We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]

Let $E$ be an elliptic curve over the rationals, and let $\chi$ be a Dirichlet character of order $\ell$ for some odd prime $\ell$. Heuristics based on the distribution of modular symbols and random matrix theory have led to conjectures predicting that the vanishing of the twisted $L$-functions $L(E, \chi, s)$ at $s = 1$ is a very rare event (David-Fearnley-Kisilevsky and Mazur-Rubin). In particular, it is conjectured that there are only finitely many characters of order $\ell > 5$ such that $L(E, \chi, 1) = 0$ for a fixed curve $E$.
We investigate the case of elliptic curves over function fields. For Dirichlet $L$-functions over function fields, Li and Donepudi-Li have shown how to use the geometry to produce infinitely many characters of order $l \geq 2$ such that the Dirichlet $L$-function $L(\chi, s)$ vanishes at $s = 1/2$, contradicting (the function field analogue of) Chowla's conjecture. We show that their work can be generalized to constant curves $E/\mathbb{F}_q(t)$, and we show that if there is one Dirichlet character $\chi$ of order $\ell$ such that $L(E, \chi, 1) = 0$, then there are infinitely many, leading to some specific examples contradicting (the function field analogue of) the number field conjectures on the vanishing of twisted $L$-functions. Such a dichotomy does not seem to exist for general curves over $\mathbb{F}_q(t)$, and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-constant $E/\mathbb{F}_q(t)$.[-]

An old open question in symplectic dynamics asks whether all normalized symplectic capacities coincide on convex domains. I will discuss this question and show that the answer is positive if we restrict the attention to domains which are close enough to a ball. The proof is based on a “quasi-invariant” normal form in Reeb dynamics, which has also implications about geodesics in the space of contact forms equipped with a Banach-Mazur pseudo-metric. This talk is based on a joined work with Gabriele Benedetti and Oliver Edtmair.[-]

This talk will cover two recent advancements in the theory of online algorithms for dynamic matching markets. The first set of results concern a stochastic model of matching with Poisson arrivals and memoryless departures over edge-weighted graphs. The second set of results focus on the incorporation of serial correlation properties in classical online stochastic matching models. We develop new mathematical programming relaxations and correlated rounding schemes, yielding the first constant-factor performance guarantees in such settings.[-]

Two-player turn-based zero-sum games on (finite or infinite) graphs are a central framework in theoretical computer science — notably as a tool for controller synthesis, but also due to their connection with logic and automata theory. A crucial challenge in the field is to understand how complex strategies need to be to play optimally, given a type of game and a winning objective. I will give a tour of recent advances aiming to characterize games where finite-memory strategies suffice (i.e., using a limited amount of information about the past). We mostly focus on so-called chromatic memory, which is limited to using colors — the basic building blocks of objectives — seen along a play to update itself. Chromatic memory has the advantage of being usable in different game graphs, and the corresponding class of strategies turns out to be of great interest to both the practical and the theoretical sides.[-]

This is the story of two distinct approaches for understanding what are 'languages of words', namely 'algebra' and 'logic'. These two approaches eventually rejoined and now irrigate a vivid community of researchers in computer science. In this talk, I will try to give a broad picture of these two perspectives and intuitions on how they nicely interact. An overview:

- The algebraic point of view: words are element in a free algebra.
The first branch, language theory, is concerned with the description of languages of words seen as elements of the free monoid (generated by some finite set traditionally called the alphabet). As such, words are simply terms in some algebra in the sense of universal algebra. After the seminal works of Kleene and Rabin&Scott, that defined the key notion of regular language, this branch developed toward the analysis of the expressive power of restricted formalisms and machines for describing languages. The leading result here is Schützenberger's theorem which states that being definable by a star-free expression is the same as being recognised by an aperiodic monoid: a brilliant algebraic insight. This algebraic description in language theory nowadays catches up with general algebra and category theory, in particular via the use of monads.

- The model-theoretic point of view: words are relational structures.
The second branch, initiated by Büchi, Elgot, and Trakhtenbrot, is the logical point of view. Words are now seen as labelled chains: linear orders equipped with unary predicates (also called monadic). Now logical sentences are used to express properties over these labelled chains. This time MSO logic (monadic second-order logic, ie first-order logic extended with the ability to quantify over monadic predicates = sets) plays the central role, and turns out to be equivalent to regularity over finite words. But, from the point of view of a logician, there is no reason to restrict our attention to finite words: indeed Büchi soon shows the decidability of MSO over omega-words (ie. labelled chains of order type omega). Rabin then proves the remarkable decidability of MSO over the infinite binary tree, and as a consequence the decidability of MSO over the class of all countable linear chains. The composition method was then introduced by Shelah in a seminal work giving another proof of this decidability over countable linear orders, and establishing at the same time undecidability of the MSO theory over the reals: a brilliant model-theoretic insight. These results were then improved by Gurevitch and Shelah, showing decidability over some restricted forms of uncountable chains, and undecidability without extra set theoretic assumptions (the original result relying on CH).

The two branches have progressively converged and are now actively developed in theoretical computer science, in particular in relation with temporal logics, verification, and algorithmic model theory.[-]