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.

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.

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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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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.

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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).

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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.

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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.

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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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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.

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 \(d \geq 3\).

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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.

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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

- the existence of a primitive element is established for fields with any number of derivations and automorphism commuting with each other (this includes, for example, partial difference and differential-difference fields);
- no extra condition on the ground field is imposed.

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.

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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.

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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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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.

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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.

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