Past

The field of transseries was introduced by Ecalle to give a solution to Dulac's problem, a weakening of Hilbert's 16th problem. They form an elementary extension of the real exponential field and have received the attention of model theorists. Another such elementary extension is given by Conway's surreal numbers, and various connections with the transseries have been conjectured, among which the possibility of introducing a Hardy type derivation on the surreal numbers. I will present a complete solution to these conjectures obtained in collaboration with Vincenzo Mantova.
 

In sieve theory, one is concerned with estimating the size of a sifted set, which avoids certain residue classes modulo many primes. For example, the problem of counting primes corresponds to the situation when the residue class 0 is removed for each prime in a suitable range. This talk will be concerned about what happens when a positive proportion of residue classes is removed for each prime, and especially when this proporition is more than a half. In doing so we will come across an algebraic question: given a polynomial f(x) in Z[x], what is the average size of the value set of f reduced modulo primes?

We consider the dynamic casino gambling model initially proposed by Barberis (2012) and study the optimal stopping strategy of a pre-committing gambler with cumulative prospect theory (CPT) preferences. We illustrate how the strategies computed in Barberis (2012) can be strictly improved by reviewing the entire betting history or by tossing random coins, and explain that such improvement is possible because CPT preferences are not quasi-convex. Finally, we develop a systematic and analytical approach to finding the optimal strategy of the gambler. This is a joint work with Prof. Xue Dong He (Columbia University), Prof. Jan Obloj, and Prof. Xun Yu Zhou.

Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this talk, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton--Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations is observed not to grow as solutions are deflated. The technique is then applied to computing distinct solutions of nonlinear PDEs, tracing bifurcation diagrams, and to computing multiple local minima of PDE-constrained optimisation problems.

Motivated by an open conjecture in anabelian geometry, we investigate which arithmetic properties of the rationals are encoded in the absolute Galois group G_Q. We give a model-theoretic framework for studying absolute Galois groups and discuss in what respect orderings and valuations of the field are known to their first-order theory. Some questions regarding local-global principles and the transfer to elementary extensions of Q are raised.

Compact F-spaces play an important role in the area of compactification theory, the prototype being w*, the Stone-Cech remainder of the integers. Two curious topological characteristics of compact F-spaces are that they don’t contain convergent sequences (apart from the constant ones), and moreover, that they often contain points that don’t lie in the boundary of any countable subset (so-called weak P-points). In this talk we investigate the space of self-maps S(X) on compact zero-dimensional F-spaces X, endowed with the compact-open topology. A natural question is whether S(X) reflects properties of the ground space X. Our main result is that for zero-dimensional compact F-spaces X, also S(X) doesn’t contain convergent sequences. If time permits, I will also comment on the existence of weak P-points in S(X). This is joint work with Richard Lupton.

In Bass-Serre theory, one derives structural properties of groups from their actions on simplicial trees. In this talk, we introduce the theory of groups acting on $\mathbb{R}$-trees. In particular, we explain how the Rips machine is used to classify finitely generated groups which act freely on $\mathbb{R}$-trees.

In particular, some nice things about branch groups, whose subgroup structure  "sees" all actions on rooted trees.

All-at-once schemes aim to solve all time-steps of parabolic PDE-constrained optimization problems in one coupled computation, leading to exceedingly large linear systems requiring efficient iterative methods. We present a new block diagonal preconditioner which is both optimal with respect to the mesh parameter and parallelizable over time, thus can provide significant speed- up. We will present numerical results to demonstrate the effectiveness of this preconditioner.

There is a beautiful problem resulting from arithmetic number theory where a continuous and compactly supported function's 3-fold autoconvolution is constant. In this talk, we optimize the coefficients of a Chebyshev series multiplied by an endpoint singularity to obtain a highly accurate approximation to this constant. Convolving functions with endpoint singularities turns out to be a challenge for standard quadrature routines. However, variable transformations inducing double exponential endpoint decay are used to effectively annihilate the singularities in a way that keeps accuracy high and complexity low.

In a quantum quench, a system is prepared in some state
$|\psi_0\rangle$, usually the ground state of a hamiltonian $H_0$, and then
evolved unitarily with a different hamiltonian $H$. I study this problem
when $H$ is a 1+1-dimensional conformal field theory on a large circle of
length $L$, and the initial state has short-range correlations and
entanglement. I argue that (a) for times $\ell/2<t<(L-\ell)/2$  the
reduced density matrix of a subinterval of length $\ell$ is exponentially
close to that of a thermal ensemble; (b) despite this, for a rational CFT
the return amplitude $\langle\psi_0|e^{-iHt}|\psi_0\rangle$ is $O(1)$ at
integer multiples of $2t/\ell$ and has interesting structure at all rational
values of this ratio. This last result is related to the modular properties
of Virasoro characters.

In particular, some nice things about branch groups, whose subgroup structure "sees" all actions on rooted trees.

We analyze wave propagation in random media in the so-called paraxial regime, which is a special high-frequency regime in which the wave propagates along a privileged axis. We show by multiscale analysis how to reduce the problem to the Ito-Schrodinger stochastic partial differential equation. We also show how to close and solve the moment equations for the random wave field. Based on these results we propose to use correlation-based methods for imaging in complex media and consider two examples: virtual source imaging in seismology and ghost imaging in optics.

In this talk I will discuss the notion of o-minimality, which can be approached from either a model-theoretic standpoint, or an algebraic one.  I will exhibit some o-minimal structures, focussing on those most relevant to number theorists, and attempt to explain how o-minimality can be used to attain an assortment of results.

Building a suitable family of walls in the Cayley complex of a finitely
presented group G leads to a nontrivial action of G on a CAT(0) cube
complex, which shows that G does not have Kazhdan's property (T).  I
will discuss how this can be done for certain random groups in Gromov's
density model.  Ollivier and Wise (building on earlier work of Wise on
small-cancellation groups) have built suitable walls at densities <1/5,
but their method fails at higher densities.  In recent joint work with
Piotr Przytycki we give a new construction which finds walls at densites
<5/24.

tba

 We describe how cross-kernel matrices, that is, kernel matrices between the data and a custom chosen set of `feature spanning points' can be used for learning. The main potential of cross-kernel matrices is that (a) they provide Nyström-type speed-ups for kernel learning without relying on subsampling, thus avoiding potential problems with sampling degeneracy, while preserving the usual approximation guarantees and the attractive linear scaling of standard Nyström methods and (b) the use of non-square matrices for kernel learning provides a non-linear generalization of the singular value decomposition and singular features. We present a novel algorithm, Ideal PCA (IPCA), which is a cross-kernel matrix variant of PCA, showcasing both advantages: we demonstrate on real and synthetic data that IPCA allows to (a) obtain kernel PCA-like features faster and (b) to extract novel features of empirical advantage in non-linear manifold learning and classification.

I will discuss how singular fibers in higher codimension in elliptically fibered Calabi-Yau fourfolds can be studied using Coulomb branch phases for d=3 supersymmetric gauge theories. This approach gives an elegent description of the generalized Kodaira fibers in terms of combinatorial/representation-theoretic objects called "box graphs", including the network of flops connecting distinct small resolutions. For physics applications, this approach can be used to constrain the possible matter spectra and possible U(1) charges (models with higher rank Mordell Weil group) for F-theory GUTs.

A form of PDE-constrained inversion is today used as an engineering tool for seismic imaging. Today there are some successful studies and good workflows are available. However, mathematicians will find some important unanswered questions: (1) robustness of inversion with highly nonconvex objective functions; (2) scalable solution highly oscillatory problem; and (3) handling of uncertainties. We shall briefly illustrate these challenges, and mention some possible solutions.

Genomic sequences contain rich evolutionary information about functional constraints on macromolecules such as proteins. This information can be efficiently mined to detect evolutionary couplings between residues in proteins and address the long-standing challenge to compute protein three-dimensional structures from amino acid sequences. Substantial progress on this problem has become possible because of the explosive growth in available sequences and the application of global statistical methods. In addition to three-dimensional structure, the improved analysis of covariation helps identify functional residues involved in ligand binding, protein-complex formation and conformational changes. We expect computation of covariation patterns to complement experimental structural biology in elucidating the full spectrum of protein structures, their functional interactions and evolutionary dynamics. Use the http://evfold.org  server to compute EVcouplings and to predict 3D structure for large sequence families. References:  http://bit.ly/tob48p - Protein 3D Structure from high-throughput sequencing;  http://bit.ly/1DSqANO - 3D structure of transmembrane proteins from evolutionary constraints; http://bit.ly/1zyYpE7 - Sequence co-evolution gives 3D contacts and structures of protein complexes.

We consider a controlled stochastic system in presence of state-constraints. Under the assumption of exponential stabilizability of the system near a target set, we aim to characterize the set of points which can be asymptotically driven by an admissible control to the target with positive probability. We show that this set can be characterized as a level set of the optimal value function of a suitable unconstrained optimal control problem which in turn is the unique viscosity solution of a second order PDE which can thus be interpreted as a generalized Zubov equation.

VerdErg Renewable Energy Ltd is developing a new hydropower unit for cost-effective energy generation at very low heads of pressure. The device is called the VETT after the underlying technology – Venturi Enhanced Turbine Technology. Flow into the VETT is split into two. The larger flow at low head transfers its energy to the smaller flow at a greater head. The smaller flow powers a conventional turbo-generator which can be a smaller, faster unit at an order of magnitude lower cost. Further, there are significant environmental benefits to fish and birds compared to the conventional hydropower solution. After several physical model test programmes* in the UK, France and The Netherlands along with CFD studies the efficiency now stands at 50%. We wish to increase that by understanding the major loss mechanisms and how they might be avoided or minimised.

The presentation will explain the VETT’s working principles and key relationships, together with some possible ideas for improvement. The comments of attendees on problem areas, potential solutions and how an enhanced understanding of key phenomena may be applied will be most welcome.

*(One was observed by Prof John Ockendon who identified a fairly extreme flow condition in a region previously thought to be benign.)

 In this talk I will present a basic introduction to conformal symmetry from a physicist perspective. I will talk about infinitesimal and finite conformal transformations and the conformal group in diverse dimensions. 

Given an abelian variety A over a number field k, its Kummer variety X is the quotient of A by the automorphism that sends each point P to -P. We study p-adic density and weak approximation on X by relating its rational points to rational points of quadratic twists of A. This leads to many examples of K3 surfaces over Q whose rational points lie dense in the p-adic topology, or in product topologies arising from p-adic topologies. Finally, the same method is used to prove that if the Brauer--Manin obstruction controls the failure of weak approximation on all Kummer varieties, then ranks of quadratic twists of (non-trivial) abelian varieties are unbounded. This last fact arises from joint work with David Holmes.

Abstract: Motivated loosely by the problem of carbon sequestration in underground aquifers, I will describe computations and analysis of one-sided two-dimensional convection of a solute in a fluid-saturated porous medium, focusing on the case in which the solute decays via a chemical reaction.   Scaling properties of the flow at high Rayleigh number are established and rationalized through an asymptotic model, that addresses the transient stability of a near-surface boundary layer and the structure of slender plumes that form beneath.  The boundary layer is shown to restrict the rate of solute transport to deep domains.  Knowledge of the plume structure enables slow erosion of the substrate of the reaction to be described in terms of a simplified free boundary problem.

Co-authors: KA Cliffe, H Power, DS Riley, TJ Ward

In this talk we will describe the steps towards self-consistent computer simulations of the evolution of the universe beginning soon after the Big Bang and ending with the formation of realistic stellar systems like the Milky Way. This is a multi-scale problem of vast proportions. The first step has been the Millennium Simulation, one of the largest and most successful numerical simulations of the Universe ever carried out. Now we are in the midst of the next step, where this is carried to a much higher level of physical fidelity on the latest supercomputing platforms. This talk will be illustrate how the role of mathematics is essential in this endeavor. Also computer simulations will be shown. This is joint work among others with Volker Springel.

The generation of functional interfaces such as superconducting and ferroelectric twin boundaries requires new ways to nucleate as many interfaces as possible in bulk materials and thin films. Materials with high densities of twin boundaries are often ferroelastics and martensites. Here we show that the nucleation and propagation of twin boundaries depend sensitively on temperature and system size. The geometrical mechanisms for the evolution of the ferroelastic microstructure under strain deformation remain similar in all thermal regimes, whereas their thermodynamic behavior differs dramatically: on heating, from power-law statistics via the Kohlrausch law to a Vogel-Fulcher law.We find that the complexity of the pattern can be well characterized by the number of junctions between twin boundaries. Materials with soft bulk moduli have much higher junction densities than those with hard bulk moduli. Soft materials also show an increase in the junction density with diminishing sample size. The change of the complexity and the number density of twin boundaries represents an important step forward in the development of ‘domain boundary engineering’, where the functionality of the materials is directly linked to the domain pattern.

Everything measurable about a quantum system, as modelled by a noncommutative operator algebra, is captured by its commutative subalgebras. We briefly survey this programme, and zoom in one specific incarnation: any bilinear associative function on the set of n-by-n matrices over a field of characteristic not two, that makes the same vectors orthogonal as ordinary matrix multiplication and gives the same trace as ordinary matrix multiplication, must in fact be ordinary matrix multiplication (or its opposite). Model-theoretic questions about the hypotheses and scope of this theorem are raised.

I will look at some decidability questions for subgroups of Aut($F_n$) for general $n$. I will then discuss semisimple actions of Aut($F_n$) on complete CAT(0) spaces proving that the Nielsen moves will act elliptically. I will also look at proving Aut($F_3$) is large and if time permits discuss the fact that Aut($F_n$) is not Kähler

In this talk we will review some of the main ideas around Hamilton's program for the Ricci flow and see how they fit together to provide a proof of the Poincaré conjecture in dimension 3. We will then analyse this tools in the context of 4-manifolds.

The classification of irreducible representations of pin double covers of Weyl groups was initiated by Schur (1911) for the symmetric group and was completed for the other groups by A. Morris, Read and others about 40 years ago. Recently, a new relation between these projective representations, graded Springer representations, and the geometry of the nilpotent cone has emerged. I will explain these connections and the relation with a Dirac operator for (extended) graded affine Hecke algebras.  The talk is partly based on joint work with Xuhua He.

I will speak about weak Fano 3-folds, K3 surfaces and their Picard lattices, and explain how to solve the matching problem in various situations

I will speak about weak Fano 3-folds, K3 surfaces and their Picard lattices, and explain how to solve the matching problem in various situations.

Erdős asked the following question: given a positive integer $n$, what is the largest integer $k$ such that any set of $n$ points in a plane, with no $4$ on a line, contains $k$ points no $3$ of which are collinear? Füredi proved that $k = o(n)$. Cardinal, Toth and Wood extended this result to $\mathbb{R}^3$, finding sets of $n$ points with no $5$ on a plane whose subsets with no $4$ points on a plane have size $o(n)$, and asked the question for the higher dimensions. For given $n$, let $k$ be largest integer such that any set of $n$ points in $\mathbb{R}^d$ with no more than $d + 1$ cohyperplanar points, has $k$ points with no $d + 1$ on a hyperplane. Is $k = o(n)$? We prove that $k = o(n)$ for any fixed $d \geq 3$.

We present an algorithm, Parallel-$\ell_0$, for {\em combinatorial compressed sensing} (CCS), where the sensing matrix $A \in \mathbb{R}^{m\times n}$ is the adjacency matrix of an expander graph. The information preserving nature of expander graphs allow the proposed algorithm to provably recover a $k$-sparse vector $x\in\mathbb{R}^n$ from $m = \mathcal{O}(k \log (n/k))$ measurements $y = Ax$ via $\mathcal{O}(\log k)$ simple and parallelizable iterations when the non-zeros in the support of the signal form a dissociated set, meaning that all of the $2^k$ subset sums of the support of $x$ are pairwise different. In addition to the low computational cost, Parallel-$\ell_0$ is observed to be able to recover vectors with $k$ substantially larger than previous CCS algorithms, and even higher than $\ell_1$-regularization when the number of measurements is significantly smaller than the vector length.

Flow thought a porous media is usually described by assuming the superficial velocity can be expressed in terms of a constant permeability and a pressure gradient. In poroelastic flows the underlying elastic matrix responds to changes in the fluid pressure. When the elastic deformation is allowed to influence the permeability through the elastic strain, it becomes possible for increased fluid pressure gradient not to result in increased flow, but to decrease the permeability and potentially this may close off or choke the flow. I will talk about a simple model problem for a number of different elastic constitutive models and a number of different permeability-strain models and examine whether there is a general criterion that can be derived to show when, or indeed if, choking can occur for different elasticity-permeability combinations.

In this talk, I develop the Hopf algebra of renormalization, as established by Connes and Kreimer. I then use the correspondence between commutative Hopf algebras and affine groups to show that the energy scale dependence of the renormalized theory can be expressed as a Maurer Cartan connection on the renormalization group.

1)      The Hardt-Lin's problem and a new approximation of a relaxed energy for harmonic maps.

We introduce a new approximation for  the relaxed energy $F$ of the Dirichlet energy and prove that the minimizers of the approximating functional converge to a minimizer $u$ of the relaxed energy for harmonic maps, and that $u$ is  partially regular without using the concept of Cartesian currents.

2)  Partial regularity in liquid crystals  for  the Oseen-Frank model:  a new proof of the result of Hardt, Kinderlehrer and Lin.

Hardt, Kinderlehrer and Lin (\cite {HKL1}, \cite {HKL2}) proved that a minimizer $u$ is smooth on some open subset
$\Omega_0\subset\Omega$ and moreover $\mathcal H^{\b} (\Omega\backslash \Omega_0)=0$ for some positive $\b <1$, where
$\mathcal H^{\b}$ is the Hausdorff measure.   We will present a new proof of Hardt, Kinderlehrer and Lin.

 3)      Global existence of solutions of the Ericksen-Leslie system for  the Oseen-Frank model.

The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of  the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model,   which generalizes the heat flow for harmonic maps  into the $2$-sphere.   In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank  model in 2D.

The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of  the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model,   which generalizes the heat flow for harmonic maps  into the $2$-sphere.   In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank  model in 2D.