13.00 -- 14.00 Peter Symonds (Manchester), Degree Bounds on Homology and a Conjecture of Derksen.
(work with M. Chardin)
We give a counterexample to a conjecture of Derksen concerning degree bounds on syzygies of rings of invariants. We also prove a modified version of the conjecture and some general results giving bounds on syzygies.
14.30 -- 15.30 Diane Maclagan (Warwick), Tropical schemes and valuated matroids
Tropical geometry is geometry over the "tropical semiring",
with multiplication replaced by addition and addition replaced by
minimum. This replaces varieties by polyhedral complexes, and cycles
by weighted complexes. Recent work of J. and N. Giansiracusa
suggested how to construct tropical schemes, as well as varieties and
cycles. I will describe joint work with Felipe Rincon that elucidates
the connection of this theory to the combinatorial notion of valuated
matroids, and will indicate some resulting commutative algebra
questions.
16.00 -- 17.00 Rodney Sharp (Sheffield), Some uses of the Frobenius skew polynomial ring in prime-characteristic commutative algebra
This year sees the 25th anniversary of the appearance of M. Hochster's and C. Huneke's first paper on tight closure, which reinvigorated the study of prime-characteristic commutative algebra. Modules with 'Frobenius actions' have played a significant role; such a module is just a left module over a Frobenius skew polynomial ring.
Let $R$ be a commutative (Noetherian) local ring of prime characteristic that is $F$-pure. The talk will explain how properties of the graded annihilators of left modules over the Frobenius skew polynomial ring over $R$ lead naturally to a finite set of prime ideals of $R$ that has connections with tight closure and related concepts.
1.00 -- 2.00 Maxim Smirnov (ICTP), Dubrovin's conjecture for IG(2,6).
For a smooth projective algebraic variety, according to a
conjecture of Dubrovin, the semisimplicity of quantum cohomology is
related to the existence of full exceptional collections in the derived
category of coherent sheaves. I will start by giving a general
introduction to quantum cohomology and, eventually, talk about a recent
joint work with S. Galkin and A. Mellit on Dubrovin's conjecture for the
symplectic isotropic Grassmannian IG(2,6). This appears to be the
simplest case where one needs to work with the big quantum cohomology to
formulate the conjecture.
2.30 -- 3.30 Alessio Corti (Imperial), On the Fano/LG correspondence for surfaces.
I will state two conjectures exploring mirror symmetry for surfaces and its implications for classification, and summarise the evidence available so far.
4.00 -- 5.00 Alexander Kasprzyk (Imperial), Maximally-mutable Laurent polynomials.
In this talk I will describe a special class of Laurent polynomials, which we call "maximally-mutable". These Laurent polynomials arise naturally in the study of Fano manifolds via mirror symmetry. In particular, I will explain why in dimension two, the rigid maximally-mutable Laurent polynomials correspond exactly, under mirror symmetry, with the 10 deformation families of smooth del Pezzo surfaces. A similar result holds in dimension 3, where the rigid maximally-mutable Laurent polynomials supported on a reflexive polytope correspond precisely with the 98 deformation families of smooth Fano 3-folds with very ample -K.
1--2.30 Y. Zarhin (Penn State) Hodge groups and Frobenius endomorphisms with special reference to K3 surfaces and cubic fourfolds.
We discuss interrelations between the computation of Hodge groups of complex K3 surfaces (and cubic fourfolds) and multiplicative independence of eigenvalues of Frobenius of of ordinary K3 surfaces (and cubic fourfolds) over finite fields.
3--4 M. Shen (Cambridge) On the Chow ring of certain hyperkahler fourfolds.
For the variety of lines on a cubic fourfold and the Hilbert scheme of two points on a K3 surface, we construct an explicit decomposition of the Chow ring and show that this decomposition is compatible with the multiplicative structure of the Chow ring. This is joint work with Charles Vial.
4--5 J. Ayoub (Zurich) Motives of rigid analytic varieties and nearby motives
We explain how to adapt the constructions of Voevodsky and Morel-Voevodsky to define triangulated categories of rigid analytic motives. In an equal-characteristic zero situation, we establish a strong link between rigid motives over a non-Archimedean field and motives over the residue field. This gives a natural construction of nearby motives
14.00 Charles Vial (Cambridge) Algebraic cycles and fibrations.
The theory of algebraic cycles can be viewed as a homology theory for
schemes. Chow groups of algebraic cycles are related to algebraic K-theory
via the Riemann-Roch theorem. A natural question is to ask,
given a fibration, i.e. a dominant morphism, from a smooth variety X to a
smooth variety B, how the Chow groups of X relate to the Chow groups of B
and of the fibers of the fibration. I will recall what Chow groups are and
give answers to this question when the Chow groups of the fibers are small,
in some sense. Time permitting, I will give motivic consequences of such
computations.
15.00 Arend Bayer (Edinburgh) Birational geometry of moduli of sheaves on K3s via Bridgeland stability.
I will explain recent results with Emanuele Macri, in which we use wall-crossing for
Bridgeland stabiltiy conditions to systematically study the birational geometry of moduli
of sheaves on K3 surfaces. In particular, we obtain descriptions of their nef cones via the Mukai lattice of the K3, their moveable cones, their divisorial contractions, and obtain counter-examples to various conjectures in the literature. We also give a proof of a well-known conjecture about Lagrangian fibrations (due to Hassett-Tschinkel/Huybrechts/Sawon). These results are new even for Hilbert schemes on the quartic surface in P^3.
Our method is based on a natural map from the space of stability conditions to the movable cone of the moduli space.
This gives a systematic connection between wall-crossing and the minimal model program of the moduli space
16.30 Andreas Langer (Exeter) De Rham-Witt complexes and p-adic cohomology.
For an algebraic variety over a finite field of char p one can consider its
zeta-function which encodes the number of rational points of the
variety over finite field extensions. It is known that this zeta-function
is rational , i.e a quotient of polymomials which arise from the Frobenius
action on certain finite-dimensional vector spaces over the field of p-adic numbers.
These vector spaces occur as suitable p-adic cohomology groups and are defined
purely algebraically if the variety is smooth and proper , but involve some p-adic
(non-archimedean) analysis otherwise, for example if X is affine.
I will explain how these vector-spaces (known as crystalline or rigid cohomology)
can be described using (overconvergent) de Rham-Witt complexes.
Then I will report on how these cohomology groups can be used to count the number
of rational points of the variety and also indicate links to general p-adic Hodge-theory.