Past

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.

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 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.

I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a
free group $F$ is exactly $Aut(F)$, provided that $F$ has rank at least $3$. I shall begin by sketching the fruitful analogy between automorphism groups of free groups, mapping class groups, and arithmetic lattices, particularly $SL_n({\mathbb{Z}})$. In this analogy, the free factor complex, introduced by Hatcher and Vogtmann, appears as the natural analogue in the $Aut(F)$ setting of the spherical Tits building associated to $SL_n $ and of the curve complex associated to a mapping class group. If $n>2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of the building is $PGL_n({\mathbb{Q}})$. Ivanov proved an analogous theorem for the curve complex, and our theorem complements this. These theorems allow one to identify the abstract commensurators of $GL_n({\mathbb{Z}})$, mapping class groups, and $Out(F)$, as I shall explain.

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The Renormalization Group (RG) was pioneered by the physicist Kenneth Wilson in the early 70's and since then it has become a fundamental tool in physics. RG remains the most general philosophy for understanding how many models in statistical mechanics behave near their critical point but implementing RG analysis in a mathematically rigorous way remains quite challenging.

I will describe how analysis of RG flows translate into statements about continuum limits, universality, and cross-over phenomena - as a concrete example I will speak about some joint work with Abdelmalek Abdesselam and Gianluca Guadagni.

The moduli space of Higgs bundles on a hyperbolic Riemann surface is a complex analytic variety which has a hyperkahler metric on its smooth locus. As such it has several associated symmetry groups including the group of complex analytic automorphisms and the group of isometries. I will discuss the classification of these and some other related groups.

Motivated by the study of supersymmetric backgrounds with non-trivial fluxes, we provide a formulation of supergravity in the language of generalised geometry, as first introduced by Hitchin, and its extensions. This description both dramatically simplifies the equations of the theory by making the hidden symmetries manifest, and writes the bosonic sector geometrically as a direct analogue of Einstein gravity. Further, a natural analogue of special holonomy manifolds emerges and coincides with the conditions for supersymmetric backgrounds with flux, thus formulating these systems as integrable geometric structures.
 

In this talk we present a new type of Soblev norm defined in the space of functions of continuous paths. Under the Wiener probability measure the corresponding norm is suitable to prove the existence and uniqueness for a large type of system of path dependent quasi-linear parabolic partial differential equations (PPDE). We have establish 1-1 correspondence between this new type of PPDE and the classical backward SDE (BSDE). For fully nonlinear PPDEs, the corresponding Sobolev norm is under a sublinear expectation called G-expectation, in the place of Wiener expectation. The canonical process becomes a new type of nonlinear Brownian motion called G-Brownian motion. A similar 1-1 correspondence has been established. We can then apply the recent results of existence, uniqueness and principle of comparison for BSDE driven by G-Brownian motion to obtain the same result for the PPDE.

Let $K\subset {\mathbb R}$ be a compact definable set in an o-minimal structure over $\mathbb R$, e.g. a semi-algebraic or a real analytic set. A definable family $\{S_\delta\ |  0<\delta\in{\mathbb R}\}$ of compact subsets of $K$, is called a monotone family if $S_\delta\subset S_\eta$ for all sufficiently small $\delta>\eta>0$. The main result in the talk is that when $\dim K=2$ or $\dim K=n=3$ there exists a definable triangulation of $K$ such that for each (open) simplex $\Lambda$ of the triangulation and each small enough $\delta>0$, the intersections $S_\delta\cap\Lambda$ is equivalent to one of five (respectively, nine) standard families in the standard simplex (the equivalence relation and a standard family will be formally defined). As a consequence, we prove the two-dimensional case of the topological conjecture on approximation of definable sets by compact families.

This is joint work with Andrei Gabrielov (Purdue).

The moduli space of G-Higgs bundles carries a natural Hyperkahler structure, through which we can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes) with respect to each structure. Notably, these A and B-branes have gained significant attention in string theory.

We shall begin the talk by first introducing G-Higgs bundles for reductive Lie groups and the associated Hitchin fibration, and sketching how to realize Langlands duality through spectral data. We shall then look at particular types of branes (BAA-branes) which correspond to very interesting geometric objects, hyperholomorphic bundles (BBB-branes). 

The presentation will be introductory and my goal is simply to sketch some of the ideas relating these very interesting areas.