Algebra and Geometry

Johan Björklund

I am interested in examining real algebraic varieties in RP^3, CP^3 from a topological point of view. A natural problem is which topological objects can be realized by a given degree. I am also interested in examining and classifying knots, seen as real algebraic curves, up to rigid isotopy, that is an isotopy where the knot is at all times a real algebraic curves.

Martin Blomgren

Valentina Chapovalova

Representation theory of Lie algebras, Categorifications of representations. I have looked at functors arising from categorifications of sl_2-modules via category O.

Georgios Dimitroglou Rizell

I am currently working with the classification of Legendrian embeddings of 2-spheres in R^5 with its standard contact structure. I am using techniques from Symplectic Field Theory, which is a new and interesting field in geometry.

Alexander Engström

Combinatorics, Discrete Geometry, Algebraic Topology, Combinatorial Commutative Algebra, Algebraic Statistics

Eric Emtander

So far I have mainly worked on problems related to Stanley-Reisner rings. In particular I am interested in edge ideals of hypergraphs. Semigroups and semigroup rings are quite new interests of mine and I am currently working on some questions about numerical semigroups.

Albin Eriksson-Östman

Symplectic- and Contact geometry, symplectic field theory.

Christian Grundh

I'm interested in the ring of vector valued Siegel modular forms of genus 2. Primarily in calculating their Fourier coefficients.

Katharina Halbhuber

Algebraic Geometry and Commutative Algebra

Isac Hedén

(Affine) Algebraic Geometry. Locally nilpotent derivations. Algebraic group actions

Uffe Heide-Jørgensen

My research is on the permanent of a quadratic matrix; the permanent defines a variety in projective space, and I'm supposed to study the dual variety. Being accepted as a PhD-student in February, I'm not very far in my research. At this point I'm primarily trying to get a grip on some important notions in algebraic geometry.

Andrea Hofmann

I am interested in curves lying on rational normal scrolls and their syzygies. A curve C lies in a natural way on rational normal scrolls which arise from linear systems on the curve. In general, if a variety X is contained in some "bigger" variety Y, then one can ask which syzygies of X are spanned by syzygies of Y. These syzygies are called geometric syzygies. One interesting issue is then to consider if all syzygies of the curve are spanned by geometric syzygies.

Andreas Holmström

I am interested in various questions in algebraic and arithmetic geometry, mainly related to cohomology theories.

Christine Jost

Operads are algebraic structures that model (e.g. associative or Lie) algebras. Its generalizations such as properads, props or wheeled operads model more general algebraic structures including co- and bialgebras, or may also code traces. Besides that, it has been shown that these generalizations of operads even may used to code structures in differential geometry, e.g. Poisson structures.

Henning Lohne

I am interested in multigraded Cohen-Macaulay modules. I have studied some special kind of multigraded S-modules that are called square-free. The annihilator of a square-free module is a square-free monomial ideal that corresponds to a simplicial complex. In this case we say that the module has support on that simplicial complex. There is an analogy between maximal Cohen-Macaulay square-free modules with support on simplicial graphs and line bundles (divisors) on curves. I study the similarities and differences from classical theory of algebraic curves using combinatorial tools. See the abstract of my talk for more details.

Recently, I have also started to study the structure of minimal free resolutions of multigraded Cohen-Macaulay modules. Cellular resolutions of monomial ideals is a natural and combinatorial way of constructing resolution of monomial ideals. I am working with resolution of other modules and searching for similar combinatorical ways for constructing resolutions (at least in low projective dimension).

Patrik Norén

Combinatorics and Algebraic Statistics.

Nils Henry Rasmussen

My main two topics of interest are algebraic coding theory and Brill-Noether theory for curves on K3 surfaces.

In algebraic coding theory, I work on defining codes from scrolls. Given a smooth, projective curve over the algebraic closure of F_q, where F_q is the finite field with q elements, and given a vector-bundle E of rank r on X, we consider the scroll P(E) (i.e., over each point P on X, we have r-1-dimensional projective space where the variables are local generators of E). We are interested in the cases where X is defined over F_q (i.e., its local equations have coefficients in F_q), and where E is defined over F_q as well (i.e., its generators are from the function field of C over F_q). We demand that E is semistable, and that the tautological line-bundle O(1) on P(E) (locally defined as all linear polynomials in the local variables of P(E)) is very ample. We then embed P(E) into projective space using O(1), and we can then define a code by regarding a set of coordinates for the F_q-rational points of the embedded P(E). A code is a subset of (F_q)^n for some positive integer n, and we want the number of coordinates in common between any two code-words must be as small as possible. In the embedded version of P(E)—suppose it is (k-1)-dimensional projective space it is embedded into—we choose a set of coordinates for each F_q-rational point and define a k-1 x n matrix (where n is the number of F_q-rational points) where the columns are the coordinates of the given points. The row-space of this matrix defines a subset of (F_q)^n.

My problem consists of finding the minimum distance of this code (n minus the smallest number of coordinates two vectors have in common). A special case is when we are just embedding the curve itself into k-1-dimensional projective space using a very ample divisor A. Then the minimum distance is at least deg(A)-g. More generally, we can just take the global sections of any divisor A and evaluate in the F_q-rational points on the curve X. However, using a more geometrical approach might yield better results.

My second research interest is Brill-Noether theory for curves on K3 surfaces, this time over the complex numbers. Given a smooth curve C on a K3 surface S, I am interested in finding "how many" g^1_d's (complete linear systems |A| of dimension 1 and degree d) we have on C. It has been proven that for general curves (not necessarily on K3 surfaces), the dimension of the moduli space of (not necessarily complete) linear systems of dimension r and degree d on C, denoted W^r_d(C), is g-(r+1)*(g-d+r). This was proved using deformation theory, but in 1986, Lazarsfeld gave another approach of finding this dimension by considering curves on K3-surfaces (since we can find curves of any genus on K3-surfaces, and we are only considering general curves). In the cases where r=1, when the gonality of C is less than maximal, this "expected dimension" g-(r+1)*(g-d+r) takes on negative values for small d, and we obviously can't assume that W^r_d(C) has the "expected dimension". My aim is to follow up on Lazarsfeld's methods and find the dimension of W^r_d(C) for these cases.

Medhi Tavakol

I am studying the tautological rings of moduli spaces of pointed curves in low genus.

Qimh Xantcha

We are currently working on different notions of polynomiality, to see how this concept may be extended to both maps between modules (not homomorphisms, obviously), and functors on module categories.

Khalid Rian

Through the last two years I have been working with problems in representation theory of Lie algebras in positive characteristic. My main research interests involve Lie algebras of Cartan type with a particular interest in Witt-Jacobson Lie algebras W(n).

Unlike the situation for classical Lie algebras, the representation theory of the reduced enveloping algebra $U_\chi(g)$ of a restricted Lie algebra g of Cartan type, is not well-known. (Here $\chi\in g^*$ denotes a functional.) Several efforts have been effective among a number of non-classical Lie algebras, but they have been far from successful in general. A classification of the irreducible $U_\chi(W)-$modules for the Witt algebra W = W(1) was first given by Chang in 1941. Many years later, a new and greatly simplified treatment was developed by Strade.

In my research, I have constructed a classification of the extensions of the irreducible $U_\chi(W)$-modules having character $\chi$ of height at most 1.

My advisor is Prof. Jens Carsten Jantzen.

Karl Rökaeus

Algebraic Geometry, in particular motivic integration.

Jan-Magnus Økland

Computational Algebra in Algebraic Geometry:

- maintain the schubert package for calculation in intersection theory/schubert calculus originally written for maple by Katz/Strømme (Now there also is a basic schubert2 for Macaulay2 by Grayson/Stillman)

(with S. A. Strømme, Bergen):
- work on adding support for equivariant intersection theory in schubert/schubert2.

(with the help of L. Gatto, Torino):
- base schubert calculus computation on ideas of Gatto and Laksov (these extend naturally to equivariant schubert calculus).

Lately have also worked in the area of special divisors on curves (with A. Hofmann, Oslo).
- have computer proof for low Clifford dimensions of a conjecture stated in a 1989 paper by Eisenbud, Lange, Martens, and Schreyer.