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

December 14, Mirco Tribastone, School for Advanced Studies Lucca
Maximal aggregation of polynomial differential equations

Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science of engineering, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. In this talk I will present an aggregation technique which rests on two notions of equivalence that relate variables of polynomial differential equations whenever they have the same solution (backward criterion), or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of this approach is the encoding of a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This encoding enables the development of a partition-refinement algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimise probabilistic models of computation according to the notion of bisimulation. I will discuss the effectiveness as well as the physical interpretability of the aggregation in applications to biochemical reaction networks, gene regulatory networks, and evolutionary game theory.
This is joint work with Luca Cardelli, Max Tschaikowski, and Andrea Vandin.

February 15, Jason Bell, University of Waterloo
Invariant hypersurfaces and ideals invariant under an endomorphism or a derivation

We prove a general geometric theorem, which in the affine case can be phrased as follows: Suppose that \(k\) is a field of characteristic zero and \(R\) and \(S\) are finitely generated commutative \(k\)-algebras, with \(R\) an integral domain, and \(f,g: R\to S\) are injective \(k\)-algebra homomorphisms with the property that \(f(R)\) and \(g(R)\) do not contain zero divisors of \(S\) other than zero. Then if the set of (pure) height one radical ideals \(I\) of \(R\) such that the radical of \(f(I)S\) is equal to the radical of \(g(I)S\) is infinite then there is some \(h\) in the field of fractions of \(R\) that is not in \(k\) such that \(f(h)=g(h)\), where we extend \(f,g\) to the fraction field of \(R\) in the natural way using the fact that \(f(R)\) and \(g(R)\) do not contain zero divisors other than zero. We show that this has numerous, somewhat unexpected applications, including recovering work of Cantat on rational dynamics and work of Jouanolou and Hrushovski on \(\delta\)-invariant ideals of a ring \(A\), where \(\delta\) is a derivation of \(A\).

March 22, Joseph Scott, Clemson University
TBA

TBA

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.
Video

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.
Video

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.
Video

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.

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.
Slides
Video

November 16, Boris Kramer, MIT
Lifting transformations in dynamical systems and model reduction

Special time: 2-3pm
Model order reduction for large-scale nonlinear systems is a key enabler for design, uncertainty quantification and control of complex systems. I will discuss a beneficial detour to deriving efficient reduced-order models for nonlinear systems. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model - it uses a change of variables, but introduces no approximations. When discretized, the lifted model yields a polynomial system of either ordinary differential equations or differential algebraic equations, depending on the problem and lifting transformation. In order to obtain computationally inexpensive models, we then proceed with reducing those lifted systems. Proper orthogonal decomposition (POD) is applied to the lifted models, yielding a reduced-order model for which all reduced-order operators can be pre-computed. We show several examples in form of a FitzHugh-Nagumo PDE and a tubular reactor PDE model, and show how this approach opens new pathways for rigorous analysis and input-independent model reduction via the introduction of the lifted problem structure.
Video

November 30, James Greene, Rutgers University
Mathematics behind induced drug resistance in cancer chemotherapy

Special time: 2-3pm
Resistance to chemotherapy is a major impediment to successful cancer treatment that has been extensively studied over the past three decades. Classically, resistance is thought to arise primarily through random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests this evolution to resistance need not occur randomly, but instead may be induced by the application of the drug. Indeed, phenotype switching via epigenetic alterations is just recently beginning to be understood. In this work, we present a mathematical model to that describes both random and induced resistance. We discuss issues related to both structural and practical identifiability of model parameters. A time-optimal control problem is formulated and analyzed utilizing differential-geometric techniques. Specifically, the control structure is precisely characterized, and therapy outcome is analyzed for different levels of resistance induction through a combination of analytic and numerical results. Existence results are also discussed, as well as further extensions to combination therapies are also considered, and questions of combination vs. sequential therapy are studied.
Slides
Video

December 7, Li Guo, Rutgers University
Rota-Baxter Algebras and Quasi-Symmetric Functions

In the 1960s, Rota applied his first construction of free Rota-Baxter algebra and his algebraic formulation of Spitzer's identity to obtain the well-known Waring formula which relates elementary symmetric functions to power symmetric functions. He later suggested that there should be a close connection between Rota-Baxter algebras and generalizations of symmetric functions. He claimed, "In short, (Rota-)Baxter algebras represent the ultimate and most natural generalization of the algebra of symmetric functions." We present some results in support of Rota's claim. We show that a free commutative Rota-Baxter algebra can be interpreted as generalized quasi-symmetric functions from weak compositions. This result also equips the free commutative Rota-Baxter algebra with a natural Hopf algebra structure.
This is joint work with Jean-Yves Thibon, Houyi Yu and Jianqiang Zhao.
Slides
Video

December 7, Gleb Pogudin, New York University
Elimination of unknowns in delay-differential equations

Special time: 2-3pm
Delay-differential equations are actively used in areas of applied mathematics ranging from mathematical biology to electrical engineering. Elimination of unknowns is a fundamental tool for studying solutions of equations (linear, polynomial, differential, etc.). In the talk, the first elimination algorithm for a system of delay-differential equations will be presented. For this algorithm, we develop new effective methods in differential algebraic geometry and combine them with parts of the approach taken in the first elimination algorithm for systems of difference equations designed recently by Ovchinnikov, Pogudin, and Scanlon.
This is a joint work with Wei Li, Alexey Ovchinnikov, and Thomas Scanlon