# 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

## Participants

• Piotr Achinger, IMPAN
• Agnieszka Bodzenta, University of Warsaw
• Agnieszka Bojanowska, University of Warsaw
• Max Briest, University of Augsburg
• Timothy De Deyn, Vrije Universiteit Brussel
• Rodion Déev, IMPAN
• Francesco Galuppi, IMPAN
• Anna Gąsior, Maria Curie-Skłodowska University in Lublin
• Christian Gleissner, University of Bayreuth
• Marek Hałenda, University of Gdańsk
• Klaus Hulek, Leibniz University Hannover
• Stefan Jackowski, University of Warsaw
• Gustavo Jasso, Lunds Universitet
• Michał Kapustka, IMPAN
• Bernhard Keller, University of Paris
• Friedrich Knop, Friedrich-Alexander Universität Erlangen-Nürnberg
• Emmanuel Kowalski, ETH Zurich
• Johannes Krah, Bielefeld University
• Dmitry Kubrak, MPIM Bonn
• Felix Kueng, Universite Libre de Bruxelles
• Marta Leśniak, University of Gdańsk
• Antonio Lorenzin, University of Pavia and University of Milano-Bicocca
• Rafał Lutowski, University of Gdańsk
• Anya Nordskova, Hasselt Universiteit
• Tudor Padurariu , Columbia University
• Jun Yong Park, MPIM Bonn
• Leonid Plachta, AGH University of Science and Technology
• Elisa Postinhel, University of Trento
• Jerzy Popko, University of Gdańsk
• Yulieth Prieto, University of Bologna
• Józef Przytycki, George Washington University and University of Gdańsk
• Fatemeh Rezaee, Maxwell Institute for Mathematical Sciences
• Witold Rosicki, University of Gdańsk
• Norbert Schappacher, IRMA
• Martin Schwald, University of Duisburg-Essen
• Peter Spacek, TU Chemnitz
• Spela Spenko, University of Brussels
• Bruno Stonek, IMPAN
• Andrzej Szczepanski, University of Gdańsk
• Błażej Szepietowski, University of Gdańsk
• Sofia Tirabassi, Stockholm University
• Bertrand Toen, Univeriste de Toulouse
• Aleksandra Utiralova, UC Berkeley
• Andreas Zastrow, University of Gdańsk
• Magdalena Zielenkiewicz, University of Warsaw

## Schedule

All talks will take place in Auditorium A of Faculty of Law and Administration (see location on a map).

### 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.

### HMS symmetries of toric boundary divisors

#### By Špela Špenko (joint work with Michel Van den Bergh)

Let $$X$$ be a complex manifold. By homological mirror symmetry one expects an action of the fundamental group of the "moduli space of Kähler structures" of $$X$$ on the derived category of $$X$$. If $$X$$ is a crepant resolution of a Gorenstein affine toric variety we obtain an action on the derived category of the toric boundary divisor of $$X$$ which leads to an action on the Grothendieck group of $$X$$.

### 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.

### Haupt-Kapovich theorem revisited

#### By Rodion Deev, IMPAN

A theorem by O. Haupt (1920) and M. Kapovich (2000) describes the set of cohomology classes on a genus $$g$$ topological surface which can be realized by holomorphic $$1$$-forms on some genus $$g$$ complex curve. Unlike in the Schottky problem, the answer is a simple linear-algebraic condition. F. Bogomolov discovered that this Haupt--Kapovich phenomenon generalizes likely to the case when we try to squeeze a given two- or three-dimensional subspace $$\tau \subset H^1$$ into $$H^{1,0}$$ for some complex structure. This matter is far from being clear, but the present results are enough to prove some interesting theorems, and to look for interesting generalizations of the problem e. g. for determining possible fibers of Lagrangian fibrations.

### Introducing LG-models for $$qH^*(G/P)$$ through examples

#### By Peter Spacek, TU Chemnitz (joint work with Charles Wang)

Using accessible examples, we introduce recent work concerning mirror symmetry for homogeneous spaces $$G/P$$. We will introduce the small quantum cohomology ($$qH^*$$), Plücker coordinates, Landau-Ginzburg (LG-) models, and time-permitting cluster structures. Finally, we will spotlight the mirror models we have constructed for the Cayley plane and the Freudenthal variety (homogeneous over groups of type $$E$$).

### Spin-structures on real Bott manifolds with Kaehler structure

#### By Anna Gąsior, Maria Curie-Skłodowska University in Lublin (joint work with Rafał Lutowski)

Real Bott manifolds occur as total spaces of specific sequences of $$\mathbb{R}P^1$$-bundles. Every such a manifold $$M$$ is a quotient of a flat torus by a free and faithful action of an elementary abelian 2-group. Up to diffeomorphism, $$M$$ is determined by a certain square matrix $$A$$ with coefficients in $$\mathbb{F}_2$$. We call $$A$$ a Bott matrix and we denote the manifold $$M$$ by $$M(A)$$. In 2011 Ishida gave necessary and sufficient conditions for existence of a Kaehler structure on $$M(A)$$, which involve properties of $$A$$ only.
In this talk we examine the existence of spin structures on real Bott manifolds with Kaehler structure. We show that, similarly as in Ishida's condition, this existence can be formulated in purely combinatorial form - in terms of some properties of Bott matrices.

### Affine cones over Grassmanians and their stringy $$E$$-functions

#### By Timothy De Deyn, Vrije Universiteit Brussel

Through the example of affine cones over Grassmanians we will discuss evidence indicating a connection between the existence of non-commutative crepant resolutions of a variety and properties of certain string-theoretic invariants associated to that variety. As a concrete incarnation hereof we will explain that non-commutative crepant resolutions, if they exist, seem to enforce the stringy $$E$$-function to be a polynomial.

### Twisted Hodge diamonds give rise to non-Fourier-Mukai functors

#### By Felix Kueng, Universite Libre de Bruxelles

We use computations of twisted Hodge Diamonds in order to compute the Hochschild cohomology of smooth degree d Hypersurfaces. Using these computations we can deduce the dimension of the kernel in Hochschild cohomology of the push forward along closed embeddings into projective space. In particular this allows the construction of new non-Fourier-Mukai functors between well behaved target and source spaces.

### The Triangulated Auslander-Iyama Correspondence

#### By Gustavo Jasso (joint work with Fernando Muro)

Previous work of Muro establishes the existence and uniqueness of (DG) enhancements for triangulated categories which admit an additive generator whose endomorphism algebra is finite-dimensional (over a perfect field). In this talk I will present a generalisation of this result that allows us to treat a larger class of triangulated categories, which instead admit a generator with a strong regularity property (a so-called dZ-cluster tilting object). I will explain our main result as well as some interesting applications.

### An introduction to the Donovan-Wemyss conjecture

#### By Bernhard Keller

Compound Du Val (=cDV) singularities were introduced by Miles Reid in the early eighties as 3-dimensional analogues of simple surface singularities. They play an important role in the minimal model program. In 2013, Donovan and Wemyss introduced the contraction algebra associated with a crepant resolution of a cDV singularity. They showed that it is a finite-dimensional (non commutative) algebra and determines many important invariants of the singularity. They conjectured that it determines the singularity itself up to isomorphism. More and more evidence for the conjecture was accumulated in work by Donovan-Wemyss, Wemyss, Toda, Hua-Toda, Hua, August, Hua-K and others. I will give a historical introduction to the conjecture and conclude by explaining how it follows by combining the work of August and Hua-K with the triangulated Auslander-Iyama correspondence, a recent theorem due to Gustavo Jasso and Fernando Muro (cf. Gustavo Jasso's talk at this Colloquium).

### 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.

### Identifiability of pairs of ternary forms

#### BY Francesco Galuppi, IMPAN

Two general quadratic forms can be simultaneously diagonalized in a unique way. For forms of higher degrees, it is a long-standing problem to classify when we get uniqueness. In this talk I'll present the solution for forms in three variables. Our strategy is to translate the problem into the study of a certain linear system on a projective bundle on the plane.

### Projective manifolds admitting Ulrich tangent bundle

#### By Yulieth Prieto, University of Bologna (joint work with V. Benedetti. P. Montero, and S. Troncoso)

One of the significant issues of the subject is whether or not smooth projective varieties admit Ulrich bundles. Inspired by the results of Casnati in the case of surfaces, we give numerical restrictions on the Chern classes of Ulrich bundles to the case of threefolds, and we observe a practical restriction concerning the first Chern class for any higher--dimensional projective manifold. Applying these facts, we show that the only projective manifolds whose tangent bundle is Ulrich are the twisted cubic and the Veronese surface.

### 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.

### How did Teaching in Braniewo affect Weierstrass’ and Killing’s Research?

#### By Norbert Schappacher

Both Weierstrass and Killing spent several years teaching in Braniewo before they obtained their positions as University professors in Berlin, resp. in Münster. The talk will reflect on the educational system that made such careers possible, and explore how these two world famous mathematicians combined their teaching duties with their mathematical research. The picture that emerges will show similarities, but also interesting differences between Weierstrass and Killing.

### 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.

### 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.

### 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.

### Harish-Chandra bimodules in complex rank

#### By Aleksandra Utiralova, UC Berkeley

The Deligne tensor categories are defined as an interpolation of the categories of representations of groups $$GL_n, O_n, Sp_{2n}$$ or $$S_n$$ to the complex values of the parameter $$n$$. One can extend many classical representation-theoretic notions and constructions to this context. These complex rank analogs of classical objects provide insights into their stable behavior patterns as n goes to infinity.
I will talk about some of my results on Harish-Chandra bimodules in the Deligne categories. It is known that in the classical case simple Harish-Chandra bimodules admit a classification in terms of W-orbits of certain pairs of weights. However, the notion of weight is not well-defined in the setting of the Deligne categories. I will explain how in complex rank the above-mentioned classification translates to a condition on the corresponding (left and right) central characters.

### Arithmetic Moduli of Weierstraß Fibrations

#### By Jun Yong Park, MPIM Bonn

We will first consider the formulation of the moduli of fibered algebraic surfaces as the Hom space of algebraic curves on moduli stacks of curves. Focusing upon the Weierstraß equations and their moduli counterparts the weighted projective stacks, we will explore some interesting consequences of a discrepancy between the Integral/Rational points on moduli stacks of elliptic curves over function fields and also correspondence of Rational maps with valuations to Twisted maps with stabilizers.

### Hodge-to-de Rham degeneration for Artin stacks

#### By Dmitry Kubrak, MPIM Bonn

I will talk about my joint work with A.Prikhodko where we prove degeneration of the Hodge-to-de Rham spectral sequence for certain class of smooth Artin stacks via a suitable refinement of the Deligne-Illusie method. This class of stacks (which we call Hodge-properly spreadable) includes all smooth proper stacks, but in fact is much more general: in particular, for the quotient stack of a scheme by an action of a reductive group it is just enough for the course moduli to be proper.

### Formality and strongly unique enhancements

#### By Antonio Lorenzin, University of Pavia and University of Milano-Bicocca

Inspired by the intrinsic formality of graded algebras, we give a characterization of strongly unique DG-enhancements for a large class of algebraic triangulated categories, linear over a commutative ring. We will discuss applications to bounded derived categories and bounded homotopy categories of complexes. For the sake of an example, the bounded derived category of finitely generated abelian groups has a strongly unique enhancement.

### Donaldson-Thomas theory of $$C^3$$ via matrix factorizations

#### By Tudor Padurariu, Columbia University (joint work with Yukinobu Toda)

Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold $$X$$. For the simplest example of $$C^3$$, the Donaldson-Thomas (DT) invariant of sheaves of zero dimensional support and length $$d$$ is $$p(d)$$, the number of plane partitions of $$d$$. The DT invariants can be recovered as the Euler characteristic of a collection of vector spaces. It is natural to ask whether for a given Calabi-Yau threefold $$X$$, there exists a categorification of these vector spaces (and thus of DT invariants), for example a category whose periodic cyclic homology recovers (a $$\mathbb{Z}/2$$-periodic version of) these vector spaces, and thus recovers the DT invariants of $$X$$. There exist such categorifications, constructed using matrix factorizations, when $$X$$ is the total space of the canonical bundle over a surface $$S$$.
I will focus on the categorification $$\mathcal C$$ of DT invariants for $$C^3$$, in particular we will construct semi-orthogonal decompositions of $$\mathcal C$$ and study the $$K$$-theory of $$\mathcal C$$, and explain how these computations relate to $$p(d)$$. The semi-orthogonal decompositions of $$\mathcal C$$ have applications in categorical DT/Pandharipande-Thomas wall-crossing.

### 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.

### 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.

### Khovanov homology and its complexity

#### By Józef Przytycki (joint work with Marithania Silvero)

We discuss our conjecture that Khovanov homology for links with fixed braid index can be computed in polynomial time (in general computing Khovanov homology is NP hard). The gentle introduction to Khovanov homology will be offered.

## Register Here

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## Do you have Questions?

Answers to some of them may be found here.

If you are in a foot-walking distance from the campus, located in district Oliwa, you can use any map/navigation provider without a problem.

For further distances you can use for example a web page jakdojade.pl (available in English) or any site/app which you find useful.

NEW: There is a possibility to stay in dormitories for a limited number of participants.

You can of course use one of many available services. Some locations nearby campus include Hotel Olivia, Brigidine guest house, Villa Lena. You can also check Where to stay at the official Gdańsk website.

## Venue

The Colloquium will take place at the Faculty of Law and Administration of University of Gdańsk.

## Supporting institutions

We kindly acknowledge support of: