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.

# Past Forthcoming Seminars

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.

We present some recent results on the study of Schatten-von Neumann properties for

operators on compact manifolds. We will explain the point of view of kernels and full symbols. In both cases

one relies on a suitable Discrete Fourier analysis depending on the domain.

We will also discuss about operators on $L^p$ spaces by using the notion of nuclear operator in the sense of

Grothendieck and deduce Grothendieck-Lidskii trace formulas in terms of the matrix-symbol. We present examples

for fractional powers of differential operators. (Joint work with Michael Ruzhansky)

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)