Algebraic Geometry/Commutative Algebra Seminar, 2019–2020

To volunteer to give a talk, or for anything else regarding the seminar, contact Claudiu Raicu.

Abstracts can be found below.

Fall Schedule

The seminar will meet on Wednesdays, 3:00–4:00pm in 258 Hurley unless otherwise noted. Related events are also listed below.

Date Speaker Title
Wednesday, Sep. 4 Mike Perlman (Notre Dame) Hodge ideals for determinantal hypersurfaces
Thursday, Sep. 12
3-4pm in 258 Hurley
Note special day/time
Chris Eur (U.C. Berkeley) Simplicial generation of Chow rings of matroids
Wednesday, Sep. 18 Eric Riedl (Notre Dame) Linear subvarieties of hypersurfaces and unirationality
Wednesday, Sep. 25 Remi Jaoui (Notre Dame) On the solutions of very general planar algebraic vector fields
Wednesday, Oct. 2
Wednesday, Oct. 9 Anand Patel (Oklahoma State) TBA
Wednesday, Oct. 16 Sayanta Mandal (UIC) TBA
Wednesday, Oct. 23 No seminar (Fall break)
Wednesday, Oct. 30 Zachary Flores (Colorado State) TBA
Wednesday, Nov. 6 Juliette Bruce (University of Wisconsin-Madison) TBA
Thursday, Nov. 14
3-4pm in 258 Hurley
Note special day/time
Antoni Rangachev (University of Chicago) TBA
Friday, Nov. 22
3-4pm in 258 Hurley
Note special day/time
Laurentiu Maxim (University of Wisconsin-Madison) TBA
Wednesday, Nov. 27 No seminar (Thanksgiving)
Wednesday, Dec. 4

Abstracts


Sep. 4, 2019

Speaker
Mike Perlman (Notre Dame)
Title
Hodge ideals for determinantal hypersurfaces
Abstract
Hodge ideals are invariants of hypersurface singularities that arise from birational geometry and M. Saito's theory of mixed Hodge modules. We compute explicitly these ideals for hypersurfaces defined by the generic determinant. To do this we take advantage of the rich symmetry of their log resolutions, the spaces of complete collineations. This is part of joint work in progress with Mircea Mustata and Claudiu Raicu.

Sep. 12, 2019

Speaker
Chris Eur (U.C. Berkeley)
Title
Simplicial generation of Chow rings of matroids
Abstract
Matroids generalize combinatorially the notion of linear independence and graphs, and has recently seen a breakthrough via the development of Hodge theory on Chow rings of matroids by Adiprasito, Huh, and Katz. We introduce a new presentation of the Chow ring of a matroid whose variables now admit a combinatorial interpretation via the theory of matroid quotients and display a geometric behavior analogous to that of nef classes on smooth projective varieties. We discuss various applications, including the recovery of the Hodge theory of matroids. This is joint work with Spencer Backman and Connor Simpson.

Sep. 18, 2019

Speaker
Eric Riedl (Notre Dame)
Title
Linear subvarieties of hypersurfaces and unirationality
Abstract
The de Jong-Debarre Conjecture predicts that the space of lines on any smooth hypersurface of degree d <= n in P^n has dimension 2n-d-3. We prove this conjecture for n > 2d, improving on the previously-known exponential bounds. We prove an analogous result for k-planes, and use this generalization to prove that an arbitrary smooth hypersurface is unirational if n > 2^{d!}. This is joint work with Roya Beheshti.

Sep. 25, 2019

Speaker
Remi Jaoui (Notre Dame)
Title
On the solutions of very general planar algebraic vector fields
Abstract
It is a theorem from Landis and Petrovskii from the fifties that the only algebraic solutions of a very general planar algebraic vector field are the stationary solutions. The non-stationary solutions of a very general vector field are therefore given by transcendental functions but few things are known about the nature of these transcendental functions. For example, can they be expressed using only “classical“ transcendental functions such as exponentials, logarithms, Weierstrass's elliptic functions? In my talk, I will describe a stronger non-integrability result (irreducibility in the sense of Painlevé-Umemura) for very general planar algebraic vector fields of degree greater or equal to three, based on a combination of methods from model-theory and from the local theory of vector fields singularities.