Seminars and Colloquia by Series

Differential Invariant Algebras

Series
Algebra Seminar
Time
Monday, February 24, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter OlverUniversity of Minnesota

A classical theorem of Lie and Tresse states that the algebra of differential invariants of a Lie group or (suitable) Lie pseudo-group action is finitely generated.  I will present a fully constructive algorithm, based on the equivariant method of moving frames, that reveals the full structure of such non-commutative differential algebras, and, in particular, pinpoints generating sets of differential invariants as well as their differential syzygies. Some applications and outstanding issues will be discussed.

TBA by Zvi Rosen

Series
Algebra Seminar
Time
Monday, February 10, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zvi RosenFlorida Atlantic University

TBA

Brill–Noether theory of Prym varieties

Series
Algebra Seminar
Time
Monday, January 13, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yoav LenGeorgia Tech

The talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian varieties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concerning Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results concerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci.

Free resolutions of function classes via order complexes

Series
Algebra Seminar
Time
Tuesday, November 19, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Justin ChenGeorgia Institute of Technology

Function classes are collections of Boolean functions on a finite set. Recently, a method of studying function classes via commutative algebra, by associating a squarefree monomial ideal to a function class, was introduced by Yang. I will describe this connection, as well as some free resolutions and Betti numbers for these ideals for an interesting collection of function classes, corresponding to intersection-closed posets. This is joint work with Chris Eur, Greg Yang, and Mengyuan Zhang.

Positively Hyperbolic Varieties

Series
Algebra Seminar
Time
Tuesday, November 12, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josephine YuGeorgia Tech

A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points.  Stable polynomials have close relations to matroids and hyperbolic programming.  We will discuss a generalization of stability to algebraic varieties of codimension larger than one.  They are varieties which are hyperbolic with respect to the nonnegative Grassmannian, following the notion of hyperbolicity studied by Shamovich, Vinnikov, Kummer, and Vinzant. We show that their tropicalization and Chow polytopes have nice combinatorial structures related to braid arrangements and positroids, generalizing some results of Choe, Oxley, Sokal, Wagner, and Brändén on Newton polytopes and tropicalizations of stable polynomials. This is based on joint work with Felipe Rincón and Cynthia Vinzant.

Tropical curves of hyperelliptic type

Series
Algebra Seminar
Time
Tuesday, November 5, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel CoreyUniversity of Wisconsin

We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. Using the tropical Torelli theorem (due to Caporaso and Viviani), this characterization may be phrased in terms of 3-edge connectiviations. We show that being of hyperelliptic type is independent of the edge lengths and is preserved when passing to genus ≥2 connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type.

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