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

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.

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.

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
TBA

Location: TBD
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+ discussions (2:00-2:30pm and 3:30-4:00pm)

March 12, 11:45am-12:45pm, Michael Wibmer, University of Notre Dame
TBA

Location: TBD
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March 12, 12:45-1:45pm, Ivan Tomašić, Queen Mary University
TBA

Location: TBD
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March 14, 11:45am-12:45pm, Wei Li, Chinese Academy of Sciences
TBA

Location: TBD
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March 14, 12:45-1:45pm, James Freitag, University of Illinois at Chicago
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Location: TBD
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March 15, 10:15-11:15am, Rémi Jaoui, University of Waterloo
TBA

Location: room 5382, CUNY Graduate Center
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March 15, 2:00-3:00pm, Rahim Moosa, University of Waterloo
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Location: room 5382, CUNY Graduate Center
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March 16, 9:00-10:00am, Gleb Pogudin, New York University
TBA

Location: room 201, 251 Mercer st., Warren Weaver Hall, Courant Institute
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March 16, 10:00-11:00am, Henry Towsner, University of Pennsylvania
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Location: room 201, 251 Mercer st., Warren Weaver Hall, Courant Institute
TBA

March 22, Joseph Scott, Clemson University
TBA

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March 22, Varadharaj Ravi Srinivasan, IISERM
TBA

Special time: 2-3pm
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March 29, Martin Hils, University of Münster
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April 5, Nigel Pynn-Coates, University of Illinois at Urbana-Champaign
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April 12, Yunnan Li, Guangzhou University
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April 12, Patrick Speissegger, McMaster University
TBA

Special time: 2-3pm
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May 3, Jorge Vitório Pereira, Instituto Nacional de Matemática Pura e Aplicada
TBA

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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.