11.00-12.00 Olivier Mathieu “On the representations of super-conformal Lie superalgebras.”
12.10-13.10 Philip Griffiths “Using Hodge theory to detect the structure of a compactified moduli space.”
16.00-16.50 Xavier Gómez-Mont “The total Dehn twist of the monodromy of a plane curve singularity using Hodge Theoretic invariants.”
17.00-17.50 Lucía López de Medrano “CSM-cycles of matroids.”
9.00-9.30 Kyoung-Seog Lee “Derived categories and motives of some Fano varieties .”
9.30-10.00 Nikon Kurnosov “Beauville-Bogomolov-Fujiki form on non-Kahler manifolds.”
10.30-11.00 Omar Antolín “Classifying spaces for commutativity.”
11.00-11.30 Jacob Mostovoy “Encoding knots by diagrams.”
12.00-12.30 Julie Decaup “Compactness of the set of preorders on a group.”
12.30-13.00 Sergei Arkhipov “Equivariant DG-modules over differential forms and coherent sheaves on derived Hamiltonian reduction.”
9.00-10.00 Olivier Mathieu “On the free Jordan algebras” José Adem Memorial Lecture
10.00-11.00 Phillip Griffiths “A Tale of Two Mathematicians” Solomon Lefschetz Memorial Lecture
11.30-12.00 Ludmil Katzarkov “D modules and Rationality.”
12.00-12.30 Alexander Petkov “The Yamabe problem in quaternionic contact geometry”
12.40-13.10 Alexey Beshenov “Weil-étale cohomology for n < 0.”
13.10-13.40 Alberto Verjovsky “Toric proalgebraic laminations”
Omar Antolín “Classifying spaces for commutativity.”
Given a Lie group $G$, we can think of the space of homomorphisms Hom($Z^n, G$) as the space of $n$-tuples of elements of $G$ that commute pairwise. These spaces are more subtle than one might think, and even basic invariants such as the number of connected components can lead to surprising results. Fixing $G$ and varying $n$ we can construct what is known as the classifying space for commutativity in $G$. I will survey what is known about these classifying spaces, whose study has just begun.
Sergei Arkhipov “Equivariant DG-modules over differential forms and coherent sheaves on derived Hamiltonian reduction.”
This is a joint work in progress with Sebastian Orsted. Given an algebraic variety X acted on by an affine algebraic group G, we make sense of the derived category of DG-modules over the DG-algebra of differential forms on X equivariant with respect to differential forms on G. The construction uses an explicit model for a certain homotopy limit of a diagram of DG-categories developed in our earlier work and generalizing a recent result of Block, Holstein and Wei. We compare the obtained category with a certain category of sheaves on the (shifted) cotangent bundle T^*X descending to the Hamiltonian reduction of the cotangent bundle. Two special cases are of interest. In the first, X is a point. Thus we compare comodules over Omega(G) with G-equivariant coherent sheaves on Lie(G). In the second case, X is a simple algebraic group, with the action of the square of the upper triangular subgroup. We obtain a category closely related to the affine Hecke category.
Alexey Beshenov “Weil-étale cohomology for n < 0.”
Stephen Lichtenbaum proposed a cohomology theory, known as Weil-étale cohomology, which for a separated scheme X of finite type over Spec Z gives the special value of the associated zeta function $\zeta_X (s)$ at s = n. Matthias Flach and Baptiste Morin have constructed Weil-étale cohomology for proper and regular X and any integer n. It turns out that the regularity and properness assumptions may be removed if n < 0.
Julie Decaup “Compactness of the set of preorders on a group.”
It is a joint work with Guillaume Rond.
In my talk, I will introduce the set ZR(G) of preorders on a group G and do some examples. Then I will define three topologies on ZR(G) and I will finish by a result of compactness of this set.
Xavier Gómez-Mont “The total Dehn twist of the monodromy of a plane curve singularity using Hodge Theoretic invariants.”
Given a plane curve singularity over the complex numbers, the geometric monodromy is a diffeomorphism of the Milnor fibre, which is a compact oriented surface with boundary, uniquely determined up to isotopy, which is the identity on the boundary. Using Resolution of Singularities, we will show that the monodromy has a decomposition into sub-surfaces with boundary. On some of this sub-surfaces it is periodic. On the other sub-surfaces, which are annulus,and are permuted by the geometric monodromy, after successive iterations of the monodromy, it is a Dehn twist. Normalizing the amount of the Dehn twist by the number of iterates, and summing over all the annulus, one obtains the total Dehn Twist of the geometric monodromy. This is a numerical invariant of the singularity of topological nature.
The algebraic monodromy map is the action of the geometric monodromy on the first homology group of rhe Milnor
Fibre. It decomposes as the product of a semisimple times a unipotent part. The semisimple part is related to the
above periodic sub-surfaces and the unipotent part is related to the Jordan Block structure of the linear monodromy.
Taking the logarithm of the unipotent part, one obtains a nilpotent endomorphism of the first homology group,
which gives rise to the weight filtration of the Mixed Hodge structure of the singularity. But more importantly, it gives
rise to a symmetric bilinear form
This is joint work with Alanis, Artal, Bonatti, González-Villa and Portilla, and in work in progress, we are generalizing to higher dimensions.
Phillip Griffiths “A Tale of Two Mathematicians.”
Phillip Griffiths “Using Hodge theory to detect the structure of a compactified moduli space.”
Nikon Kurnosov “Beauville-Bogomolov-Fujiki form on non-Kahler manifolds.”
I will talk about the BBF-form for the non-Kahler holomorphically symplectic manifolds, and will explain, why such manifolds have this form, which is analogous to the form for hyperkahler manifolds.
Kyoung-Seog Lee “Derived categories and motives of some Fano varieties .”
Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. In this talk, I will discuss these invariants for certain Fano varieties.
Alexander Petkov “The Yamabe problem in quaternionic contact geometry”
We show that the quaternionic contact (qc) Yamabe problem has a solution on any compact qc manifold which is non-locally qc equivalent to the standard 3-Sasakian sphere. More precisely, we show that on a compact non-locally spherical qc manifold there exists a qc conformal qc structure with constant qc scalar curvature. The latter is a consequence of our basic result that the qc Yamabe constant of the considered qc manifold is strictly less than the corresponding constant on the 3-Sasakian sphere.
Alberto Verjovsky “Toric proalgebraic laminations”
This is Joint work with Juan Manuel Burgos. We present several results about toric proalgebraic laminations. These “solenoidal laminations” are obtained as inverse limits of branched self-coverings of toric manifods. A paradigmatic example is the space ${\hat{\mathbb C}}_\mathbb{Q}$ obtained as the inverse limit of the maps of the Riemann sphere ${\hat{\mathbb C}}_\mathbb{Q}=P_{\mathbb C}^1$ given by $z\mapsto{z^n}$ ($n\in\mathbb{N}$). Then ${\hat{\mathbb C}}_\mathbb{Q}$ is homeomorphic to the topological suspension of the one-dimensional solenoid $S_{\mathbb{Q}}$ which is the Pontryagin dual of the additive rationals $\mathbb Q$ with its discrete topology. Outside the two suspension points (which correspond to $0,\infty\in\hat{\mathbb{C}}$) the space is a Riemann surface lamination with leaves densely immersed copies of ${\mathbb{C}}^*$. In fact this set is the profinite completion ${\mathbb{C}}^*_{\mathbb Q}$ of the algebraic torus ${\mathbb{C}}^*$. We call this singular lamination the adèlic projective line.
We present results that show that this type of construction applies to any projective toric variety (or toric orbifold).