Past

Magnus expansion based methods are an efficient class of integrators for solving Schrödinger equations that feature time dependent potentials such as lasers. These methods have been found to be highly effective in computational quantum chemistry since the pioneering work of Tal Ezer and Kosloff in the early 90s. The convergence of the Magnus expansion, however, is usually understood only for ODEs and traditional analysis suggests a much poorer performance of these methods than observed experimentally. It was not till the work of Hochbruck and Lubich in 2003 that a rigorous analysis justifying the application to PDEs with unbounded operators, such as the Schrödinger equation, was presented. In this talk we will extend this analysis to the semiclassical regime, where the highly oscillatory solution conventionally suggests large errors and a requirement for very small time steps.
 

In everyday language, this talk studies the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and for short times we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the well-posedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using a new decomposition of $L^2(\Rd)$ into heat packets from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated numerically by solving a sequence of finite-dimensional optimization problems. (joint with Alden Waters)
 

The (kinetic) Langevin equation is an SDE with degenerate noise that describes the motion of a particle in a force field subject to damping and random collisions. It is also closely related to Hamiltonian Monte Carlo methods. An important open question is, why in certain cases kinetic Langevin diffusions seem to approach equilibrium faster than overdamped Langevin diffusions. So far, convergence to equilibrium for kinetic Langevin diffusions has almost exclusively been studied by analytic techniques. In this talk, I present a new probabilistic approach that is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. The approach yields rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime, and it may help to shed some light on the open question mentioned above.

 There is a routine for obtaining formulae for derivatives of smooth heat semigroups,and for certain heat semigroups acting on differential forms etc, established some time ago by myself, LeJan, & XueMei Li.  Following a description of this in its general form, I will discuss its applicability in some sub-Riemannian situations and to higher order derivatives.

Transport properties of liquids and gases in the regime of weak coupling (or effective weak coupling) are determined by the solutions of relevant kinetic equations for particles or quasiparticles, with transport coefficients being proportional to the minimal eigenvalue of the linearized kinetic operator. At strong coupling, the same physical quantities can sometimes be determined from dual gravity, where quasinormal spectra enter as the eigenvalues of the linearized Einstein's equations. We discuss the problem of interpolating between the two regimes using results from higher derivative gravity.

In this session we will refresh our understanding of the purpose of an interview, review some top tips, and practise answering some typical interview questions. Rachel will also signpost further resources on interview preparation available at the Careers Service.

Over the last few decades, a number of authors have discussed the analogy between linking numbers in three manifold topology and symbols in arithmetic. This talk will outline some results that make this precise in terms of natural complexes associated to arithmetic duality theorems. In particular, we will describe a ‘finite path integral’ formula for power residue symbols.

Dedalus is a new open-source framework for solving general partial differential equations using spectral methods.  It is designed for maximum extensibility and incorporates features such as symbolic equation entry, custom domain construction, and automatic MPI parallelization.  I will briefly describe key algorithmic features of the code, including our sparse formulation and support for general tensor calculus in curvilinear domains.  I will then show examples of the code’s capabilities with various applications to astrophysical and geophysical fluid dynamics, including a compressible flow benchmark against a finite volume code, and direct numerical simulations of turbulent glacial melting

In the first part of my presentation, I will briefly summarize a dynamic view of the cell cycle created in collaboration with Prof John Tyson over the past 25 years. 
In our view, the decisions a cell must make during DNA synthesis and mitosis are controlled by bistable switches, which provide abrupt and irreversible transition 
between successive cell cycle phases. In addition, bistability provides the foundation for 'checkpoints' that can stop cell proliferation if problems arise 
(e.g., DNA damage by UV irradiation). In the second part of my talk, I will highlight a few representative examples from our ongoing BBSRC Strategic LoLa grant 
(http://cellcycle.org.uk/) in which we are testing the predictions of our theoretical ideas in human cells in collaboration with four experimental groups.

I will discuss how relations among natural generators of the  Hall algebra of vector bundles on a curve over F_q are related to the zeroes of the zeta function of the curve.

According to a classical result of Bertoin (1998), if the initial data for Burgers equation is a Levy Process with no positive jump, then the same is true at later times, and there is an explicit equation for the evolution of the associated Levy measures. In 2010, Menon and Srinivasan published a conjecture for the statistical structure of solutions to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe the solution as a stochastic process in x with t fixed (or in t with x fixed). In a joint work with Dave Kaspar, we have been able to establish this conjecture. Our argument uses a particle system representation of solutions.

Given a representation of Gal_{Q_p} with coefficients in a p-adically complete local ring R, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex.  When R=Z_p and the representation is a lattice in a de Rham representation, this trivialization should be related to the \varepsilon-factor of the corresponding Weil--Deligne representation.  Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations.  This is joint work with Otmar Venjakob.

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Soft active materials are largely employed to realize devices (actuators), where deformations and displacements are triggered by a wide range of external stimuli such as electric field, pH, temperature, and solvent absorption. The effectiveness of these actuators critically depends on the capability of achieving prescribed changes in their shape and size and on the rate of changes. In particular, in gel–based actuators, the shape of the structures can be related to the spatial distribution of the solvent inside the gel, to the magnitude and the rate of solvent uptake.

In the talk, I am going to discuss some results obtained by my group regarding surface patterns arising in the transient dynamics of swelling gels [1,2], based on the stress diffusion model we presented a few years ago [3]. I am also going to show our extended stress diffusion model suited for investigating swelling processes in fiber gels, and to discuss shape formation issues in presence of fiber gels [4-6].

[1]   A. Lucantonio, M. Rochè, PN, H.A. Stone. Buckling dynamics of a solvent-stimulated stretched elastomeric sheet. Soft Matter 10, 2014.

[2]   M. Curatolo, PN, E. Puntel, L. Teresi. Full computational analysis of transient surface patterns in swelling hydrogels. Submitted, 2017.

[3]   A. Lucantonio, PN, L. Teresi. Transient analysis of swelling-induced large deformations in polymer gels. JMPS 61, 2013.

[4]   PN, M. Pezzulla, L. Teresi. Anisotropic swelling of thin gel sheets. Soft Matter 11, 2015.

[5]   PN, M. Pezzulla, L. Teresi. Steady and transient analysis of anisotropic swelling in fibered gels. JAP 118, 2015.

[6]   PN, L. Teresi. Actuation performances of anisotropic gels. JAP 120, 2016.

The talk will describe accelerated algorithms for computing full or partial matrix factorizations such as the eigenvalue decomposition, the QR factorization, etc. The key technical novelty is the use of  randomized projections to reduce the effective dimensionality of  intermediate steps in the computation. The resulting algorithms execute faster on modern hardware than traditional algorithms, and are particularly well suited for processing very large data sets.

The algorithms described are supported by a rigorous mathematical analysis that exploits recent work in random matrix theory. The talk will briefly review some representative theoretical results.

The Constantin-Lax-Majda model is a 1d system which shares certain features (related to vortex stretching) with the 3d Euler equation. The model is explicitly solvable and exhibits finite-time blow-up for an open subset of smooth initial data. In 1990s De Gregorio suggested adding a transport term to the system, which is analogous to the transport term in the Euler equation. It turns out the transport term has some regularizing effects, which we will discuss in the lecture.

Moduli spaces attempt to classify all mathematical objects of a particular type, for example algebraic curves or vector bundles, and record how they 'vary in families'. Often they are constructed using Geometric Invariant Theory (GIT) as a quotient of a parameter space by a group action. A common theme is that in order to have a nice (eg Hausdorff) space one must restrict one's attention to a suitable subclass of 'stable' objects, in effect leaving certain badly behaved objects out of the classification. Assuming no prior familiarity, I will elucidate the structure of instability in GIT, and explain how recent progress in non-reductive GIT allows one to construct moduli spaces for these so-called 'unstable' objects. The particular focus will be on the application of this principle to the GIT construction of the moduli space of stable curves, leading to moduli spaces of curves of fixed singularity type.
 

Hash Proof Systems or Smooth Projective Hash Functions (SPHFs) are a
form of implicit arguments introduced by Cramer and Shoup at
Eurocrypt’02. They have found many applications since then, in
particular for authenticated key exchange or honest-verifier
zero-knowledge proofs. While they are relatively well understood in
group settings, they seem painful to construct directly in the lattice
setting.
Only one construction of an SPHF over lattices has been proposed, by
Katz and Vaikuntanathan at Asiacrypt’09. But this construction has an
important drawback: it only works for an ad-hoc language of ciphertexts.
Concretely, the corresponding decryption procedure needs to be tweaked,
now requiring q many trapdoor inversion attempts, where q is the modulus
of the underlying Learning With Error (LWE) problem.
Using harmonic analysis, we explain the source of this limitation, and
propose a way around it. We show how to construct SPHFs for standard
languages of LWE ciphertexts, and explicit our construction over a
tag-CCA2 encryption scheme à la Micciancio-Peikert (Eurocrypt’12).

If there is enough time, we will conclude with applications of these
SPHFs to password-authenticated key exchange, honest-verifier
zero-knowledge and a variant of witness encryption.

There are many natural questions one can ask about presentations of finite groups- for instance, given two presentations of the same group with the same number of generators, must the number of relations also be equal? This question, and closely related ones, are unsolved. However if one asks the same question in the category of profinite groups, surprisingly strong properties hold- including a positive answer to the above question. I will make this statement precise and give the proof of this and similar results due to Alex Lubotzky.

Lagrangian Floer cohomology groups are extremely hard compute in most situations. In this talk I’ll describe two ways to extract information about the self-Floer cohomology of a monotone Lagrangian possessing certain kinds of symmetry, based on the closed-open string map and the Oh spectral sequence. The focus will be on a particular family of examples, where the techniques can be combined to deduce some unusual properties.

A central theorem in combinatorics is Sperner’s Theorem, which determines the maximum size of a family in the Boolean lattice that does not contain a 2-chain. Erdos later extended this result and determined the largest family not containing a k-chain. Erdos and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result.

This question was resolved for 2-chains by Kleitman in 1966, who showed that amongst families of size M in the Boolean lattice, the number of 2-chains is minimized by a family whose sets are taken as close to the middle layer as possible. He also conjectured that the same conclusion should hold for all k, not just 2. The best result on this question is due to Das, Gan and Sudakov who showed roughly that Kleitman’s conjecture holds for families whose size is at most the size of the k+1 middle layers of the Boolean lattice. Our main result is that for every fixed k and epsilon, if n is sufficiently large then Kleitman’s conjecture holds for families of size at most (1-epsilon)2^n, thereby establishing Kleitman’s conjecture asymptotically (in a sense). Our proof is based on ideas of Kleitman and Das, Gan and Sudakov.

Joint work with Jozsef Balogh.

We present some dimensionality reduction techniques for global optimization algorithms, in order to increase their scalability. Inspired by ideas in machine learning, and extending the approach of random projections in Zhang et al (2016), we present some new algorithmic approaches for global optimisation with theoretical guarantees of good behaviour and encouraging numerical results.
 

Derivative-free optimisation (DFO) algorithms are a category of optimisation methods for situations when one is unable to compute or estimate derivatives of the objective, such as when the objective has noise or is very expensive to evaluate. In this talk I will present a flexible DFO framework for unconstrained nonlinear least-squares problems, and in particular discuss its performance on noisy problems.

In this talk we consider the limiting behaviour of the strong solution of the Navier--Stokes equation as the viscosity goes to zero, on a three--dimensional region with curved boundary. Under the Navier and kinematic boundary conditions, we show that the solution converges to that of the Euler equation (in suitable topologies). The proof is based on energy estimates and differential--geometric considerations. This is a joint work with Profs. Gui-Qiang Chen and Zhongmin Qian, both at Oxford. 

Some recent applications of supertwistors to superparticle mechanics will be reviewed.
First: Supertwistors allow a simple quantization of the  N-extended 4D massless superparticle, and peculiarities of massless 4D supermultiplets can then be explained by considering the quantum fate of a classical ``worldline CPT'' symmetry. For N=1 there is a global CPT anomaly, which explains why there is no CPT self-conjugate supermultiplet. For N=2 there is no anomaly but a Kramers degeneracy explains the doubling of states in the CPT self-conjugate hypermultiplet.
Second: the bi-supertwistor formulation of the N-extended massive superparticle in 3D, 4D and 6D makes manifest a ``hidden’’ 2N-extended supersymmetry. It also has a simple expression in terms of hermitian 2x2 matrices over the associative division algebras R,C,H.
Third: omission of the mass-shell constraint in this 3D,4D,6D bi-supertwistor action yields, as suggested  by holography, the action for a supergraviton in 4D,5D,7D AdS. Application to the near horizon AdSxS geometries of the M2,D3 and M5 brane confirms that the graviton supermultiplet has 128+128 polarisation states. 

I will report some recent analytical results on microstructures in low-hysteresis shape memory alloys. The modelling assumption is that the width of the thermal hysteresis is closely related to the minimal energy that is necessary to build a martensitic nucleus in an austenitic matrix. This energy barrier is typically modeled by (singularly perturbed) nonconvex elasticity functionals. In this talk, I will discuss recent results on the resulting variational problems, including stress-free inclusions and microstructures in the case of almost compatible phases. This talk is partly based on joint works with S. Conti, J. Diermeier, M. Klar, and D. Melching.

The theory of connections on curves and Hitchin systems is something like a “global theory of Lie groups”, where one works over a Riemann surface rather than just at a point. We’ll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of “Dynkin diagrams” as a step towards classifying some examples of such objects.

Erik Panzer

Feynman integrals, graph polynomials and zeta values

Where do particle physicists, algebraic geometers and number theorists meet?

Feynman integrals compute how elementary particles interact and they are fundamental for our understanding of collider experiments. At the same time, they provide a rich family of special functions that are defined as period integrals, including special values of certain L functions.

In the talk I will give the definition of Feynman integrals via graph polynomials and discuss some examples that evaluate to values of the Riemann zeta function. Then I will discuss some of the interesting questions in this field and mention some of the techniques that are used to study these.

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Yuji Nakatsukasa

Computing matrix eigenvalues

The numerical linear algebra community solves two main problems: linear systems, and eigenvalue problems. They are both vastly important; it would not be too far-fetched to say that most (continuous) problems in scientific computing eventually boil down to one or both of these.

This talk focuses on eigenvalue problems. I will first describe some of their applications, such as Google's PageRank, PCA, finding zeros and poles of functions, and global optimization. I will then turn to algorithms for computing eigenvalues, namely the classical QR algorithm---which is still the basis for state-of-the-art. I will emphasize that the underlying mathematics is (together with the power method and numerical stability analysis) rational approximation theory.

Proteomics is seen as the next logical step after genomics to understand life processes at the molecular level. With increasing capabilities of modern mass spectrometers the deep proteome (>8000 proteins detected) has become widely accessible, only to be replaced recently by the "Ultra-deep proteome" with ~14000 proteins detected in a single cell line. Furthermore, new data search algorithms and sample preparation methods allow not only to achieve comprehensive sequence coverage for the majority of proteins, but also to detect protein variations and single amino acid polymorphisms in proteins, further linking genomic variation to protein phenotypes. The combination of genomic and proteomic information on individual (patient) level could mark the beginning of the "Post-Proteomic Era".

Please register via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2017-ticke…

The ability to study the transcriptome, proteome – and other aspects – of many individual cells represents one of the most important technical breakthroughs and tools in biology and medical science of the past few years. They are revolutionising study of biological systems and human disease, enabling for example: hypothesis-free identification of rare pathogenic (or protective) cell subsets in chronic diseases, routine monitoring of patient immune phenotypes and direct discovery of mole cular targets in rare cell populations. In parallel, new computational and analytical approaches are being intensively developed to analyse the vast data sets generated by these technologies. However, there is still a huge gap between our ability to generate the data, analyse their technical soundness and actually interpret them. The QBIOX network may provide for a unique opportunity to complement recent investments in Oxford technical capabilities in single-cell technologies with the development of revolutionary, visionary ways of interpreting the data that would help Oxford researchers to compete as leaders in this field.

Please register via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2017-ticke…

We consider the differentiability of definable functions in tame expansions
of algebraically closed valued fields.
As the Frobenius inverse shows such a function may be nowhere
differentiable.
We prove differentiability almost everywhere in valued fields of
characteristic 0
that are C-minimal, definably complete and such that, in the valuation
group,
definable functions are strongly eventually linear.
This is joint work with Pablo Cubides-Kovacsics.

Manifolds with ordinary boundary/corners have found their presence in differential geometry and PDEs: they form Man^b or Man^c category; and for boundary value problems, they are nice objects to work on. Manifolds with analytical corners -- a-corners for short -- form a larger category Man^{ac} which contains Man^c, and they can in some sense be viewed as manifolds with boundary at infinity.
In this talk I'll walk you through the definition of manifolds with corners and a-corners, and give some examples to illustrate how the new definition will help.

Understanding the outcome of a collision between liquid drops (merge or bounce?) as well their impact and spreading over solid surfaces (splash or spread?) is key for a host of processes ranging from 3d printing to cloud formation. Accurate experimental observation of these phenomena is complex due to the small spatio-temporal scales or interest and, consequently, mathematical modelling and computational simulation become key tools with which to probe such flows.

Experiments show that the gas surrounding the drops can have a key role in the dynamics of impact and wetting, despite the small gas-to-liquid density and viscosity ratios. This is due to the formation of gas microfilms which exert their influence on drops through strong lubrication forces.  In this talk, I will describe how these microfilms cannot be described by the Navier-Stokes equations and instead require the development of a model based on the kinetic theory of gases.  Simulation results obtained using this model will then be discussed and compared to experimental data.

Dynatomic curves parametrize n-periodic orbits of a one-parameter family of polynomial dynamical systems. These curves lack the structure of their arithmetic-geometric analogues (modular curves of level n) but can be studied dynamically.  Morton and Silverman conjectured a dynamical analogue of the uniform boundedness conjecture (theorems of Mazur, Merel), asserting uniform bounds for the number of rational periodic points for such a family.  I will discuss recent work towards the function field version of their conjecture, including results on the reduction mod p of dynatomic curves for the quadratic polynomial family z^2+c.

For many applications in science and engineering, the ability to efficiently and accurately approximate solutions to elliptic PDEs dictates what physical phenomena can be simulated numerically.  In this seminar, we present a high-order accurate discretization technique for variable coefficient PDEs with smooth coefficients.  The technique comes with a nested dissection inspired direct solver that scales linearly or nearly linearly with respect to the number of unknowns.  Unlike the application of nested dissection methods to classic discretization techniques, the constant prefactors do not grow with the order of the discretization.  The discretization is robust even for problems with highly oscillatory solutions.  For example, a problem 100 wavelengths in size can be solved to 9 digits of accuracy with 3.7 million unknowns on a desktop computer.  The precomputation of the direct solver takes 6 minutes on a desktop computer.  Then applying the computed solver takes 3 seconds.  The recent application of the algorithm to inverse media scattering also will be presented.

I will  address the study of decay rates of solutions to dissipative equations. The characterization of these rates will first be given for a wide class of linear systems by the decay character, which is a number associated to the initial datum that describes the behavior of the datum near the origin in frequency space. The understanding of the behavior of the linear  combined with the decay character and the Fourier Splitting method is then used to obtain some  upper and lower bounds for decay of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case. 

I will talk about a result on meromorphic continuation of Euler products over primes p of definable p-adic or motivic integrals, and applications to zeta functions of groups. If time permitting, I'll state an analogue for counting rational points of bounded height in some adelic homogeneous spac

The goal of this talk will be to give an overview of recent work, joint with Kim Moore, on a short time existence problem in Lagrangian mean curvature flow. More specifically, we consider a compact initial Lagrangian submanifold with a finite number of singularities, each asymptotic to a pair of transversely intersecting planes. We show it is possible to construct a smooth Lagrangian mean curvature flow, existing for positive times, that attains the singular Lagrangian as its initial condition in a suitable weak sense.  The construction uses a family of smooth solutions whose initial conditions approximate the singular Lagrangian. In order to appeal to compactness theorems and produce the desired solution, it is necessary to first establish uniform curvature estimates on the approximating family. As time allows I hope to focus in particular on the proof of these estimates, and their role in the proof of the main theorem.

Each year Prof. Trefethen gives the Problem Solving Squad a sequence of problems with no hints, one a week, where the solution of each problem is a single real number to be computed by any method available.  We will present this year's three problems, involving (1) an S-shaped bifurcation curve, (2) shortest path around a random web, and (3) switching a time-varying system to maximize a matrix norm.

The 14 students this year are Simon Vary plus InFoMM cohort 2: Matteo Croci, Davin Lunz, Michael McPhail, Tori Pereira, Lindon Roberts, Caoimhe Rooney, Ian Roper, Thomas Roy, Tino Sulzer, Bogdan Toader, Florian Wechsung, Jess Williams, and Fabian Ying.  The presentations will be by (1) Lindon Roberts, (2) Florian Wechsung, and (3) Thomas Roy.

Delsarte-Goethals frames are a popular choice for deterministic measurement matrices in compressive sensing. I will show that it is possible to construct extremely sparse matrices which share precisely the same row space as Delsarte-Goethals frames. I will also describe the combinatorial block design underlying the construction and make a connection to Steiner equiangular tight frames.