Topology of non-archimedean manifolds
By Piotr Achinger
Rigid geometry is the analog of complex analytic geometry over
non-archimedean fields such as the \(p\)-adic numbers \(\mathbb Q_p\) or complex
formal Laurent series \(C((t))\). I will review some recent results
concerning the "homotopy types" of rigid analytic spaces, mostly
dealing with different aspects of the fundamental group.
Moduli of elliptic K3 surfaces and the Weierstrass normal form: monodromy versus Shimada strata
By Klaus Hulek (joint work with Michael Lönne)
The Weiertrass normal form can be used to construct a moduli space of non-isotrivial elliptic K3.
Alternately, one can use Hodge structures. This moduli space admits several stratifications. Here
we consider two of them. The first is the monodromy stratification introduced by Bogomolov, Petrov
and Tschinkel and the second is the Shimada stratification obtained by fixing the root lattice of
the trivial Neron–Severi group. Neither of these stratifications is a refinement of the other.
There are, however, a number of strata which are common to both stratifications. Here we classify
these so called ambi-typical strata, of which there are 50.
Cluster algebras and representation theory
By Bernhard Keller
Cluster algebras were invented by Fomin-Zelevinsky around the
year 2000.
They are certain commutative algebras endowed with a rich combinatorial
structure. Fomin-Zelevinsky's motivations came from Lie theory but it
soon turned out that cluster algebras are linked to an astounding variety
of subjects both in mathematics and in physics. In this survey talk, we
will present the definition of cluster algebras, which is completely
elementary, and show how their interaction with representation theory
has had a deep and fruitful impact on both subjects. We will conclude with
a glimpse of some of the most exciting ongoing research in the area.
On tensor envelopes of regular categories
By Friedrich Knop
A classical construction assigns to every regular category the
tensor category of relations. Twisting this construction with a degree
function one arrives at the tensor enveloping categories which are under
certain conditions semisimple abelian. In the talk we determine the
simple objects, calculate their categorical dimension and their tensor
product multiplicities.
Quantitative sheaf theory and arithmetic Fourier transforms over finite fields
By Emmanuel Kowalski (joint work with A. Forey, J. Fresán and W. Sawin)
The applications of Deligne's Riemann Hypothesis over finite fields to
analytic number theory have long been restricted by the problem of
estimating ell-adic Betti numbers independently of the characteristic.
W. Sawin has recently developed a form of quantitative sheaf theory
which solves this problem almost entirely. The talk will explain the
basic background and ideas, and present applications to Fourier
transforms and equidistribution problems for exponential sums
parameterized by characters of commutative algebraic groups,
generalizing results of Deligne and Katz.
The geometry of Weyl orbits on blow-ups of projective spaces
By Eliza Postinghel (joint work with C. Brambilla, O. Dumitrescu and L. Santana Sánchez)
Linear systems of divisors on blow-ups of projective spaces in points in general positions are
connected to certain polynomial interpolation problems. While for the case of plane curves and
of surfaces in 3-space there are conjectures, although long standing, formulated by M. Nagata,
B. Segre and others, in the higher dimensional case we are in the dark. However, when the number
of points is not too large and the blow-ups are Mori dream spaces, an action of the Weyl group on
cycles of any codimension governs the birational behaviour of the space on the one hand, and the
stable base locus of divisors on the other hand, and it yields a solution to the interpolation problem.
Wall-crossing and its applications in moduli theory
By Fatemeh Rezaee
We will start by introducing the notion of wall-crossing with respect to
Bridgeland stability conditions in dimension three. Then we will apply
this machinery to the moduli problems in algebraic geometry, such as
the moduli space of stable pairs for specific space curves. Time permitting,
we will explain how incorporating derived algebraic geometry into
the picture can extend it to handle the general case.
Deformations of K3 twistor families
By Martin Schwald (joint work with Daniel Greb)
We aim for establishing a moduli theory for families of K3 surfaces over \(CP^1\),
most importantly, twistor families. While the moduli space of K3 surfaces is non-Hausdorff,
the deformations of twistor families are classified by a smooth connected component
in a cycle space. Furthermore, K3 surfaces over the same cycle share a geometric
property. This is a joint work with Daniel Greb, building on previous work
with Ana-Maria Franzen-Brecan and Tim Kirschner.
Characterization of quasi-abelian surfaces
By Sofia Tirabassi (joint work with R. Pardini and M. Mendes Lopes)
We give an effective charactrization of quasi-abelian surfaces,
analogue to the characterizations of abelian surfaces of Enriques and Chen-Hacon.
Grothendieck classes of quadric hypersurfaces and involution varieties
By Gonçalo Tabuada
The Grothendieck ring of varieties, introduced in a letter
from Alexander Grothendieck to Jean-Pierre Serre (August 16th 1964),
plays an important role in algebraic geometry. However, despite the
efforts of several mathematicians, the structure of this ring still
remains poorly understood. In this talk, in order to better understand
the Grothendieck ring of varieties, I will describe some new structural
properties of the Grothendieck classes of quadric hypersurfaces and
involution varieties.
Algebraic foliations and derived geometry
By Bertrand Toën (joint work with G. Vezzosi)
Foliations defined on algebraic varieties
are rarely without singularities. These singularities can be studied using
derived techniques via a notion of "derived foliations", in the same way than
singularities of algebraic varieties can be studied using the notion of derived
schemes. In this talk, I will explain the notion of derived foliations and
report on recent applications for the study of singular foliations. These
include results in the complex case, as well as foliations defined over
base fields of arbitrary characteristics.
