Past

Divergence-measure fields are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose divergence is a Radon measure. Such vector fields form a new family of function spaces, which in a sense generalize the $BV$ fields, and were introduced at first by Anzellotti, before being rediscovered in the early 2000s by many authors for different purposes.
Chen and Frid were interested in the applications to the theory of systems of conservation laws with the Lax entropy condition and achieved a Gauss-Green formula for divergence-measure fields, for any $1 \le p \le \infty$, on open bounded sets with Lipschitz deformable boundary. We show in this talk that any Lipschitz domain is deformable.
Later, Chen, Torres and Ziemer extended this result to the sets of finite perimeter in the case $p = \infty$, showing in addition that the interior and exterior normal traces of the vector field are essentially bounded functions.
The Gauss-Green formula for $1 \le p \le \infty$ has been also studied by Silhavý on general open sets, and by Schuricht on compact sets. In such cases, the normal trace is not in general a summable function: it may even not be a measure, but just a distribution of order 1. However, we can show that such a trace is the limit of the integral of classical normal traces on (smooth) approximations of the integration domain.

 I will discuss my ongoing project towards a version of the Modular Andre-Oort Conjecture incorporating the derivatives of the j function.  The work originates with Jonathan Pila, who formulated the first "Modular Andre-Oort with Derivatives" conjecture.  The problem can be approached via o-minimality; I will discuss two categories of result.  The first is a weakened version of Jonathan's conjecture.  Under an algebraic independence conjecture (of my own, though it follows from standard conjectures), the result is equivalent to the statement that Jonathan's conjecture holds.  
The second result is conditional on the same algebraic independence conjecture - it specifies more precisely how the special points in varieties can occur in this context.  
If time permits, I will discuss my most recent work towards making the two results uniform in algebraic families.

Direct Anonymous Attestation (DAA) is a protocol that allows a security chip embedded in a platform such as laptop to authenticate itself as a genuine chip.  Different authentications are not linkeable, thus the protocol protects the privacy of the platform. The first DAA protocol was proposed by Brickell, Chen, and Camenisch and was standardized in 2004 by the Trusted Computing Group (TCG). Implementations of this protocols were rather slow because it is based on RSA. Later, alternative and faster protocols were proposed based on elliptic curves. Recently the specification by the TCG was updated to allow for DAA protocols based on elliptic curves. Unfortunately, the new standard does not allow for provably secure DAA protocols. In this talk, we will review some of the history of DAA and  then discuss the latest protocols, security models, and finally a provably secure realization of DAA based on elliptic curves.

It is well known that there is a finite colouring of the natural numbers such that there is no infinite set X with X+X (the pairwise sums from X, allowing repetition) monochromatic. It is easy to extend this to the rationals. Hindman, Leader and Strauss showed that there is also such a colouring of the reals, and asked if there exists a space 'large enough' that for every finite colouring there does exist an infinite X with X+X monochromatic. We show that there is indeed such a space. Joint work with Imre Leader.

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.