The ideal membership problem, the subalgebra membership problem, and the structure of automorphism groups are well studied in the case of free groups, polynomial algebras, free associative algebras, and free Lie algebras. I will talk on the state of these problems for differential polynomial algebras: formulate some results and open questions. I will also talk about the relations between differential algebras and Novikov algebras.

Let K be an o-minimal expansion of a real closed ordered field. We equip K with a derivation δ which is compatible with the definable functions on K. For instance, if K is the real exponential field, then we require that δexp(a) = exp(a)δa for all a in K. Derivations which satisfy this compatibility condition are called T-derivations, where T is the elementary theory of the structure K. I will outline some basic facts about T-derivations and introduce some motivating examples, including Hardy fields and fields of transseries. Then I will turn to “generic” T-derivations. If δ is generic, then the structure (K,δ) eliminates quantifiers (relative to K), eliminates imaginaries, and exhibits tame model-theoretic and topological behavior (like distality and o-minimal open core). These structures generalize the closed ordered differential fields introduced by Singer. This is joint work with Antongiulio Fornasiero.

Abstract in pdf

Throughout this talk, we consider polynomial state-space models, which are used in various fields such as systems biology and control. Such models have three types of variables; state, input and output variables. In some application such as identifiability analysis of biological models, the models are analysed by eliminating state variables in order to specify the relationships between input and output variables. The polynomial equations obtained through such differential elimination are called input-output equations. In this talk, we introduce two applications of the input-output equations.

The first application is analysis of structurally unidentifiable models. This is an extension of structural identifiability analysis but we apply them in the context of parameter estimations; we utilize the equations so as not to overlooking feasible parameters coincident with given time-series data. Examples of analysis of biological models and related data are shown.

The second application is analysis of physical reservoir computing. Reservoir computing is a method for training recurrent neural networks, which has loops inside the network. In reservoir computing, a recurrent neural network called a reservoir is randomly created and fixed in advance. Then the target time-series data is predicted by, e.g., a linear weighted sum of the outputs of the reservoir where only the weights are learned. Recently, a framework that replaces the reservoir to physical systems has been proposed, which is called physical reservoir computing. In order to investigate computational capabilities of physical reservoirs theoretically, we apply the input-output equations of the models of physical reservoirs.

In this talk, I will outline a new technique for showing that nonlinear algebraic differential equations are strongly minimal. This is used to prove the strong minimality of generic differential equations with sufficiently large degree, answering a question of Poizat (1980). Time permitting, I will also discuss ongoing work in applying this method to differential equations of interest whose coefficients are not generic. This is joint work with James Freitag.

During this talk, we will recall the definition of generalized Wronskians, and exhibit a sub-family, whose elements are called geometric. Those geometric generalized Wronskians have two advantages: on the one hand, they allow global geometric constructions, that we will describe, and on the other hand, they still allow to detect linear independance of holomorphic functions (which is the fundamental property of generalized Wronskians, known since the work of Roth in the 1950s).

We will then present applications of this construction in hyperbolicity (more precisely in the study of families of entire curves in Fermat hypersurfaces) and, if time allows, in foliation theory.

This is a joint work with Ronald Bustamente Medina and Zoé Chatzidakis.

In this talk we are interested in differential and difference fields from the model-theoretic point of view. A differential field is a field with a set of commuting derivations and a difference-differential field is a differential field equipped with an automorphism which commutes with the derivations.

Cassidy studied definable groups in differentially closed fields, in particular she studied Zariski dense definable subgroups of simple algebraic groups and showed that they are isomorphic to the rational points of an algebraic group over some definable field. In this talk we study groups definable in existentially closed difference-differential fields. In particular, we study Zariski dense definable subgroups of simple algebraic groups, and show an analogue of Phyllis Cassidy's result for partial differential fields.

A partition of a positive integer n is a decreasing sequence of strictly positive integers whose sum is equal to n. My research is based on the study of the integer partitions and the identities between them. One important example of these identities is as follows:

Theorem. (The first Rogers-Ramanujan identity) The number of partitions of a positive integer n with no equal or consecutive parts is equal to the number of partitions of n into parts congruent to 1 or 4 modulo 5.

Using the relation between the combinatorics of partitions and the combinatorics of graded algebras associated to arc spaces, we will prove a theorem which adds a new member to this partition identity. This is a joint work with H. Mourtada. Then we will prove a theorem which adds another new member to Rogers-Ramanujan identities. This new member counts partitions with different type of constraints on even and odd parts.

The density theorem of Schlesinger ensures that the monodromy group of a differential equation with regular singular points is Zariski-dense in its differential Galois group. An analog of this theorem was obtained for q-difference equations around 2000. For the particular case of regular equations, Etingof constructed a dense subgroup thanks to local solutions at 0 and at infinity. Then, Zariski-dense subgroups for the regular singular q-difference equations were obtained by Sauloy on the one hand and by van der Put and Singer on the other hand using the theory of Tannakian categories and the theory of Picard-Vessiot respectively. After an introduction to these results, we will present the difficulties of the Mahlerian case and, in particular, how they can be overcome to obtain an analog of the Schlesinger’s density theorem for Mahler equations.

Motivated by the well-known differential basis theorem, we recently explored Poisson noetherianity in the category of Poisson algebras rising as symmetric algebras of noetherian Lie algebras. In a recent paper, we conjectured that, under certain suitable assumptions, these Poisson algebras have ACC on radical Poisson ideals. As a first step, we proved that this holds when the Lie algebra in simple; and, more generally, when it satisfies a property that we call "Dicksonianity". This is joint work with Sue Sierra.