First Workshop 2015

Dates: 10-13 August 2015

Organizers
  • Gavin Brown (Warwick)
  • Damian Maingi (Nairobi)
  • Jared Ongaro (Nairobi)
  • Balazs Szendroi (Oxford)
Monday 10 August
9am Opening ceremony
9.30am Gavin Brown (University of Warwick): Riemann-Roch and graded rings I
11am Ben Davison (EPFL Lausanne): The cohomology of the space of semistable representations of a quiver I
2pm Andre Saint Eudes Mialebama Bouesso (AIMS SA): Groebner bases over some quotient rings
3.30pm Damian Maingi (University of Nairobi): Vector bundles associated to monads on multiprojective spaces

Tuesday 11 August
9am Gavin Brown (University of Warwick): Riemann-Roch and graded rings II
10.30am Ben Davison (EPFL Lausanne): The cohomology of the space of semistable representations of a quiver II
1.30pm Renzo Cavalieri (Colorado State University): From Hurwitz Numbers to Tropical Geometry I
2.30pm Junior talks
4pm David Stern: Algebraic geometry in Africa? (talk and discussion)

Wednesday 12 August
9am Balazs Szendroi (University of Oxford): Hilbert schemes of points on surfaces I
10.30am Renzo Cavalieri (Colorado State University): From Hurwitz Numbers to Tropical Geometry II
1.30pm Jared Ongaro (University of Nairobi): Plane Hurwitz numbers
3pm Ben Kikwai (ICTP): Refined node polynomials via long edge graphs

Thursday 13 August
9am Balazs Szendroi (University of Oxford): Hilbert schemes of points on surfaces II
10.30am Praise Adeyemo (University of Ibadan): Cohomological Consequences of the Pattern Map
1.30pm Junior talks/discussions/next steps/etc

Abstracts

Praise Adeyemo: Cohomological Consequences of the Pattern Map
Billey and Braden defined maps on flag manifolds that are the geometric counterpart of permutation patterns. A section of their pattern map is an embedding of the flag manifold of a Levi subgroup into the full flag manifold. We give two expressions for the induced map on cohomology. One is in terms of generators and the other is in terms of the Schubert basis. We show that the coefficients in the second expression are naturally Schubert structure constants and therefore positive. These formulas also hold for K-theory, and generalize known formulas in type A for cohomology and K-theory (This is a joint work with Frank Sottile).

Balazs Szendroi: Hilbert schemes of points on surfaces
I will discuss aspects of the Hilbert scheme of points on algebraic surfaces. After a general introduction, I will concentrate on some Hilbert schemes arising in the context of the McKay correspondence, and some recent topological results concerning these. 
References:
Diane Maclagan: Notes on Hilbert schemes 
Ezra Miller: Hilbert scheme of points in the plane

Andre Saint Eudes Mialebama Bouesso: Groebner bases over some quotient rings
We explain the main idea of the theory of Groebner bases and we propose a method for computing a Groebner basis over some quotient rings that we called "dual Bezout domain". Our method will cover other rings such as: principal ideal rings, valuation rings, Dedekind rings, Gaussian rings, ....

Ben Davison: The cohomology of the space of semistable representations of a quiver
Although the moduli space of semistable representations of a quiver (with no relations) is quite easy to define, it turns out that the cohomology of this space satisfies several remarkable theorems, proven in recent years.  I will explain some of the geometry that underlies these results, specifically the construction of equivariant cohomology directly from smooth moduli spaces, and the subsequent role of the decomposition theorem of Beilinson, Bernstein and Deligne.

Gavin Brown: Riemann-Roch and graded rings
I'll review standard material on applying Riemann-Roch for a (very) ample divisor on a curve to predict the homogeneous coordinate rings of the corresponding embedding, and then move on to harder questions for (some or all of) K3 surfaces, Fano 3-folds and Calabi-Yau 3-folds.
Links:
Colloquial introduction: http://arxiv.org/abs/math/0202092
Fano 3-folds in codimension 4: http://arxiv.org/abs/1009.4313
General overview of graded ring methods: http://homepages.warwick.ac.uk/~masda/3folds/Ki/Ki.pdf


Ben Kikwai (ICTP): Refined node polynomials via long edge graphs
The generating functions of the Severi degrees for suciently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. In the talk I shall present some results on the multiplicativity of the refined Severi degrees for P^2 and rational ruled surfaces.