Kolchin Seminar in Differential Algebra

Fall 2019 and Spring 2020

All talks take place at the CUNY Graduate Center, 10:15-11:30 am, in Room 5382 unless something else is specified.
The seminar activities are partially supported by the National Science Foundation.
Talks of the Spring 2019 semester are available here.
Talks of the Fall 2018 semester are available here.
Talks of the Spring 2018 semester are available here.
For earlier seminars, see the old webpage.

Past talks

February 21, 10:15-11:30 am, room 5382, Valery Romanovski, University of Maribor
Integrability and limit cycles in polynomial systems of ODE's

We discuss two problems related to the theory of polynomial plane differential systems, that is, systems of the form \begin{equation}\tag{1} \frac{dx}{dt}=P_{n}(x,y), \ \ \ \frac{dy}{dt}=Q_{n}(x,y), \end{equation} where $P_{n}(x,y), Q_{n}(x,y)$ are polynomials of degree $n$, $x$ and $y$ are real unknown functions. The first one is the problem of local integrability, that is, the problem of finding local analytic integrals in a neighborhood of singular points of system (1). We present a computational approach to find integrable systems within given parametric families of systems and describe some mechanisms of integrability. The second problem is called the cyclicity problem, or the local 16th Hilbert problem, and is related to the estimation of the number of limit cycles arising in system (1) after perturbations of integrable systems. The approach is algorithmic and is based on algorithms of computational commutative algebra relying on the Groebner bases theory.

February 21, 12:30 - 1:45 pm, room 6417, Patrick Speissegger, McMaster University
A new Hardy field of relevance to Hilbert's 16th problem

In our paper, we construct a Hardy field that embeds, via a map representing asymptotic expansion, into the field of transseries as described by Aschenbrenner, van den Dries and van der Hoeven in the recent seminal book. This Hardy field extends that of the o-minimal structure generated by all restricted analytic functions and the exponential function, and it contains Ilyashenko’s almost regular germs. I will describe how this Hardy field arises quite naturally in the study of Hilbert’s 16th problem and give an outline of its construction.

February 21, 2-4 pm, room 5382, Boris Adamczewski, CNRS, Institut Camille Jordan
Algebraic independence of G-functions via reductions modulo primes

Siegel G-functions form an important class of analytic functions which are solutions to some arithmetic linear differential equations. In this talk, I will discuss a new method for proving algebraic independence of such functions. It is based on the following observation: G-functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with the Frobenius that are obtained by reduction modulo prime ideals. I will explain how such difference equations can be used for our purpose. This is a joint work with Jason Bell and Eric Delaygue.

February 10-14, Differential Algebra and Related Topics X workshop

January 31, Amir Ali Ahmadi, Princeton University
Two Problems at the Interface of Optimization and Dynamical Systems

We propose and/or analyze semidefinite programming-based algorithms for two problems at the interface of optimization and dynamical systems:

In part (i), we study the power and limitations of sum of squares optimization and semialgebraic Lyapunov functions for proving asymptotic stability of polynomial dynamical systems. We give the first example of a globally asymptotically stable polynomial vector field with rational coefficients that does not admit a polynomial (or even analytic) Lyapunov function in any neighborhood of the origin. We show, however, that if the polynomial vector field is homogeneous, then its asymptotic stability is equivalent to existence of a rational Lyapunov function whose inequalities have sum of squares proofs. This statement generalizes the classical result in control on the equivalence between asymptotic stability of linear systems and existence of a quadratic Lyapunov function satisfying a certain linear matrix inequality.

In part (ii), we study a new class of optimization problems that have constraints imposed by trajectories of a dynamical system. As a concrete example, we consider the problem of minimizing a linear function over the set of points that remain in a given polyhedron under the repeated action of a linear, or a switched linear, dynamical system. We present a hierarchy of linear and semidefinite programs that respectively lower and upper bound the optimal value of such problems to arbitrary accuracy. We show that on problem instances where the spectral radius of the linear system is bounded above by some constant less than one, our hierarchies find an optimal solution and a certificate of optimality in polynomial time. Joint work with Bachir El Khadir (part (i)) and with Oktay Gunluk (part (ii)).

December 13, Yi Zhang, University of Texas at Dallas
Apparent Singularities of D-Finite Systems

We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities. This is joint work with Shaoshi Chen, Manuel Kauers, and Ziming Li.

December 6, Sam Coogan, Georgia Tech
Probabilistic guarantees for autonomous systems

For complex autonomous systems subject to stochastic dynamics, providing absolute assurances of performance may not be possible. Instead, probabilistic guarantees that assure, for example, desirable performance with high probability are often more appropriate. In this talk, we first describe how interval-valued Markov Decision Processes (IMDP) are able to model stochastic dynamical systems. Unlike classical Markov Decision Processes, IMDPs allow for a range of transition intervals between any two states. We then show that such IMDPs arise naturally when computing finite state abstractions of discrete-time, nonlinear stochastic dynamics. In general, computing such IMDP abstractions can be computationally challenging. However, we present a class of mixed monotone systems for which such abstractions can be efficiently computed. Mixed monotonicity extends the classical notion of monotonicity for dynamical systems to allow for dynamics that have cooperative and competitive effects among the state variables.

November 22, Thomas Dreyfus, Université de Strasbourg
Differential transcendence of solutions of difference equations (remote presentation)

A function is said to be differentially algebraic if it satisfies a non trivial algebraic differential equation. It is said to be differentially transcendent otherwise. Example of differentially transcendent functions are known, for instance, the Gamma function, or the generating series of automatic sequences. All these functions have in common to satisfy a linear functional equation. In this framework, the difference Galois theory provides tools to prove the differential transcendence of the functions. This strategy has given many recent papers presenting results that get more and more general. In this talk we are going to present a new result for which the hypothesizes are very minimal. This is a joint work with B. Adamczewki and C. Hardouin.

November 15, Jonathan Kirby, University of East Anglia
Local Definability of Holomorphic Functions (remote presentation)

Given a collection $\mathcal{F}$ of complex or real analytic functions, one can ask what other functions are obtainable from them by finitary algebraic operations. If we just mean polynomial operations we get some field of functions. If we include as algebraic operations such things as taking implicit functions, maybe in several variables, we get a much more interesting framework, which is closely related to the theory of local definability in an o-minimal setting, starting with suitable restrictions of the functions in $\mathcal{F}$. O-minimality is a setting for tame topology of real- or complex-analytic functions which does not allow for "bad" singularities. However some more tame singularities can occur. In this talk I will explain work showing what singularities we have to consider to get a characterisation of the locally definable functions in terms of complex analytic operations. Ax’s theorem on the differential algebra version of Schanuel’s conjecture is important to give one counterexample, and also for some applications to exponential and elliptic functions.

This is joint work with Gareth Jones, Olivier Le Gal, and Tamara Servi.

November 8, Léo Jimenez, University of Notre Dame
Strengthenings of C-algebraicity in differentially closed fields of characteristic zero

In model theory, the notion of internality to a fixed family of types plays an important role. During this talk, I will focus on one of its differential algebraic manifestations: being C-algebraic, where C is the field of constants of a differentially closed field. An irreducible differential-algebraic variety is C-algebraic if it is, roughly speaking, differentially birational to an algebraic variety in C. I will discuss a new property, uniform C-internality, and discuss examples, non-examples, and applications.

November 1, Carsten Schneider, Johannes Kepler University
An Algorithmic Difference Ring Theory for Symbolic Summation

Inspired by Karr's pioneering work (1981) we developed an algorithmic difference ring theory for symbolic summation that enables one to rephrase indefinite nested sums and products in formal difference rings. An important outcome of this representation is that one obtains a simplified expression where the arising sums and products are algebraically independent among each other. In this talk the main ideas of these algorithmic constructions and crucial features of the underlying difference ring theory are presented. Combining such optimal representations in combination with definite summation algorithms, like creative telescoping and recurrence solving in the setting of difference rings, yield a strong summation toolbox for practical problem solving. We will demonstrate this machinery implemented in the summation package Sigma by concrete examples coming from particle physics.

October 25, Fabian Immler, Carnegie Mellon University
Formal mathematics and a proof of chaos

Formal proof has been successfully applied to the verification of hardware and software systems. But formal proof is also applicable to mathematics: proofs can be checked with ultimate rigor and one can build libraries of computer-searchable, formalized mathematics.

I will talk about formalization of mathematics and my formalization of ordinary differential equations in the Isabelle/HOL theorem prover. This underpins the formal verification of the computer-assisted part of Tucker's proof of Smale's 14th problem, a proof that relies on numerical bounds to certify chaos for the Lorenz system of ordinary differential equations.

October 18, Omar Leon Sanchez, University of Manchester
Differentially large fields

Recall that a field $K$ is large if it is existentially closed in the field of Laurent series $K((t))$. Examples of such fields are the complex, the real, and the p-adic numbers. This class of fields has been exploited significantly by F. Pop and others in inverse Galois-theoretic problems. In recent work with Tressl we introduced and explored a differential analogue of largeness, that we conveniently call "differentially large". I will present some properties of such fields and characterise them using formal Laurent series and to even construct “natural” examples (which ultimately yield examples of DCFs and CODFs... acronyms that will be explained in the talk). Time permitting I will mention some applications to Parameterized PV theory.

October 11, Yi Zhou, Florida State University
Algorithms on p-Curvatures of Linear Difference Operators

In the study of factoring linear difference operators, we have found p-curvature a powerful tool. I will talk about algorithms for computing p-curvatures and the math behind them.

October 4, Anand Pillay, University of Notre Dame
Finiteness theorems for Kolchin's constrained cohomology

This is joint work with Omar Leon Sanchez. Working under a certain general assumption on the differential field $K$ (which includes the case where $K$ is a closed order differential field in the sense of Michael Singer) we prove finiteness of the "constrained cohomology sets" $H^{1}_{\partial}(K,G)$, for $G$ any linear differential algebraic group over $K$. I will define everything and touch on some applications.

September 13, Daniel Robertz, University of Plymouth
Algorithmic Approach to Strong Consistency Analysis of Finite Difference Approximations to PDE Systems

The most common numerical method for solving partial differential equations is the finite difference method. Consistency of a finite difference scheme with a given PDE is a basic requirement for this method. Earlier work by V. P. Gerdt and the speaker introduced the notion of strong consistency that takes into account the differential ideal and the difference ideal associated with the PDE system and the approximating difference system, respectively. We present an algorithmic approach to strong consistency for polynomially nonlinear PDE systems based on a new decomposition technique for nonlinear partial difference systems that is analogous to the differential Thomas decomposition. This is joint work with Vladimir P. Gerdt (JINR, Dubna).