### 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.

### 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.