# The Fourth W. Killing and K. Weierstrass Colloquium

### Welcome to the homepage of The Fourth W. Killing and K. Weierstrass Colloquium

The core of the Fourth W. Killing and K. Weierstrass Colloquium (KWWK 2022) concerns geometry, algebraic geometry, algebra, number theory, Lie theory, group theory and other branches of mathematics which are related to the work of W. Killing and K. Weierstrass.

The (first) W. Killing and K. Weierstrass Colloquium was connected with the unveiling of the memorial plate in Braniewo (more information can be found in this article). In the next four years the second and the third Colloquium took place. All of these events gathered leading mathematicians working in the areas which may be thought of as a legacy of work both of Weierstrass and Killing.

Text on the plate:

In memory of stay and work in Braniewo
of the famous mathematicians
Karl Weierstrass (1815-1897)
and Wilhelm Killing (1847-1923)
who taught here
at the Catholic Gymnasium and the Lyceum Hosianum

Polish Mathematical Society
German Mathematical Society

Braniewo, 24.07.2008

## Talks and abstracts

### 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

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.

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## Venue

The Colloquium will take place at the Institute of Mathematics of University of Gdańsk.

## Supporting institutions

We kindly acknowledge support of: