Foredragsholdere, Nasjonalt algebramøte 2016
Gunnar Fløystad, Bergen: Rigid ideals by deforming letterplace ideals
Abstract: Given a finite poset P, we form the polynomial ring k[xp,yp]p∈P , and the monomial ideal L(2,P) generated by monomials xpyq. (These ideals are precisely those edge ideals of bipartite graphs, which are Cohen-Macaulay ideals.) We get a complete understanding of all polynomial ideals which specialize to the monomial ideal L(2,P), when the Hasse diagram of P is a rooted tree:
- The deformations of L(2,P) are unobstructed
- We compute explicitly the full deformation ideal J(2,P) of L(2,P).
- The full deformation family has a polynomial ring as a base ring.
- The ideal J(2,P) defining the full deformation ideal, is rigid.
- When these ideals are on a Hilbert scheme they are smooth points, and we describe the general point on the Hilbert scheme.
- In simple example cases J(2,P) is the ideal of maximal minors of a generic matrix, and the Pfaffians of a skew-symmetric matrix.
This is joint work with Amin Nematbakhsh.
Olav Gravir Imenes, HiOA: Illuminating non-commutative algebraic geometry by using input from electroweak theory.
Abstract: Let A be an associative k-algebra. A point in the non-commutative affine sheaf Sch(A), is a simple module ρ0 : A → Endk(V). Understanding the differentiable structure, or the dynamics, of Sch(A), requires the introduction of a non-commutative phase space functor, an injecitve homomorphism i : A → Ph(A), with a universal derivation d : A → Ph(A) in the category of A-algebras, together with the choice of a dynamical structure, A(σ), a quotient of Ph(A) containing A and with a derivation δ, extending d.
A representation ρ0 : A(σ) → Endk(V) corresponds to a measurement of the parameters of A together with a tangent of A(σ) at the point ρ0. The moduli space of isomorphism classes, Simp(A(σ)), of finite dimensional (simple modules), ρ : A(σ) → Endk(V ), is therefore a kind of phase space of Sch(A). The choice of σ leads to a specific derivation, δ of A(σ), acting as a vector field [δ] on the moduli space Simp(A(σ)), a kind of a time-evolution.
In this talk we will use a method of modeling physics, due to Laudal (Geometry of Time-Spaces, World Scientific, 2011), where the generators of the algebra A(σ) represent the parameters of the physical phenomena we are interested in. We will be inspired by electroweak theory and shall therefore consider algebras generated by position, momentum, electric charge and weak charge operators. We will have to include infinite dimensional representations, and specialise to stable points. Borrowing notions and examples from physics give us a better understanding of parts of non-commutative algebraic geometry, and in some cases a better understanding of physics. But borrowing notions from physics, and comparing the resulting model to physics, is obviously not the same as doing physics, even though, in some cases, it looks remarkably similar.
Riccardo Moschetti, Stavanger: Space of non uniform points of projections
Abstract: Fix a projective surface S of degree d greater than two and consider a point P not lying on this surface. The projection from P to a plane defines a finite map from S to the plane, and then we can consider the monodromy of such a map. We call P uniform if the monodromy group is the whole symmetric group Sd, otherwise P is called non-uniform. Our goal is to study the codimension of the locus of non-uniform points. This is a joint work with A. Cuzzucoli and M. Serizawa started in the PRAGMATIC school 2016.
Andrea Tobia Ricolfi, Stavanger: Local contributions to Donaldson-Thomas invariants
Abstract: Donaldson-Thomas invariants are integers counting stable sheaves on Calabi-Yau 3-folds Y. In the rank 1 case, they can be interpreted as counting holomorphic curves on Y in a fixed homology class. We will define the "contribution" of a single smooth curve on Y to the full DT invariant of its homology class. We show how to compute these contributions and exhibit an explicit wall-crossing formula relating these local DT invariants to the local stable pair invariants, defined and computed by Pandharipande-Thomas.
Felipe Rincon, Oslo: Tropical Ideals
Abstract: The past few years have seen a significant effort to give tropical geometry a solid algebraic foundation. In this talk I will introduce tropical ideals, which are ideals over the tropical semiring that are "matroidal". I will discuss joint work with Diane Maclagan studying some of their main properties, and in particular showing that their underlying varieties are always finite polyhedral complexes.
Sammy A. Soulimani, Stavanger: On the space of abelian four-folds of type (1,1,2,2)
Abstract: There is a natural involution on the moduli space of polarized abelian varieties of dimension 4 and type (1,1,2,2) (consequence of a result by Birkenhake and Lange). During this talk, I will show how to construct two different divisors of this moduli space and study their invariance with respect to the said involution. The aim is to understand the Picard group of this moduli space as a first step in studying its geometry. This is a joint work with P. Porru which began during the Pragmatic summer school 2016.
Bernd Sturmfels, Berkeley: Cameras and their Distortions
Abstract: We discuss two current projects at the interface of computer vision and algebraic geometry. Work with Joe Kileel, Zuzana Kukelov and Tomas Pajdla introduces the distortion varieties of a given projective variety. These are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions, the case of most interest for modeling cameras with image distortion. Work with Jean Ponce and Matthew Trager develops multi-view geometry for algebraic cameras that are represented by congruences in the Grassmannian of lines in 3-space.