Kolchin Seminar in Differential Algebra

Spring 2019

All talks take place at 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 Fall 2018 semester are available here.
Talks of the Spring 2018 semester are available here.
For earlier seminars, see the old webpage.

Upcoming talks

April 12, Richard Gustavson, Manhattan College
Algebraic Structures from Integral Equations

Integral calculus is in general more complicated than differential calculus. For functions of one variable, integral equations involving integrals of the form \(\displaystyle \int_a^x f(t)\,dt\), for some unknown function \(f\), have been studied algebraically using the theory of Rota-Baxter algebras. In this talk we will discuss the algebraic structures of Volterra integral equations, which are equations involving integrals of the form \(\displaystyle \int_a^x K(x,t)f(t)\,dt\) for some unknown function \(f\) and given kernel \(K\). While there are methods for finding solutions to Volterra equations, the presence of the kernel \(K(x,t)\) as a function of \(x\) and \(t\) makes these equations much more difficult to study from an algebraic perspective. This talk is based on joint work with Li Guo and Yunnan Li.

April 12, Mengxiao Sun, City University of New York
On the Complexity of Computing Galois Groups of Differential Equations (Ph.D. defense)

Special time and place: 12:15-1:45 pm, room 6114
The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential Galois group of a general linear differential equation. Recently, Feng approached finding a complexity bound of the algorithm, which is the degree bound of the polynomials used in the first step of the algorithm for finding a proto-Galois group. The bound given by Feng is quintuply exponential in the order \(n\) of the differential equation. The complexity, in the worst case, of computing a Gröbner basis is doubly exponential in the number of variables. Feng chose to represent the radical of the ideal generated by the defining equations of a proto-Galois group by its Gröbner basis. Hence, a double-exponential degree bound for computing Gröbner bases was involved when Feng derived the complexity bound of computing a proto-Galois group.

Triangular decomposition provides an alternative method for representing the radical of an ideal. It represents the radical of an ideal by the triangular sets instead of its generators. The first step of Hrushovski's algorithm is to find a proto-Galois group which can be used further to find the differential Galois group. So it is important to analyze the complexity for finding a proto-Galois group. We represent the radical of the ideal generated by the defining equations of a proto-Galois group using the triangular sets instead of the generating sets. We apply Szántó's modified Wu-Ritt type decomposition algorithm and make use of the numerical bound for Szántó's algorithm to adapt to the complexity analysis of Hrushovski's algorithm. We present a triple exponential complexity bound for finding a proto-Galois group in the first step of Hrushovski's algorithm.

May 3, Jorge Vitório Pereira, Instituto Nacional de Matemática Pura e Aplicada
TBA

TBA

May 10, Yunnan Li, Guangzhou University
Extension of Gröbner-Shirshov basis of an algebra to its generating free differential algebra

The free differential algebra on a set is well-understood as the polynomial algebra on the differential variables. More generally, a free differential algebra on an algebra can be defined, giving the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. Little is known for such free objects. In this talk, we show that generator-relation properties of a base algebra can be extended to the free differential algebra on this algebra. More precisely, Gröbner-Shirshov basis property of the base algebra can be extended, allowing construction of these more general free differential algebras in concrete terms. Examples are given as illustrations. Finally, the free differential Lie algebra on a Lie algebra \(\mathfrak{g}\) are introduced and are shown to be generated within the free differential associative algebra generated by the enveloping algebra \(U(\mathfrak{g})\) of \(\mathfrak{g}\), similar to the classical case of generating free Lie algebras in a free associative algebra. This is joint work with Li Guo.

Past talks

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 \(A\).
Video

February 15, Harm Derksen, University of Michigan
Singular Values of Tensors

Special time: 2-3pm
This is a joint talk with the Courant/CUNY symbolic-numeric seminar
Tensor decompositions have many applications, including chemometrics and algebraic complexity theory. Various notions, such as the rank and the nuclear norm of a matrix, have been generalized to tensors. In this talk I will present a new generalization of the singular value decomposition to tensors that shares many of the properties of the singular value decomposition of a matrix.
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Slides

February 22, Malabika Pramanik, University of British Columbia
Analysis and geometry of sparse sets

Special time: 2-3pm
Pattern identification in sets has long been a focal point of interest in analysis, geometry, combinatorics and number theory. No doubt the source of inspiration lies in the deceptively simple statements and the visual appeal of these problems. For example, when does a given set contain a copy of your favourite pattern (say specially arranged points on a line or a spiral, the vertices of a polyhedron or solutions of a functional equation)? Does the answer depend on how thin the set is in some quantifiable sense?
Here is another problem. Curves and surfaces form a class of thin sets in Euclidean space that is rich in analytic and geometric structure. They form the central core in many problems in harmonic and complex analysis (such as restriction phenomena and integral transforms) and play an important role in the study of partial differential equations with a geometric flavour. How well do properties of surfaces and submanifolds carry over to the setting of an arbitrary sparse set with no differential-geometric structure?
Problems of this flavour fall under the category of geometric measure theory. Under varying interpretations of size, they have been vigorously pursued both in the discrete and continuous setting, often with spectacular results that run contrary to intuition. Yet many deceptively simple questions remain open. I will survey the literature in this area, emphasizing some of the landmark results that focus on different aspects of the problem.
Video

March 1, Carlos Arreche, University of Texas at Dallas
Differential transcendence of elliptic hypergeometric functions through Galois theory

Elliptic hypergeometric functions arose roughly 10 years ago as a generalization of classical hypergeometric functions and q-hypergeometric functions. These special functions enjoy remarkable symmetry properties, like their more classical counterparts, and find applications in mathematical physics. After interpreting one of these symmetries as a linear difference equation over an elliptic curve, we apply the differential Galois theory of difference equations to show that these functions are always differentially transcendental for “generic” values of the parameters. This is joint work with Thomas Dreyfus and Julien Roques.
Video
Slides

Workshop on Model Theory, Differential/Difference Algebra, and Applications, CUNY/Courant, March 11-16

March 11, 2:30-3:30pm, Joel (Ronnie) Nagloo, City University of New York
The generic Schwarz triangle equations

Location: room 3207, CUNY Graduate Center
In this talk, I will focus on the ODEs satisfied by the Schwarz triangle functions. These are the conformal mappings from the circular triangles (in \(\mathbb{C}\)) onto the complex unit disk. I will explain how, building on my recent work joint with Casale and Freitag on the genus zero Fuchsian groups, one can give a full description of the structure of the set of solutions of a generic Schwarz triangle equation. More precisely, I will explain how one can show that the solution set is strongly minimal and also strictly disintegrated, i.e there are no algebraic relations between distinct solutions (including their derivatives).
Video
+ discussions (2:00-2:30pm and 3:30-4:00pm)

March 12, 11:45am-12:45pm, Michael Wibmer, University of Notre Dame
On the dimension of systems of algebraic difference equations

Location: room 3207, CUNY Graduate Center
We introduce and study a notion of dimension for the solution set of a system of algebraic difference equations. This dimension measures the degrees of freedom when determining a solution in the ring of sequences. This number need not be an integer, but as we show, it satisfies properties suitable for a notion of dimension. We also show that the dimension of a difference monomial is given by the covering density of its set of exponents.
Video

March 12, 12:45-1:45pm, Ivan Tomašić, Queen Mary University
A topos-theoretic view of difference algebra

Location: room 3207, CUNY Graduate Center
Abstract difference algebra was founded by Ritt in the 1930s as the study of algebraic structures equipped with distinguished endomorphisms. This approach has a long and productive history, but attempts to develop methods of homological algebra within this context quickly reach insurmountable obstacles.
We will show how to use the methods of topos theory and categorical logic to resolve these issues and to elevate the study of difference algebraic geometry to the level of classical algebraic geometry.
Slides
Video

March 13, 1:30-5:00pm

Location: rooms 402 and 805, 251 Mercer st., Warren Weaver Hall, Courant Institute
Discussions

March 14, 11:45am-12:45pm, Wei Li, Chinese Academy of Sciences
Sparse resultants in differential and difference algebra: an overview

Location: room 3309, CUNY Graduate Center
The (sparse) resultant, which gives conditions for an over-determined system of polynomial equations to have common solutions, is a basic concept in algebraic geometry, and emerges to be one of the most powerful computational tools in (sparse) elimination theory due to its ability to eliminate several variables simultaneously. In the recent years, a theory has been developed for these analogous concepts in differential and difference algebra, and many new problems have arisen. In this talk, I will give an overview of the progress we have made in this area, and present several open problems.
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March 14, 12:45-1:45pm, James Freitag, University of Illinois at Chicago
Algebraic relations between solutions of Painlevé equations

Location: room 3309, CUNY Graduate Center
In this talk we will explain the origin and importance of Painlevé equations, before addressing the central question of the talk. What are the algebraic relations between solutions of Painlevé equations? The work of Pillay and Nagloo brought this question into focus, and following recent work of Nagloo on the sixth Painlevé equation, we can now give a complete answer when at least one coefficient in one of the equations we consider is transcendental. This is joint work with Ronnie Nagloo.

March 15, 10:15-11:15am, Rémi Jaoui, University of Waterloo
Disintegration and planar algebraic vector fields

Location: room 5382, CUNY Graduate Center
A differential equation is disintegrated (or geometrically trivial) if any algebraic relation between an arbitrary number of its solutions can be decomposed into algebraic relations between couples of solutions. I will explain that disintegration is a typical property for complex planar algebraic vector fields of degree \(d \geq 3\). This implies, for example, that the set of parameters for which this property holds has full Lebesgue measure in the parameter space of algebraic planar vector fields of degree \(d \geq 3\).
Video

March 15, 2:00-3:00pm, Rahim Moosa, University of Waterloo
Pullbacks under the logarithmic derivative

Location: room 6417, CUNY Graduate Center
Let \(X\) be the Kolchin closed set defined by an algebraic differential equation of the form \(Dx=f(x)\), where \(f\) is a rational function over constant parameters. Rosenlicht's theorem gives us a condition on \(f\) that tells us when \(X\) is (in model-theoretic terms) internal to the constants. In this talk I will describe a criterion in a similar spirit answering the question of when the pullback of \(X\) under the logarithmic derivative is internal to the constants. The case of nonconstant parameters will also be discussed. These are results from my student Ruizhang Jin's recent thesis, as well as further joint work.
Video

March 16, 9:00-10:00am, Gleb Pogudin, New York University
Primitive Element Theorem for fields with commuting derivations and automorphisms

Location: room 201, 251 Mercer st., Warren Weaver Hall, Courant Institute
Primitive Element Theorem says that every finitely generated algebraic extension of fields of zero characteristic is generated by a single element. It is a classical tool in field theory and symbolic computation. It has been generalized to partial differential fields by Kolchin in 1942 and to difference fields (with a single automorphism) by Cohn in 1965. These theorems guarantee that if an extension \(F \subset E\) is finitely generated and algebraic in an appropriate sense and the groud field \(F\) is "nonconstant", then the extension can be generated by a single element. These generalizations played an important role in differential/difference algebra and its applications.

However, both theorems by Kolchin and Cohn imposed an extra condition for the ground field \(F\) to be "nonconstant" that made them not applicable to many important extensions coming from autonomous differential/difference equations or algebraic variaties equipped with a vector field or an automorphism. In 2015, I have partially resolved this issue by strengthening Kolchin's theorem in the case of one derivation so that the condition that \(F\) contains a nonconstant was replaced by a natural condition that \(E\) contains a nonconstant (otherwise, the derivation would be zero).

In this talk, I will describe my recent result that generalizes the primitive element theorems by Kolchin, Cohn, and myself in two directions Video

March 16, 10:00-11:00am, Henry Towsner, University of Pennsylvania
Ultraproducts: What are they good for?

Location: room 201, 251 Mercer st., Warren Weaver Hall, Courant Institute
The use of ultraproducts as a technique for proving results in algebra and differential algebra is well established. We will discuss how ultraproduct arguments can be transformed into explicit, constructive arguments. Along the way, we will be able to identify what features of a proof can make them suitable for simplifying using an ultraproduct.
Video

March 22, Joseph Scott, Clemson University
Rapid and Accurate Reachability Analysis for Nonlinear Systems by Exploiting Model Redundancy

The presentation will cover recent advances in techniques for rapidly and accurately propagating rigorous uncertainty bounds through complex dynamic models. In applications from autonomous aircraft to biochemical networks, the ability to quantify the effects of uncertainty is essential for designing systems that are passively robust to uncertainty, as well as for making optimal, real-time control decisions under uncertainty. Moreover, methods that can provide rigorous bounds on the system states achievable under uncertainty are uniquely useful in their ability to guarantee that a particular course of action will satisfy all relevant constraints (e.g., in aircraft collision avoidance). Although it has long been possible to compute such bounds efficiently using interval methods, the results are often too conservative to be of any practical use (i.e., the upper and lower bounds tend to \(\pm\infty\) over short time-scales). In contrast, modern bounding strategies can achieve remarkably sharp bounds, even for highly nonlinear systems with large uncertainties, but are far too costly for real-time decision making when the number of states and uncertain parameters exceeds ∼ 5. Thus, there is a critical need for an alternative approach to uncertainty propagation in nonlinear dynamic systems that is simultaneously rigorous, accurate, fast enough for real-time applications, and scalable to much larger systems.

Toward this end, our key insight is that the conservatism of fast interval methods can be dramatically reduced through the use of model redundancy. Indeed, our recent work shows that bounds produced by these methods often enclose large regions of state-space that violate redundant relations implied by the dynamics, such as conservation laws, and that these can be exploited to obtain much sharper bounds for a limited class of systems. Motivated by these observations, we have developed an innovative new approach for arbitrary systems based on the deliberate introduction of model redundancy to reduce conservatism. This technique lies at interface of numerical and symbolic computing and has been shown to lead to remarkably sharp bounds at low cost in a variety of challenging applications. We will discuss the mechanisms by which redundancy leads to improved bounds, strategies for introducing redundant equations that are effective in this context, and preliminary results on automating the construction of these equations. Finally, our methods will be demonstrated on uncertain dynamic system arising in the chemical and aerospace domains.
Slides
Video

March 22, Varadharaj Ravi Srinivasan, IISERM
Integration in Finite Terms: Error Functions, Logarithmic Integrals and Polylogarithmic Integrals

Special time: 2-3pm
The talk concerns the problem of integration in finite terms with special functions. Our main result extends the classical theorem of Liouville in the context of elementary functions to various classes of special functions: error functions, logarithmic integrals, dilogarithmic and trilogarithmic integrals. The results are important since they provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension generated by transcendental elementary functions and special functions. A special case of our main result simplifies and generalizes a theorem of Baddoura on integration in finite terms with dilogarithmic integrals. Our results can be naturally generalized to include polylogarithmic integrals and to this end, a conjecture will be stated for integration in finite terms with transcendental elementary functions and polylogarithmic integrals.

This is a joint with Yashpreet Kaur.

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Video

March 29, Martin Hils, University of Münster
Imaginaries in separably closed valued fields

Let \(p\) be a fixed prime number and let \(SCVF_p\) be the first order theory of separably closed non-trivially valued fields of characteristic \(p\). In the talk, we will see that, in many ways, from a model-theoretic point of view, the step from algebraically closed VALUED fields in characteristic \(p\) to \(SCVF_p\) is not more complicated than the one from algebraically closed fields to separably closed fields in characteristic \(p\).

At a basic level, this is true for quantifier elimination (Delon), for which it suffices to add parametrized \(p\)-coordinate functions to any of the usual languages for valued fields. At a more sophisticated level, in finite degree of imperfection, when a \(p\)-basis is named by constants or when one just works with Hasse derivations, the imaginaries (i.e. definable quotients) are classified by so-called the geometric sorts of Haskell-Hrushovski-Macpherson, certain higher-dimensional analogs of the residue field and the value group. This classification is proved by a reduction to the algebraically closed case, using prolongations.

This is joint work with Moshe Kamensky and Silvain Rideau.

Video

March 29, Peter Thompson, City University of New York
A differential algebra approach to commuting polynomial vector fields and to parameter identifiability in ODE models (Ph.D. defence)

Special time and place: 11:45am-1:45pm, room 6114
In the first part, we study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field. One motivating factor is that we can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. We first show that a linear vector field admits a full complement of commuting vector fields. Then we study a type of planar vector field for which there exists an upper bound on the degree of a commuting polynomial vector field. Finally, we turn our attention to conservative Newton systems and show the following result. Let \(f \in K[x]\), where \(K\) is a field of characteristic zero, and \(d\) the derivation that corresponds to the differential equation \(\ddot x = f(x)\) in a standard way. We show that if \(\deg f\geqslant 2\), then any \(K\)-derivation commuting with \(d\) is equal to \(d\) multiplied by a conserved quantity. For example, the classical elliptic equation \(\ddot x = 6x^2+a\), where \(a \in \mathbb{C}\), falls into this category.

In the second part, we study structural identifiability of parameterized ordinary differential equation models of physical systems, for example, systems arising in biology and medicine. A parameter is said to be structurally identifiable if its numerical value can be determined from perfect observation of the observable variables in the model. Structural identifiability is necessary for practical identifiability. We study structural identifiability via differential algebra. In particular, we use characteristic decompositions. A system of ODEs can be viewed as a set of differential polynomials in a differential ring, and the consequences of this system form a differential ideal. This differential ideal can be described by a finite set of differential equations called a characteristic decomposition. The technique of studying identifiability via a set of special equations, sometimes called "input-output" equations, has been in use for the past thirty years. However it is still a challenge to provide rigorous justification for some conclusions that have been drawn in published studies. Our work provides justification for some cases, and provides a computable condition that can be used to justify the others. We present a computable condition on the elements of the characteristic decomposition such that if this condition is satisfied, then the conclusions about identifiability drawn from this decomposition are correct. We proceed to show that all linear systems of ODEs with one observable variable satisfy this condition.

April 5, Nigel Pynn-Coates, University of Illinois at Urbana-Champaign
Asymptotic valued differential fields

The general goal is to do valuation theory for differential fields given an appropriate condition on the interaction between the valuation and the derivation. In this talk, I will consider asymptotic valued differential fields, introduced by Aschenbrenner, van den Dries, and van der Hoeven during their work on transseries, extending work of Rosenlicht. I will present analogues of three fundamental results from valuation theory that go through in this setting, concerning (differential-algebraically) maximal immediate extensions and their connection with differential-henselianity.
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Video