Algebra seminar

Previous

Time and place: Jan. 20, 2022 2:15 PM4:00 PM, NHA B1120
Hilbert schemes of points for a surface are a well studied subject with many famous results like Göttsche’s formula for its Betti numbers. A natural generalization comes from studying Grothendieck’s Quot-schemes and the associated enumerative invariants. Unlike the former, punctual Quot-schemes are smooth only virtually admitting perfect obstruction theories and virtual fundamental classes. This has recently been used to study invariants counting zero-dimensional quotients of trivial vector bundles by multiple authors who used virtual localization and therefore could not treat the case of a general vector bundle. We rely on other techniques which use a general wall-crossing framework of D. Joyce to study these. Our methods rely on existence of a Lie algebra coming from vertex algebras constructed out of topological data. I will explain how these arise naturally in the Quot-scheme setting and how one can obtain explicit invariants and study their symmetries.
Time and place: Nov. 18, 2021 2:15 PM4:00 PM, NHA B1120
When does the Zariski topology determine a variety? This certainly does not hold for curves, and examples of Wiegand and Krauter show it is neither true for countable surfaces. The cardinality assumption is important: The reconstruction theorem says that two homeomorphic (normal, projective) varieties of dimension at least two over non-countable fields of characteristic zero  K and L (a priori different) are in fact isomorphic (as schemes).
I shall present my version (a slight simplification of the original proof) of the cluster of ideas leading up to the reconstruction theorem (and maybe a miniscule extension to positive characteristic)
Time and place: Nov. 4, 2021 2:15 PM4:00 PM, NHA B1120
Time and place: Oct. 14, 2021 2:15 PM4:00 PM, NHA B1120
Planar polypols - “polygons with curved sides” - were proposed by Eugene Wachspress as generalized algebraic finite elements. In order to define barycentric coordinates for polypols, he introduced the adjoint curve of a rational polypol. In recent work by physicists, positive geometries are defined as certain semialgebraic sets together with a meromorphic differential form called the canonical form. We show that a rational regular polypol gives a positive geometry and give an explicit expression for its canonical form in terms of the adjoint and boundary curves of the polypol. In the special case that the polypol is a convex polygon, we show that the adjoint curve is hyperbolic and describe its nested ovals. 
 
This talk is based on joint work with K. Kohn, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn,  M.-S. Sorea, and S. Telen.
Time and place: Oct. 7, 2021 2:15 PM4:00 PM, NHA B1120

Stable polynomials are a multivariate generalization of real-rooted univariate polynomials. This notion of stability for hypersurfaces can be extended to lower-dimensional varieties, giving rise to positively hyperbolic varieties. I will present results showing that tropicalizations of positively hyperbolic varieties are very special polyhedral complexes with a rich combinatorial structure. This, in particular, generalizes a result of P. Brändén showing that the support of a stable polynomial must be an M-convex set.

Time and place: Sep. 30, 2021 2:00 PM3:45 PM, NHA B1120

In a famous paper, Geir Ellingsrud and Stein Arild Strømme use the Atiyah-Bott localization theorem in equivariant cohomology to compute the number of complex twisted cubics on a complete intersection. Motivated by results from A1-homotopy theory there is a new way of doing such enumerative counts which works over an arbitrary base field, not only the complex numbers. Recently, Marc Levine proved a version of Atiyah-Bott localization for this new way of counting.

In the talk I will recall the classical Atiyah-Bott localization theorem and explain how one can use it in enumerative geometry. Furthermore, I will explain how this new way of counting works and present some results about twisted cubics on complete intersections counted this way. This is based on joint work with Marc Levine.

Time: Sep. 23, 2021 2:15 PM4:00 PM

For the second talk, I will talk about how to relate relative Gromov--Witten invariants with relative periods via relative mirror symmetry and, given a degeneration, how relative periods and (absolute) periods are related on the mirror side.  

Time and place: Sep. 16, 2021 2:15 PM4:00 PM, NHA B1120
For the first talk, I will talk about the story of Gromov--Witten theory on the A-side. Relative Gromov--Witten invariants play a central role in computing Gromov--Witten invariants via the degeneration formula. I will give a summary of some recent progress of basic structures in relative Gromov--Witten theory.
Image may contain: Forehead, Selfie.
Time and place: Sep. 9, 2021 2:15 PM4:00 PM, NHA B1120

A cohomology class of a smooth complex variety of dimension n has coniveau ≥c if it vanishes in the complement of a closed subvariety of codimension ≥c, and has strong coniveau ≥c if it comes by proper pushforward from the cohomology of a smooth variety of dimension ≤n−c. We show that these two notions differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties. This is joint work with Olivier Benoist.

Et bilde av Kristian Ranestad
Time and place: Sep. 2, 2021 2:15 PM4:00 PM, NHA B1120

A graded Artinian Gorenstein ring A is a quotient of a polynomial ring S with the apolar ideal of a homogeneous form. The Betti numbers of the resolution of A as an S-module are invariants to the homogeneous form. In joint work with Michal and Gregorz Kapustka, Hal Schenck, Mike Stillman and Beihui Yuan, we use these Betti numbers to describe a stratification of the space of quartics in four variables.

Time and place: May 9, 2019 2:15 PM4:00 PM, NHA B1120
Time and place: Apr. 11, 2019 2:15 PM4:00 PM, NHA B1120
Time and place: Apr. 11, 2019 2:15 PM4:00 PM, NHA B1120