Time and place: Dec. 1, 2022 12:30 PM2:15 PM, NHA B1120
Specialization of (stable) birational types is an important tool when studying (stable) rationality in families. A crucial ingredient is to cook up one parameter degenerations such that the limit has certain combinatorial and geometric properties. Nicaise-Ottem studied these questions for hypersurfaces in algebraic tori, and used tropical geometry to construct degenerations that would have been hard (impossible) to construct geometrically. Even after these are constructed one must carefully study the limit in order to apply specialization techniques, this involves both combinatorics and questions about variation of stable birational types. I will talk about the specialization technique in the setup of Nicaise-Ottem, explain some natural questions that appear through the combinatorics, and give some positive results in this direction.
Time and place: Dec. 8, 2022 2:15 PM4:00 PM, NHA B1120


Time and place: Nov. 24, 2022 2:15 PM4:00 PM, NHA B1120
The variety of sums of powers, VSP(F, r) of a homogeneous form F of rank r is the closure in the Hilbert scheme of apolar schemes of length r. A bad limit is a scheme in the closure that is not apolar to F. I will discuss examples of bad limits, including examples for quadrics found by Joachim Jelisiejew that contradicts earlier results on polar simplicies. This is report on work in progress with Jelisiejew and Schreyer and with Grzegorz and Michal Kapustka.
Time and place: Nov. 17, 2022 2:15 PM4:00 PM, NHA B1120
Counterexamples to the integral Hodge conjecture can arise either from torsion cohomology classes (as in Atiyah's and Hirzebruch's original counterexample from 1961) or from non-torsion classes (as first seen in Kollár's counterexample from 1991). After Voisin proved the IHC for uniruled threefolds, Schreieder found a unirational fourfold where the IHC fails. His construction of a non-algebraic Hodge class relies on abstract arguments with unramified cohomology. It was an open question whether this class is of torsion type. In this talk, I want to explain a new method that gives an explicit geometric description of the unramified cohomology class appearing in his argument. In particular, this approach allows to prove that Schreieder's unirational counterexample is of torsion type.
Time and place: Nov. 10, 2022 2:15 PM4:00 PM, NHA B1120

As a consequence of the S-duality conjecture, Vafa and Witten conjectured certain symmetries concerning invariants derived from spaces of vector bundles on a closed Riemannian four-manifold. For a smooth complex projective surface X, a satisfying mathematical definition of Vafa-Witten invariants has been given by Tanaka and Thomas. Their invariants are a sum of two parts, one of which can be defined in terms of moduli spaces of stable vector bundles on X. Focusing on this instanton part of the VW invariants one can ask how it changes under blowing up the surface X. I will discuss joint work with Oliver Leigh and Yuuji Tanaka that answers this question.

Time and place: Nov. 3, 2022 2:15 PM4:00 PM, NHA B1120

I will explain how a recent “universal wall-crossing” framework of Joyce works in equivariant K-theory, which I view as a multiplicative refinement of equivariant cohomology. Enumerative invariants, possibly of strictly semistable objects living on the walls, are controlled by a certain (multiplicative version of) vertex algebra structure on the K-homology groups of the ambient stack. In very special settings like refined Vafa-Witten theory, one can obtain some explicit formulas. For moduli stacks of quiver representations, this geometric vertex algebra should be dual in some sense to the quantum loop algebras that act on the K-theory of stable loci.

Time and place: Oct. 20, 2022 2:15 PM4:00 PM, NHA B1120
Time and place: Oct. 6, 2022 2:15 PM4:00 PM, NHA B1120

Abstract (PDF)

Time and place: Sep. 29, 2022 2:15 PM4:00 PM, NHA B1120

In 80s Weibel observed that K-theory is homotopy invariant on Fp-schemes up to p-torsion. His main tool was the action of the ring Witt vectors on nil-K-groups: NKi(R) = Ker(Ki(R[t]) → Ki(R)). We will revisit the proof and check that the same result holds for all finitary localizing invariants.

Time and place: Sep. 22, 2022 2:15 PM4:00 PM, NHA B1120

I will explain how motivic homotopy theory can be used to attack problems regarding finite projective modules over smooth affine k-algebras. I will recall in particular the foundational theorem of Morel and Asok-Hoyois-Wendt, and the construction of the Barge-Morel Euler class. Time permitting, I will explain recent progress on Murthy's splitting conjecture.

Time and place: Sep. 15, 2022 2:15 PM4:00 PM, NHA B1119

Abstract (PDF)

Time and place: Aug. 25, 2022 2:15 PM4:00 PM, NHA B1119

I will discuss the question in the title. This is joint work with Alex Degtyarev and Ilia Itenberg. This will be a talk involving very classical topics in algebraic geometry. I will try to make the talk accessible to students at master- and PhD level.

Nordfjordeid landscape, sky, fjord, mountains
Time and place: June 20, 2022June 24, 2022, Sophus Lie Conference Center, Nordfjordeid, Norway.

Nordfjordeid Summer school 2022

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: May 13, 2020 9:30 AMMay 15, 2020 5:00 PM

The 5th Scandinavian Gathering Around Remarkable Discrete Mathematics