L4

Hawking radiation and particle creation by an expanding Universe
are paradigmatic predictions of quantum field theory in curved spacetime.
Although the theory is a few decades old, it still awaits experimental
demonstration. At first sight, the effects predicted by the theory are too
small to be measured in the laboratory. Therefore, current experimental
efforts have been directed towards siumlating Hawking radiation and
studying quantum particle creation in analogue spacetimes.
In this talk, I will present a proposal to test directly effects of
quantum field theory in the Earth's spacetime using quantum technologies.
Under certain circumstances, real spacetime distortions (such as
gravitational waves) can produce observable effects in the state of
phonons of a Bose-Einstein condensate. The sensitivity of the phononic
field to the underlying spacetime can also be used to measure spacetime
parameters such as the Schwarzschild radius of the Earth.

We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has produced asymptotics for the growth of the number of simple closed curves and curves with one self-intersection (respectively) with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result

Law in mathematics and mathematics in law: Probability theory and the fair price in contracts in England and France 1700–1850

From the middle of the eighteenth century, references to mathematicians such as Edmond Halley and Abraham De Moivre begin to appear in judgments in English courts on the law of contract and French mathematicians such as Antoine Deparcieux and Emmanuel-Etienne Duvillard de Durand are mentioned in French treatises on contract law in the first half of the nineteenth century. In books on the then nascent subject of probability at the beginning of the eighteenth century, discussions of legal problems and principally contracts, are especially prominent. Nicolas Bernoulli’s thesis at Basle in 1705 on The Use of the Art of Conjecturing in Law was aptly called a Dissertatio Inauguralis Matematico-Juridica. In England, twenty years later, De Moivre dedicated one of his books on probability to the Lord Chancellor, Lord Macclesfield and expressly referred to its significance for contract law.

The objective of this paper is to highlight this textual interaction between law and mathematics and consider its significance for both disciplines but primarily for law. Probability was an applied science before it became theoretical. Legal problems, particularly those raised by the law of contract, were one of the most frequent applications and as such played an essential role in the development of this subject from its inception. In law, probability was particularly important in contracts. The idea that exchanges must be fair, that what one receives must be the just price for what one gives, has had a significant influence on European contract law since the Middle Ages. Probability theory allowed, for the first time, such an idea to be applied to the sale of interests which began or terminated on the death of certain people. These interests, particularly reversionary interests in land and personal property in English law and rentes viagères in French law were very common in practice at this time. This paper will consider the surprising and very different practical effects of these mathematical texts on English and French contract law especially during their formative period in the late eighteenth and nineteenth centuries.

We consider the problem of unveiling the implicit network structure of user interactions in a social network, based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by $\ell_1$ and trace norms. We provide a first theoretical analysis of the generalization error for this problem, that includes sparsity and low-rank inducing priors. This result involves a new data-driven concentration inequality for matrix martingales in continuous time with observable variance, which is a result of independent interest. The analysis is based on a new supermartingale property of the trace exponential, based on tools from stochastic calculus. A consequence of our analysis is the construction of sharply tuned $\ell_1$ and trace-norm penalizations, that leads to a data-driven scaling of the variability of information available for each users. Numerical experiments illustrate the strong improvements achieved by the use of such data-driven penalizations.

Tropicalization replaces a variety by a polyhedral complex that is a "combinatorial shadow" of the original variety.  This allows algebraic geometric problems to be attacked using combinatorial and
polyhedral techniques.  While this idea has proved surprisingly effective over the last decade, it has so far been restricted to the study of varieties and algebraic cycles.  I will discuss joint work with Felipe Rincon, building on work of Jeff and Noah Giansiracusa, to understand tropicalizing schemes, and more generally the concept of a tropical scheme.

I will talk about recent joint work with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. These intersection numbers, together with the generating series of Noether-Lefschetz numbers, will provide the ingrediants to prove modularity of the above DT invariants over the quintic threefold.