Kolchin Seminar in Differential Algebra

Fall 2018

All talks take place at 10-11 am in Room 5382 unless something else is specified.
The seminar activities are partially supported by the National Science Foundation.
Talks of the Spring 2018 semester are available here.
For earlier seminars, see the old webpage.

Upcoming talks

October 12, Peter Thompson, City University of New York
Input-output equations for parameter identifiability in rational ODE models, Part 2

We continue our discussion of input-output equation methods for the problem of parameter identifiability in ODE modeling. It is commonly assumed that the functions of parameters appearing as coefficients in input-output equations are identifiable. We discuss the validity of this in the single-output and multiple-output cases.

October 19, Françoise Point, University of Mons
Differential expansions of topological large fields and transfer results

Given a theory \(T\) of large topological fields of characteristic \(0\) admitting quantifier elimination (in some relational expansion \(L\) of the language of fields), we consider its (generic) expansion \(T_D\) to a theory of differential fields. Under some natural hypotheses, that we will detail, it is known that the class of existentially closed models of such expansions is axiomatizable and that its theory \(T_D^*\) admits quantifier elimination in \(L_D\) (the language \(L\) to which we add the derivation \(D\)). For instance if one starts with the class of real-closed fields, M. Singer showed that one obtains the class of closed ordered differential fields (CODF). We will first review a number of known transfer results between \(T\) and \(T_D^*\) and their consequences for the theory of dense pairs of models of \(T\). Then we will concentrate on elimination of imaginaries, a property that allows one to associate with any definable set a code (for instance, the theory of differentially closed fields of characteristic zero has that property). Under the hypothesis that \(T_D^*\) has open core, namely any open \(L_D\)-definable set is already \(L\)-definable, we will show transfer of elimination of imaginaries between \(T\) and \(T_D^*\), using a topological argument due to M. Tressl in the case of CODF. This is a joint work with Pablo Cubid├Ęs Kovacsics (Caen).
There will be no prerequisites in model theory.

November 16, Boris Kramer, MIT

Special time: 2-3pm

November 30, James Greene, Rutgers University

Special time: 2-3pm

December 14, Mirco Tribastone, School for Advanced Studies Lucca


Past talks

September 7, Stephen Melczer, University of Pennsylvania
Symbolic Computation and Analytic Combinatorics in Several Variables

The field of analytic combinatorics in several variables (ACSV) is at the forefront of computational combinatorics - a subject dedicated to computability and complexity questions in enumeration. Drawing from singularity theory, computational topology, and algebraic geometry, ACSV raises a wide range of interesting computer algebra questions. This talk will survey ACSV from a computer algebra viewpoint, discuss current work, and highlight remaining open problems and generalizations.

September 14, Joel (Ronnie) Nagloo, City University of New York
The Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian Groups

The works of Pila and later Freitag and Scanlon, give the Ax-Lindemann-Weierstrass with derivatives for the Hauptmoduls of arithmetic subgroups of \(\mathrm{PSL}_2(\mathbb{Z})\). A challenge has been to prove similar transcendence results for the Hauptmoduls of all Fuchsian groups of genus zero. In this talk I will explain recent progress towards the resolution of those problems. This is report of joint work with Guy Casale and James Freitag.

September 21, Thomas Dreyfus, University of Strasbourg
Differential transcendence of special functions

One of the goal of the difference Galois theory is to understand the algebraic relations between solutions of a linear functional equation. Recently, Hardouin and Signer developed a Galois theory that aims at understanding what are the algebraic and differential relations among solution of such equations. In this talk we are going to see recent results ensuring that in many situations, such solutions satisfy no algebraic differential relations.

September 28, Doron Zeilberger, Rutgers University
The C-finite ansatz

A sequence belongs to the C-finite ansatz if it satisfies a linear recurrence equation with constant coefficients. For example, \(2^n\), and the sequence of Fibonacci numbers, \(F_n\). After describing some applications to enumerative combinatorics, I will describe yet another approach to the Ising model, different than the one Manuel Kauers talked about three weeks earlier (September 6, at the CUNY/NYU symbolic-numeric computing seminar). This is also joint work with Manuel Kauers.

October 5, Peter Thompson, City University of New York
Input-output equations for parameter identifiability in rational ODE models

The problem of parameter identifiability is of great importance in modeling, for example in biological systems. One technique used in studying identifiability is the notion of input-output equations. Let S be a system of ordinary differential equations in several variables, some of which are observable and others of which are unobservable. The input-output equations are a subset of the consequences of S in which only observable variables appear, and from which information about the identifiability of certain parameters can be gained. We discuss input-output equation methods.