Algebra and Geometry

Karl Rökaeus

We begin by giving the definition and basic properties of this ring (the Grothendieck ring of varieties). We then proceed to discuss some known structure result (e.g., existence of zero-divisors), and some conjectured ones.

Carel Faber

I will attempt to give a survey of what is known about the cohomology of moduli spaces of curves and of abelian varieties. I will discuss both algebraic and non-algebraic classes.

Nils Henry Rasmussen

Let F_q be the finite field with q elements. A code C over F_q is defined to be a subset of (F_q)^n where n is a positive integer. The aim in coding theory is to have the elements of C "spread out" such that the number of coordinates in common between any two elements is as small as possible. In this talk, we will attempt to define such a code using scrolls.

Let X be a smooth, projective curve over the algebraic closure of F_q, and let E be a semistable vector bundle of rank r defined over F_q, where r is a positive integer, and such that the tautological line bundle O(1) on the scroll P(E) is very ample. We then embed P(E) into projective space and let a choice of the coordinates of the F_q-rational points define the code C.

I will give an example of such a curve and a scroll, including how to find semistable vector bundles. We will also get to see what P(E) and O(1) look like and how we find the equations for the embedding of P(E) into projective space.

Dan Petersen

Inside the moduli space M_{2,n} of n-pointed curves of genus two, we can consider the locus of curves which are double covers of curves of genus one. These are called bi-elliptic. It turns out that any bi-elliptic curve of genus two also covers a second elliptic curve. We describe how the genus two curve can be recovered from data coming from the two covered elliptic curves, how this gives a description of (the normalization of) the bi-elliptic locus as essentially a fibered product of the moduli space of elliptic curves with full level two structure with itself, and try to indicate implications for some local systems on the bi-elliptic locus.

Andrea Hofmann

Let C be a curve of genus 2, linearly normal embedded in P^(d-2) by a complete linear system of degree d. There exists exactly one linear system g^1_2 on C, and this is the canonical divisor K_C. The family of g^1_3's on C is two-dimensional and this surface is isomorphic to the Jacobian of C.

Rational normal scrolls arise from linear systems on C in a natural way: The unique g^1_2(C) gives rise to a surface scroll S, and each g^1_3(C) gives rise to a threefold-scroll. Now, by construction, C is contained in S and in each threefold-scroll V. Thus, the ideals I_S and I_V are contained in I_C. It is known that all the ideals involved are generated by quadrics. One interesting issue is now to investigate if the ideal I_C is generated by the quadrics in I_S and I_V where V is a threefold-scroll that does not contain S. The answer is positive for d=6,7,8. In this talk, we will sketch the proofs of these results and also look at minimal free resolutions of I_C and discuss if the i'th syzygies of I_C can be generated by the i'th syzygies of I_S and the i'th syzygies of the ideals of all the threefold-scrolls that contain C.

Qimh Xantcha

Numerical rings were invented/discovered by the celebrated Professor Ekedahl in 2002 (he uses himself the word "introduced", but humility has always been his among his chief virtues); they may best be described as rings with binomial coefficients. These rings turn out to come equipped with an array of pleasant properties (and are fundamental to our study of polynomial functors, for those who know what that is).

Patrik Norén

I will explain what algebraic statistics is and give some motivating examples. A certain type of statistical model correspond to certain toric ideals. The goal of algebraic statistics is then to use algebraic methods on the ideals to answer statistical questions, for example hypothesis testing.

Valentina Chapovalova

Permutation modules are essential in the study of representation theory of the symmetric group S_n. There is a natural action of a certain direct sum of symmetric groups on a permutation module. This gives rise to a decomposition of the permutation module into special submodules (which correspond to irreducible modules for certain semigroup generalization of S_n). A natural problem is to find multiplicities of Specht modules in these special submodules. For some cases, the solution is known and will be presented.

Jan-Magnus Økland

The following is a classical result due to G. Castelnuovo in the formulation of [ELMS1989]:

Let $C$ be an irreducible curve of degree $d$ and geometric genus $g$ in $\PP^r$. If $C$ has only finitely many $(2r - 2)$-secant $(r - 2)$-planes then their number is $C(d,g,r)$ (counted with multiplicities):

\[C(d,g,r)=\sum_{i=0}^{r-1} \frac{(-1)^i}{r-i} {d-r-i+1 \choose r-1-i} {d-r-i \choose r-1-i} {g \choose i}.\]

For fixed $r$, $C(d,g,r)$ is a polynomial in \(d, g\). I refer to this as the $r$-th Castelnuovo polynomial.

These are interesting polynomials that appear in the proof for low Clifford dimensions $r$ of a conjecture stated in [ELMS1989].

I'll talk about work with A. Hoffman where we pushed the proof from \(r\leq g\) to \(r \leq 12=\) and strategies for the conjecture proper.

[ELMS1989] David Eisenbud, Herbert Lange, Gerriet Martens, and Frank-Olaf Schreyer. The Clifford dimension of a projective curve. Compositio Math., 72(2):173–204, 1989

Mehdi Tavakol

I will give an introduction to the tautological ring of moduli space of curves as a subalgebra of their rational Chow rings. Then there will be a discussion about the conjectural structures of these rings for the Deligne-Mumford compactification of M_{g,n} and other partial compactifications.

Johan Björklund

Two real algebraic knots are said to be rigidly isotopic if there exists a path in the parameter space of real algebraic curves connecting them such that the curve is at each moment a knot. A classification up to rigid isotopy for knots of genus 0 and degree < 6 will be presented. Viros encomplexed writhe is then a complete knot invariant for these degrees. This is not true for higher degrees, a counterexample in degree 6 will be shown. An algorithm to construct real algebraic knots of given degree/writhe is presented.

Alexander Engström

Building on ideas from graph theory, the dichtonomy between structure and randomness have been used for example in number theory to prove that there are arbitrary long arithmetic progressions in the primes. To prove Szemerdi type theorems about ideals we need a notion of random ideals and how to compactify them. In a sense, we want to ask "what are the real numbers?" if the rational numbers are a natural class of ideals. We will build on work by Diaconis, Janson, and Kallenberg on probability theory; and by Dochtermann and Engström on resolutions of monomial and toric ideals. This is partly joint work with Oscar Andersson Forsman.

Georgios Dimitroglou Rizell

We provide an introduction to different types of knottedness of Lagrangian submanifolds and some standard results in the area. We also sketch some techniques involving pseudo-holomorphic curves to construct isotopies of the Lagrangian submanifolds.

Henning Lohne

Let S=k[x_1,...,x_n] be the polynomial ring. Kohji Yanagawa introduced the notion of square-free N^n-graded S-modules. Every square-free module has support on a simplicial complex. There is an analogy between certain square-free modules on simplicial graphs and line bundles (divisors) on curves. We illustrate this by proving a corresponding Riemann-Roch theorem for certain square-free Cohen--Macaulay modules on a simplicial graph. Furthermore, if the simplicial graph is 2-Cohen-Macaulay we define the gonality for certain square-free modules, and show some similarities and differences from the classical theory of algebraic curves.

Isac Hedén

The aim of my talk is to give a geometric argument which shows that the Makar-Limanov invariant of Russell's hypersurface $X$ is nontrivial and thus $X$ is only diffeomorphic, but not isomorphic to affine 3-space. Russell's hypersurface is the affine variety in $\mathbb C^4$ defined by $x+x^2y+z^3+t^2=0$, and the Makar-Limanov invariant is by definition equal to the intersection of all kernels of locally nilpotent derivations on its coordinate ring. To see why the ML-invariant is nontrivial, we consider $X$ as an open part of a blow up $Z\longrightarrow\mathbb C^3$, and show that every $\mathbb C^+$-action on $X$ descends to $\mathbb C^3$. This in turn follows from the fact that any nontrivial $\mathbb C^+$-action on $X$ induces a nontrivial $\mathbb C^+$-action on $W:=Sp(B)$, where $B$ is the graded algebra associated to the filtration of $\mathcal O(X)$ given by the pole order along the divisor $Z\setminus X$ at infinity. The induced corresponding locally nilpotent derivati on is homogenous, and we prove that the degree of any such derivation is negative - the result concerning the ML-invariant follows from this.

Khalid Rian

Witt's discovery of a non-classical simple Lie algebra started the interest in modular Lie algebras. The Witt algebra, as it has been named, was distinguished from the classical Lie algebras by the fact that it is not associated to a smooth algebraic group. Subsequently yet more non-classical Lie algerbas were discovered and a new class of restricted simple Lie algebras was established and distinguished by the name of Cartan. They have been classified into four categories: Contact Lie algebras K(n), Hamiltonian Lie algebras H(2n), special Lie algebras S(n) and Witt-Jacobson Lie algebras W(n).

The interest in modular Lie algebras was motivated by the Kostrikin- Shafarevich Conjecture, which states that a finite dimensional restricted simple Lie algebra is either classical or of Cartan type. The conjecture was proved for p > 5, where p denotes the characteristic of the ground field.

In the talk, I will introduce the Witt-Jacobson Lie algebras W(n). If time permits, I will give a very brief summary of the representation theory of the smallest Witt-Jacobson Lie algebra W(1).

Christine Jost

The story of operads begins in the 70's in Chicago, where a group of topologists (May, Stasheff, Boardman, Vogt et.al.) used them to study iterated loop spaces. This was followed by a long silence until 1994, the year of the renaissance of operads. Ginzburg and Kapranov succeeded in generalizing Koszul duality—a well known theory for associative algebras—to operads. Since then, operads have have never ceased to find applications in various fields of mathematics, mainly in algebra, topology and mathematical physics.

This talk focuses on a small detail in this long story: In topology and homological algebra, one often finds that the well-known structures of associative algebras and Lie algebras are not the right notions - one needs the more general notions of an associative algebra or Lie algebra up to homotopy. Even other types of algebras have their respective up to homotopy equivalent. And here operads enter the scene. They not only allow us to specify the notion "type of algebra". This language allows us also to get some idea about the general nature of algebras up to homotopy. And furthermore, Koszul duality gives us an algorithm that allows us to compute the algebra up to homotopy for many common types of algebras.

Andreas Holmström

A cohomology theory is a tool for constructing invariants of spaces, and it typically takes the form of a functor from some category of spaces to the category of abelian groups. The word "spaces" can refer to many different things, such as manifolds, CW complexes, algebraic varieties, and schemes. In algebraic geometry, there are many different kinds of cohomology, and many famous unsolved problems are formulated in terms of cohomology. In this talk I will present some examples of such problems, and try to indicate how the notion of simplicial sheaves can work as a simplifying and unifying language for different kinds of cohomology.