible to introduce multiscale modelling to them.

The equidistribution theorem for rational points on expanding horospheres with fixed denominator in the space of d-dimensional Euclidean lattices has been derived in the work by M. Einsiedler, S. Mozes, N. Shah and U. Shapira. The proof of their theorem requires ergodic theoretic tools, including Ratner's measure classification theorem. In this talk I will present an alternative approach, based on harmonic analysis and Weil's bound for Kloosterman sums. In the case of d=3, unlike the ergodic-theoretic approach, this provides an explicit estimate on the rate of convergence. This is a joint work with Jens Marklof. 

Backward SDEs have proven to be a useful tool in mathematical finance. Their applications include the solution to various pricing and equilibrium problems in complete and incomplete markets, the estimation of value adjustments in the presence of funding costs, and the solution to many utility/risk optimisation type of problems.
In this work, we prove an explicit error expansion for the approximation of BSDEs. We focus our work on studying the cubature  method of solution. To profit fully from these expansions in this case, e.g. to design high order approximation methods, we need in addition to control the complexity growth of the base algorithm. In our work, this is achieved by using a sparse grid representation. We present several numerical results that confirm the efficiency of our new method. Based on joint work with J.F. Chassagneux.
 

The smooth representation theory of a p-adic reductive group G

with characteristic zero coefficients is very closely connected to the

module theory of its (pro-p) Iwahori-Hecke algebra H(G). In the modular

case, where the coefficients have characteristic p, this connection

breaks down to a large extent. I will first explain how this connection

can be reinstated by passing to a derived setting. It involves a certain

differential graded algebra whose zeroth cohomology is H(G). Then I will

report on a joint project with

R. Ollivier in which we analyze the higher cohomology groups of this dg

algebra for the group G = SL_2.

The smooth representation theory of a p-adic reductive group G with characteristic zero coefficients is very closely connected to the module theory of its (pro-p) Iwahori-Hecke algebra H(G). In the modular case, where the coefficients have characteristic p, this connection breaks down to a large extent. I will first explain how this connection can be reinstated by passing to a derived setting. It involves a certain differential graded algebra whose zeroth cohomology is H(G). Then I will report on a joint project with R. Ollivier in which we analyze the higher cohomology groups of this dg algebra for the group G = SL_2.

Tbd

Gauss was the first to give a formula for the number of monic irreducible polynomials of degree n over a finite field. A natural problem is to determine the number of such polynomials for which certain coefficients are prescribed. While some asymptotic and existence results have been obtained, very few exact results are known. In this talk I shall present an algorithm which for any finite field GF(q) of characteristic p expresses the number of monic irreducibles of degree n for which the first l < p coefficients are prescribed, for n >= l and coprime to p, in terms of the number of GF(q^n)-rational points of certain affine varieties defined over GF(q). 
The GF(2) base field case is related to the distribution of binary Kloosterman sums, which have numerous applications in coding theory and cryptography, for example via the construction of bent functions. Using a variant of the algorithm, we present varieties (which are all curves) for l <= 7 and compute explicit formulae for l <= 5; before this work such formulae were only known for l <= 3. While this connection motivates the problem, the talk shall focus mainly on computational algebraic geometry, with the algorithm, theoretical questions and computational challenges taking centre stage.

Two polyhedra are said to be scissors congruent if they can be subdivided into the same finite number of polyhedra such that each piece in the first polyhedron is congruent to one in the second. In 1900, Hilbert asked if there exist tetrahedra of the same volume which are not scissors congruent. I will give a history of this problem and its proofs, including an incorrect 'proof' by Bricard from 1896 which was only rectified in 2007.

Ever since moduli spaces of polarised K3 surfaces were constructed in the 1980's, people have wondered about the question of compactification: can one make the moduli space of K3 surfaces compact by adding in some boundary components in a "nice" way? Ideally, one hopes to find a compactification that is both explicit and geometric (in the sense that the boundary components provide moduli for degenerate K3's). I will present on joint work in progress with V. Alexeev, which aims to solve the compactification problem for the moduli space of K3 surfaces of degree 2.

The Turán number of an $r$-graph $G$, denoted by $ex(n,G)$, is the maximum number of edges in an $G$-free $r$-graph on $n$ vertices. The Turán density  of an $r$-graph $G$, denoted by $\pi(G)$, is the limit as $n$ tends to infinity of the maximum edge density of an $G$-free $r$-graph on $n$ vertices.

During this talk I will discuss a method, which we call  local stability method, that allows one to obtain exact Turán numbers from Turán density results. This method can be thought of as an extension of the classical stability method by  generically utilising the Lagrangian function. Using it, we obtained new hypergraph Turán numbers. In particular, we did so for a hypergraph called generalized triangle, for uniformities 5 and 6, which solved a conjecture of Frankl and Füredi from 1980's.

This is joint work with Sergey Norin.

Nested commutators of differential operators appear frequently in the numerical solution of equations of quantum mechanics. These are expensive to compute with and a significant effort is typically made to avoid such commutators. In the case of Magnus-Lanczos methods, which remain the standard approach for solving Schrödinger equations featuring time-varying potentials, however, it is not possible to avoid the nested commutators appearing in the Magnus expansion.

We show that, when working directly with the undiscretised differential operators, these commutators can be simplified and are fairly benign, cost-wise. The caveat is that this direct approach compromises structure -- we end up with differential operators that are no longer skew-Hermitian under discretisation. This leads to loss of unitarity as well as resulting in numerical instability when moderate to large time steps are involved. Instead, we resort to working with symmetrised differential operators whose discretisation naturally results in preservation of structure, conservation of unitarity and stability
 

What are the irreducible constituents of a smooth representation of a p-adic group that is constructed through parabolic induction? In the case of GL_n this is the study of the multiplicative behaviour of irreducible representations in the Bernstein-Zelevinski ring. Strikingly, the same decomposition problem can be reformulated through various Lie-theoretic settings of type A, such as canonical bases in quantum groups, representations of affine Hecke algebras, quantum affine Lie algebras, or more recently, KLR algebras. While partially touching on some of these phenomena, I will present new results on the problem using mostly classical tools. In particular, we will see how introducing a width invariant to an irreducible representation can circumvent the complexity involved in computations of Kazhdan-Lusztig polynomials.

In this talk I will present a class of super-Wilson loops in N=4 super Yang-Mills theory. The expectation value of these operators has been shown previously to be invariant under a Yangian symmetry. I will show how the kinematics of such super-Wilson loops can be described in a twistorial way and how this leads to compact, manifestly super-conformal invariant expressions for some two-point functions.
 

In this talk I will present a recent result on the characterisation of the equilibrium measure for a nonlocal and non-radial energy arising as the Gamma-limit of discrete interacting dislocations.

A cornerstone of the study of mapping class groups is the
Nielsen--Thurston classification theorem. I will outline a
polynomial-time algorithm that determines the Nielsen--Thurston type and
the canonical curve system of a mapping class. Time permitting, I shall
describe a polynomial-time algorithm to compute the quotient orbifold of
a periodic mapping class, and I shall discuss the conjugacy problem for
the mapping class group. This is joint work with Mark Bell.

A system of interacting particles described by stochastic differential equations is considered. As opposed to the usual model, where the noise perturbations acting on different particles are independent, here the particles are subject to the same space-dependent noise, similar to the (no interacting) particles of the theory of diffusion of passive scalars. We prove a result of propagation of chaos and show that the limit PDE is stochastic and of in viscid type, as opposed to the case when independent noises drive the different particles. Moreover, we use this result to derive a mean field approximation of the stochastic Euler equations for the vorticity of an incompressible fluid.

Ambitious mathematical models of highly complex natural phenomena are challenging to analyse, and more and more computationally expensive to evaluate. This is a particularly acute problem for many tasks of interest and numerical methods will tend to be slow, due to the complexity of the models, and potentially lead to sub-optimal solutions with high levels of uncertainty which needs to be accounted for and subsequently propagated in the statistical reasoning process. This talk will introduce our contributions to an emerging area of research defining a nexus of applied mathematics, statistical science and computer science, called "probabilistic numerics". The aim is to consider numerical problems from a statistical viewpoint, and as such provide numerical methods for which numerical error can be quantified and controlled in a probabilistic manner. This philosophy will be illustrated on problems ranging from predictive policing via crime modelling to computer vision, where probabilistic numerical methods provide a rich and essential quantification of the uncertainty associated with such models and their computation. 

We will discuss two recent short-time existence results for (1) mean curvature of surface clusters, where n-dimensional surfaces in R^{n+k}, are allowed to meet at equal angles along smooth edges, and (2) for planar networks, where curves are initially allowed to meet in multiple junctions that resolve immediately into triple junctions with equal angles. The first result, which is joint work with B. White, follows from an elliptic regularisation scheme, together with a local regularity result for flows with triple junctions, which are close to a static flow of the half-planes. The second result, which is joint work with T. Ilmanen and A.Neves, relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow. 
 

In this talk I will discuss some features of interacting conformal higher spin theories in 2+1 dimensions. This is done in the context of Chern-Simons theory (giving e.g. the complete spin 2 covariant  spin 3 sector) and a higher spin coupled unfolded equation for the scalar singleton. One motivation for studying these theories is that their non-linear properties are rather poorly understood contrary to the situation for the Vasiliev type theory in this dimension which is under much better control. Another reason for the interest in these theories comes from AdS4/CFT3 and the possibility that Neumann/mixed bc for bulk higher spin fields may lead to conformal higher spin fields governed by Chern-Simons terms on the boundary. These theories generalise the spin 2 gauged BLG-ABJ(M)  theories found a few years ago to higher spins than 2.

Following the spirit of Grothendieck’s Esquisse d’un Programme, the Ihara/Oda-Matsumoto conjecture predicted a combinatorial description of the absolute Galois group of Q based on its action on geometric fundamental groups of varieties. This conjecture was resolved in the 90’s by Pop using anabelian techniques. In this talk, I will discuss some satronger variants of this conjecture, focusing on the more recent solutions of its pro-ell and mod-ell two-step nilpotent variants.
 

What is the minimal size of a separating set? -- Emilie Dufresne

[[{"fid":"23334","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Emilie Dufresne","field_file_image_title_text[und][0][value]":"Emilie Dufresne"},"type":"media","attributes":{"alt":"Emilie Dufresne","title":"Emilie Dufresne","height":"258","width":"258","class":"media-element file-media-square"}}]]

Abstract: The problem of classifying objects up to certain allowed transformations figures prominently in almost all branches of Mathematics, and Invariants are used to decide if two objects are equivalent. A separating set is a set of invariants which achieve the desired classification. In this talk we take the point of view of Invariant Theory, where the objects correspond to points on an affine variety (often a vector space) and equivalence is given by the action of an algebraic group on this affine variety. We explain how the geometry and combinatorics of the group action govern the minimal size of separating sets.

Predator-Prey-Subsidy Dynamics and the Paradox of Enrichment on Networks -- Robert Van Gorder

[[{"fid":"23335","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Robert Van Gorder","field_file_image_title_text[und][0][value]":"Robert Van Gorder"},"type":"media","attributes":{"alt":"Robert Van Gorder","title":"Robert Van Gorder","height":"258","width":"258","class":"media-element file-media-square"}}]]

Abstract: The phrase "paradox of enrichment" was coined by Rosenzweig (1971) to describe the observation that increasing the food available to prey participating in predator-prey interactions can destabilize the predator's population. Subsequent work demonstrated that food-web connectance on networks can stabilize the predator-prey dynamics, thereby dampening the paradox of enrichment in networked domains (such as those used in stepping-stone models). However, when a resource subsidy is available to predators which migrate between nodes on such a network (as is actually observed in some real systems), we may show that predator-prey systems can exhibit a paradox of enrichment - induced by the motion of predators between nodes - provided that such networks are sufficiently densely connected. 

Strombolian volcanoes are thought to maintain their semi-permanent eruptive style by means of counter-current two-phase convective flow in the volcanic conduit leading from the magma chamber, driven by the buoyancy provided by exsolution of volatiles such as water vapour and carbon dioxide in the upwelling magma, due to pressure release. A model of bubbly two-phase flow is presented to describe this, but it is found that the solution breaks down before the vent at the surface is reached. We propose that the mathematical breakdown of the solution is associated with the physical breakdown of the two-phase flow regime from a bubbly flow to a churn-turbulent flow. We provide a second two-phase flow model to describe this regime, and we show that the solution can be realistically continued to the vent. The model is also in keeping with observations of eruptive style.

We will introduce variants of the optimal transport problem, namely martingale optimal transport problem and subharmonic martingale transport problem. Their motivation is partly from mathematical finance. We will see that in dimension greater than one, the additional constraints imply interesting and deep mathematical subtlety on the attainment of dual problem, and it also affects heavily on the geometry of optimal solutions. If time permits, we will introduce still another variant of the martingale transport problem, called the multi-martingale optimal transport problem.

A Fueter map between two hyperKaehler manifolds is a solution of a Cauchy-Riemann-type equation in the quaternionic context. In this talk I will describe relations between Fueter maps, generalized Seiberg-Witten equations, and Yang-Mills instantons on G2-manifolds (so called G2-instantons).

Currently all medical x-ray imaging is performed using point-like sources which produce cone or fan beams. In planar radiology the source is fixed relative to the patient and detector array and therefore only 2D images can be produced. In CT imaging, the source and detector are rotated about the patient and through reconstruction (such as Radon methods), a 3D image can be formed. In Tomosynthesis, a limited range of angles are captured which greatly reduces the complexity and cost of the device and the dose exposure to the patient while largely preserving the clinical utility of the 3D images. Conventional tomosynthesis relies on mechanically moving a source about a fixed trajectory (e.g. an arc) and capturing multiple images along that path. Adaptix is developing a fixed source with an electronically addressable array that allows for a motion-free tomosynthesis system. The Adaptix approach has many advantages including reduced cost, portability, angular information acquired in 2D, and the ability to shape the radiation field (by selectively activating only certain emitters).

The proposed work would examine the effects of patient motion and apply suitable corrections to the image reconstruction (or raw data). Many approaches have been considered in the literature for motion correction, and only some of these may be of use in tomosynthesis. The study will consider which approaches are optimal, and apply them to the present geometry.

A related but perhaps distinct area of investigation is the use of “structured light” techniques to encode the x-rays and extract additional information from the imaging. Most conventional structured light approaches are not suitable for transmissive operation nor for the limited control available in x-rays. Selection of appropriate techniques and algorithms, however, could prove very powerful and yield new ways of performing medical imaging.

Adaptix is a start-up based at the Begbroke Centre for Innovation and Enterprise. Adaptix is transforming planar X-ray – the diagnostic imaging modality most widely used in healthcare worldwide. We are adding low-dose 3D capability – digital tomosynthesis - to planar X-ray while making it more affordable and truly portable so radiology can more easily travel to the patient. This transformation will enhance patient’s access to the world’s most important imaging technologies and likely increases the diagnostic accuracy for many high incidence conditions such as cardiovascular and pulmonary diseases, lung cancer and osteoporosis. 
 

We are increasingly better at predicting things about our environment. Modern weather forecasts are a lot better than they used to be, and our ability to predict climate change illustrates our better understanding of our effect on our environment. But what about predicting our collective effect on ourselves?  We now use tools like Google maps to predict how long it will take us to drive to work, and other small things, but we fail miserably when it comes to many of the big things. For example, the recent financial crisis cost the world tens of trillions of pounds, yet our ability to forecast, understand and mitigate the next economic crisis is very low. Is this inherently impossible? Or perhaps we are just not going about it the right way? The complex systems approach to economics, which brings in insights from the physical and natural sciences, presents an alternative to standard methods. Doyne will explain what this new approach is and give a few examples of its successes so far. He will then present a vision of the economics of the future which will need to confront the serious problems that the world will soon face.
 

Please email @email to register

In 2000, Vojta solved the n-squares problem under the Bombieri-Lang conjecture, by explicitly finding all the curves of genus 0 or 1 on the surfaces related to this problem. The fundamental notion used by him is $\omega$-integrality of curves. 
In this talk, I will show a generalization of Vojta's method to find all curves of low genus in some surfaces, with arithmetic applications.
I will also explain how to use $\omega$-integrality to obtain a bound of the height of a non-constant morphism from a curve to $\mathbb{P}^2$ in terms of the number of intersections (without multiplicities) of its image with a divisor of a particular kind. This proves some new special cases of Vojta's conjecture for function fields.
 

One of the fundamental tools in numerical analysis and PDE
is the finite element method (FEM). A main ingredient in
FEM are splines: piecewise polynomial functions on a
mesh. Even for a fixed mesh in the plane, there are many open
questions about splines: for a triangular mesh T and
smoothness order one, the dimension of the vector space
  C^1_3(T) of splines of polynomial degree at most three
is unknown. In 1973, Gil Strang conjectured a formula
for the dimension of the space C^1_2(T) in terms of the
combinatorics and geometry of the mesh T, and in 1987 Lou
Billera used algebraic topology to prove the conjecture
(and win the Fulkerson prize). I'll describe recent progress
on the study of spline spaces, including a quick and self
contained introduction to some basic but quite useful tools
from topology.

In 2000, Vojta solved the n-squares problem under the Bombieri-Lang conjecture, by explicitly finding all the curves of genus 0 or 1 on the surfaces related to this problem. The fundamental notion used by him is $\omega$-integrality of curves. 

In this talk, I will show a generalization of Vojta's method to find all curves of low genus in some surfaces, with arithmetic applications.

I will also explain how to use $\omega$-integrality to obtain a bound of the height of a non-constant morphism from a curve to $\mathbb{P}^2$ in terms of the number of intersections (without multiplicities) of its image with a divisor of a particular kind.
This proves some new special cases of Vojta's conjecture for function fields.
 

We consider the problem of computing a nonnegative low rank factorization to a given nonnegative input matrix under the so-called "separabilty condition".  This assumption makes this otherwise NP hard problem polynomial time solvable, and we will use first order optimization techniques to compute such a factorization. The optimization model use is based on sparse regression with a self-dictionary, in which the low rank constraint is relaxed to the minimization of an l1-norm objective function.  We apply these techniques to endmember detection and classification in hyperspecral imaging data.

Stallings' theorem states that a finitely generated group splits over a finite subgroup if and only if it has more than one end. As a consequence of this, group splittings over finite subgroups are invariant under quasi-isometry. I will discuss a generalisation of Stallings' theorem which shows that under suitable hypotheses, group splittings over classes of infinite groups, namely coarse $PD_n$ groups, are also invariant under quasi-isometry.

Not considering classified work, the first person to have asked and solved the problem of secure communication over insecure communication channels was Ralph Merkle, in a project for a Computer securitjohn y course at UC Berkeley in 1974. In this work, he gave a protocol that allow two legitimate parties to establish a secret key with an effort of the order of N, but such that an eavesdropper can not discover the secret key with non-vanishing probability if he is not willing to spend an effort of at least the order of N^2.
In this talk, we will consider key exchange protocols in the presence of a quantum eavesdropper. Unfortunately, it is easy to see that in this case, breaking Merkle’s original protocol only requires an effort of the order of N, similar to the one of the legitimate parties. We will show how to restore the security by presenting two sequences of protocols with the following properties:
- In the first sequence, the legitimate parties have access to a quantum computer, and the eavesdropper's effort is arbitrarily close to N^2.
- In the second sequence, the protocols are classical, but the eavesdropper’s effort is arbitrarily close to N^{3/2}.
We will show the key exchange protocols, the quantum attacks with the proof of their optimality. We will focus mostly on the techniques from quantum algorithms and complexity theory used to devise quantum algorithms and to prove lower bounds. The underlying tools are the quantum walk formalism, and the quantum adversary lower bound method, respectively. Finally, we will introduce a new method to prove average-case quantum query complexity lower bounds.

The problem of computing the Galois group of an irreducible, rational polynomial has been studied for many years. I will discuss the methods developed over the years to approach this problem, and give some examples of them in practice. These methods mainly involve constructing and factorising resolvent polynomials, and thereby determining better upper bounds for the conjugacy class of the Galois group within the symmetric group, i.e. describe its action on the roots of the polynomial explicitly. I will describe how using approximations to the zeros of the polynomial allows us to construct resolvents, and in particular, how using p-adic approximations can be advantageous over numerical approximations, and how this can yield a direct and systematic method of determining the Galois group.

The Hall algebra can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how it extends naturally to give a bi-algebraic structure. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.

For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show that for fixed $k\geq2$ and $n$ odd and sufficiently large,
$$
R_k(C_n)=2^{k-1}(n-1)+1.
$$
This resolves a conjecture of Bondy and Erdős for large $n$. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation.  This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal $k$-colourings for this problem and perfect matchings in the $k$-dimensional hypercube $Q_k$.

It has long been expected, and is now proved in many important cases, 
that quantum algebras are more rigid than their classical limits. That is, they 
have much smaller automorphism groups. This begs the question of whether this 
broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative 
projective geometry, from which we see that the correct object to study is a 
groupoid, rather than a group, and maps in this groupoid are the replacement 
for automorphisms. I will illustrate this with the example of quantum 
projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

Thin glass sheets are used in smartphone, battery and semiconductor technology, and may be manufactured by first producing a relatively thick glass slab (known as a preform) and subsequently redrawing it to a required thickness. Theoretically, if the sheet is redrawn through an infinitely long heater zone, a product with the same aspect ratio as the preform may be manufactured. However, in reality the effect of surface tension and the restriction to factories of finite size prevent this. In this talk I will present a mathematical model for a viscous sheet undergoing redraw, and use asymptotic analysis in the thin-sheet, low-Reynolds-number limit to investigate how the product shape is affected by process parameters. 

I will survey of some of the many significant connections between integrable many-body physics and mathematics. I exploit an algebraic structure called a fusion category, familiar from the study of conformal field theory, topological quantum field theory and knot invariants. Rewriting statistical-mechanical models in terms of a fusion category allows the derivation of combinatorial identities for the Tutte polynomial, the analysis of discrete ``holomorphic'' observables in probability, and to defining topological defects in lattice models. I will give a little more detail on topological defects, explaining how they allows exact computations of conformal-field-theory quantities directly on the lattice, as well as a greatly generalised set of duality transformations.