### Research team

### Expertise

Research in mathematics in the broad area of algebra, geometry and topology. More specifically research in noncommutative geometry, homological and homotopical algebra, deformation theory.

## Operadic approaches to deforming higher categories.

#### Abstract

With the current project proposal, we will further the development of noncommutative algebraic geometry by establishing highly structured deformation complexes for a variety of higher categorical structures. We will realise this goal through the following three main objectives. In a first objective, we will develop operadic approaches to encode higher linear categories and (higher) prestacks, leading to the definition of L infinity structured Gerstenhaber-Schack type complexes based upon endomorphism operads. For this, inspired by Leinster's free completion multicategories, we will develop an underlying operadic framework of ``cubical box operads'' in a linearly enriched setup. In the second objective, we will develop an alternative approach based upon the classical operadic cohomology due to Markl. In doing so we will answer an open question posed by Markl regarding the cohomology complex for presheaves. These two approaches will be compared and combined, and we will show that they calculate a natural notion of Hochschild cohomology as desired. This will facilitate our third objective, in which we will develop Keller's arrow category argument in the newly defined setup, leading to new cohomology computation and comparison tools.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Hermans Lander

#### Research team(s)

## Higher linear topoi and curved noncommutative spaces.

#### Abstract

Broadly, this project can be summarized as looking for connections between noncommutative algebraic geometry (NCAG) and higher category theory. NCAG is the modern understanding, and a drastic abstract generalization, of classical geometry. To known geometrical spaces, one can associate commutative (i.e. x*y = y*x) algebraic structures. However, in algebra, noncommutative structures are just as common. The idea of NCAG is to study new ``geometric spaces'' associated to these noncommutative algebraic structures. Higher category theory and in particular so-called infinity-topoi generalize the following idea. Consider the familiar example of sets, and maps that describe relations between those sets. Further, we can also describe relations between the maps, which we could call "2-maps". We then have 3-maps between 2-maps and so on, yielding an infinite hierarchy of maps. In relation to NCAG, the most important abelian categories correspond to linear topoi, in which the "maps" have some additional structure. One goal of the project is to establish a suitable notion of linear infinity-topoi, using ideas from NCAG. Another goal is to use ideas from higher categories to investigate the so-called "curvature problem" from NCAG. This involves "curved objects", which are slightly tweaked versions of some original object, that turn out difficult to grasp using the familiar tools of homological algebra.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Mertens Arne

#### Research team(s)

## Derived categories and Hochschild cohomology in (noncommutative) algebraic geometry.

#### Abstract

Algebraic geometry is an old subject, going back to the ancient Greeks who studied the geometry of ellipses, parabolas and hyperbolas using the language of conic sections. In the 16th century, Descartes rephrased everything in terms of coordinates. The conic sections of the Greeks became solutions to quadratic polynomial equations. Finally, in the 1960s the current framework of algebraic geometry was introduced by Grothendieck, with the advent of scheme theory. An important question in the context of conic sections is their classification: how many different types are there, and how can one be related, or "deformed", into another? It is possible to consider this problem in the three settings introduced above, giving equivalent answers. But the high level of abstraction in the last setting allows one to really explain what is specific to the situation of conic sections, and what is true more generally. My research proposal concerns these classification and deformation problems in (non-commutative) algebraic geometry: Hochschild cohomology describes previously unknown ways of deforming objects in algebraic geometry, making them non-commutative. My goal is to study and obtain exciting and unexpected connections: Can we understand symmetries of deformations? Can we translate noncommutativity back to commutativity? How can we relate geometric objects in new ways? How are deformations similar or different?#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Belmans Pieter

#### Research team(s)

## Foundations for Higher and Curved Noncommutative Algebraic Geometry (FHiCuNCAG).

#### Abstract

With this research programme, inspired by open problems within noncommutative algebraic geometry (NCAG) as well as by actual developments in algebraic topology, it is our aim to lay out new foundations for NCAG. On the one hand, the categorical approach to geometry put forth in NCAG has seen a wide range of applications both in mathematics and in theoretical physics. On the other hand, algebraic topology has received a vast impetus from the development of higher topos theory by Lurie and others. The current project is aimed at cross-fertilisation between the two subjects, in particular through the development of "higher linear topos theory". We will approach the higher structure on Hochschild type complexes from two angles. Firstly, focusing on intrinsic incarnations of spaces as large categories, we will use the tensor products developed jointly with Ramos González and Shoikhet to obtain a "large version" of the Deligne conjecture. Secondly, focusing on concrete representations, we will develop new operadic techniques in order to endow complexes like the Gerstenhaber-Schack complex for prestacks (due to Dinh Van-Lowen) and the deformation complexes for monoidal categories and pasting diagrams (due to Shrestha and Yetter) with new combinatorial structure. In another direction, we will move from Hochschild cohomology of abelian categories (in the sense of Lowen-Van den Bergh) to Mac Lane cohomology for exact categories (in the sense of Kaledin-Lowen), extending the scope of NCAG to "non-linear deformations". One of the mysteries in algebraic deformation theory is the curvature problem: in the process of deformation we are brought to the boundaries of NCAG territory through the introduction of a curvature component which disables the standard approaches to cohomology. Eventually, it is our goal to set up a new framework for NCAG which incorporates curved objects, drawing inspiration from the realm of higher categories.#### Researcher(s)

- Promotor: Lowen Wendy

#### Research team(s)

## Tensor products in non-commutative geometry and higher deformation theory

#### Abstract

Algebraic Geometry is a mathematical discipline based on a symbiotic two-directional dictionary between the fields of Algebra and Geometry. Roughly, it is a dictionary from equations (algebra) to geometrical figures (geometry) and vice versa. For example, given the equation y=x^2, we can draw its corresponding figure, in this case a parabola. We can add a third language to that dictionary, which is the highly abstract field of category theory. To each equation (or figure) we can associate a category, and given the category, we can recover its equations or figure. This dictionary is very useful when we work in commutative algebra, where the multiplication of our equations is commutative. But there exist algebraic structures were the multiplication is no longer commutative with the issue that "drawing" is no longer possible. However the dictionary algebra-category theory is still available. In algebra there is an operation called tensor product, which corresponds to taking the product of geometrical figures in an appropriate sense. In previous research we introduced a tensor product at the level of categories, in order to translate the algebraic operation to the categorical language. In this project we want to analyse further this tensor product of categories and use it to try to understand how the deformation of geometrical figures (both in commutative and "not-drawable" non-commutative directions) behaves.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Ramos González Julia

#### Research team(s)

## Hochschild cohomology, non-commutative deformations and mirror symmetry.

#### Abstract

Our research programme addresses several interesting current issues in non-commutative algebraic geometry, and important links with symplectic geometry and algebraic topology. Non-commutative algebraic geometry is concerned with the study of algebraic objects in geometric ways. One of the basic philosophies is that, in analogy with (derived) categories of (quasi-)coherent sheaves over schemes and (derived) module categories, non-commutative spaces can be represented by suitable abelian or triangulated categories. This point of view has proven extremely useful in non-commutative algebra, algebraic geometry and more recently in string theory thanks to the Homological Mirror Symmetry conjecture. One of our main aims is to set up a deformation framework for non-commutative spaces represented by "enhanced" triangulated categories, encompassing both the non-commutative schemes represented by derived abelian categories and the derived-affine spaces, represented by dg algebras. This framework should clarify and resolve some of the important problems known to exist in the deformation theory of derived-affine spaces. It should moreover be applicable to Fukaya-type categories, and yield a new way of proving and interpreting instances of "deformed mirror symmetry". This theory will be developed in interaction with concrete applications of the abelian deformation theory developed in our earlier work, and with the development of new decomposition and comparison techniques for Hochschild cohomology. By understanding the links between the different theories and fields of application, we aim to achieve an interdisciplinary understanding of non-commutative spaces using abelian and triangulated structures.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Lowen Wendy

#### Research team(s)

## Higher linear topoi and curved noncommutative spaces.

#### Abstract

Broadly, this project can be summarized as looking for connections between noncommutative algebraic geometry (NCAG) and higher category theory. NCAG is the modern understanding, and a drastic abstract generalization, of classical geometry. To known geometrical spaces, one can associate commutative (i.e. x*y = y*x) algebraic structures. However, in algebra, noncommutative structures are just as common. The idea of NCAG is to study new ``geometric spaces'' associated to these noncommutative algebraic structures. Higher category theory and in particular so-called infinity-topoi generalize the following idea. Consider the familiar example of sets, and maps that describe relations between those sets. Further, we can also describe relations between the maps, which we could call "2-maps". We then have 3-maps between 2-maps and so on, yielding an infinite hierarchy of maps. In relation to NCAG, the most important abelian categories correspond to linear topoi, in which the "maps" have some additional structure. One goal of the project is to establish a suitable notion of linear infinity-topoi, using ideas from NCAG. Another goal is to use ideas from higher categories to investigate the so-called "curvature problem" from NCAG. This involves "curved objects", which are slightly tweaked versions of some original object, that turn out difficult to grasp using the familiar tools of homological algebra.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Mertens Arne

#### Research team(s)

## A study of the impact of stopping rules on conventional estimation with probabilistic and approach theoretic techniques

#### Abstract

In a sequential trial, the data collector is allowed to take intermediate looks at the data. After each intermediate look, he can decide, based on the data observed so far and a prescribed stopping rule, if the trial is stopped or continued. Stopping a trial early is potentially beneficial for economic and ethical reasons in e.g. a clinical study. The existing literature on sequential trials has reported that simple conventional estimators, such as the sample mean, become biased in the presence of observation based stopping rules. However, very recently, Molenberghs and colleagues have taken away some of this concern by showing that in many cases the bias, caused by stopping rules, vanishes quickly as the number of collected data increases. Building upon the insights obtained by Molenberghs and colleagues, we will provide a theoretical underpinning of the fact that conventional estimation remains in many important cases legitimate in a sequential trial. More precisely, we seek to establish Berry-Esseen type inequalities in sequential analysis that justify the use of confidence intervals based on conventional estimators. Also, the implications for hypothesis testing will be investigated. To this end, we will use mathematical techniques from probability, e.g. Stein's method, and approach theory, a topological theory pioneered by R. Lowen, which has already been useful in the theory of probability metrics, central limit theory, and estimation with contaminated data.#### Researcher(s)

- Promotor: Lowen Wendy
- Co-promotor: Molenberghs Geert
- Fellow: Berckmoes Ben

#### Research team(s)

## Mac Lane cohomology and deformations of exact and triangulated models.

#### Abstract

The research proposal deals with the investigation of an invariant to distinguish between rings, by which I mean objects with an addition and multiplication. The invariant in question, the so called Mac Lane cohomology, characterizes moreover some special types of maps between these rings, which helps us even more in understanding their nature and possible difference. This contributes to the desire of mathematicians in classifying things. Bearing this in mind, I hope to elaborate the theory to settings other than that of rings, namely to the world of triangulated and exact categories. Moreover, the Mac Lane cohomology can be seen itself as an improvement of yet another invariant, the Hochschild cohomology, with interactions between both theories. Thus the question arises whether this improvement holds in the new settings and what relations remain intact between both notions.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Caenepeel Frederik

#### Research team(s)

## Hochschild cohomology and deformation theory of triangulated categories.

#### Abstract

Our general objective in this project is to study deformations of pre-triangulated categories, as models for non-commutative spaces. Examples are not only to be found in non-commutative algebraic geometry, but also in symplectic geometry, with Fukaya categories as prime examples. Our basic aim is to connect deformations of pre-triangulated categories to Hochschild cohomology classes, and more generally to solutions of the Maurer-Cartan equation in the Hochschild complex.#### Researcher(s)

- Promotor: Lowen Wendy

#### Research team(s)

## Stacks and duality in non-commutative algebraic geometry.

#### Abstract

An important class of results in algebraic geometry are so-called duality statements, by which we try to relate less-understood objects to more well-known ones. In the development of modern algebraic geometry, stacks have played a crucial role. In this project we will develop a theory of stacks from the point of view of non-commutative algebraic geometry, which should enhance our understanding of Grothendieck duality, and allow the investigation of interesting new applications within (non-commutative) algebraic geometry and cluster theory.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Belmans Pieter

#### Research team(s)

## Mac Lane cohomology and deformations of exact and triangulated models.

#### Abstract

The research proposal deals with the investigation of an invariant to distinguish between rings, by which I mean objects with an addition and multiplication. The invariant in question, the so called Mac Lane cohomology, characterizes moreover some special types of maps between these rings, which helps us even more in understanding their nature and possible difference. This contributes to the desire of mathematicians in classifying things. Bearing this in mind, I hope to elaborate the theory to settings other than that of rings, namely to the world of triangulated and exact categories. Moreover, the Mac Lane cohomology can be seen itself as an improvement of yet another invariant, the Hochschild cohomology, with interactions between both theories. Thus the question arises whether this improvement holds in the new settings and what relations remain intact between both notions.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Caenepeel Frederik

#### Research team(s)

## Stacks and duality in non-commutative algebraic geometry.

#### Abstract

An important class of results in algebraic geometry are so-called duality statements, by which we try to relate less-understood objects to more well-known ones. In the development of modern algebraic geometry, stacks have played a crucial role. In this project we will develop a theory of stacks from the point of view of non-commutative algebraic geometry, which should enhance our understanding of Grothendieck duality, and allow the investigation of interesting new applications within (non-commutative) algebraic geometry and cluster theory.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Belmans Pieter

#### Research team(s)

## Algebraic deformation techniques in geometric contexts.

#### Abstract

The main aim of our proposed research project is twofold: (a) we will apply our current understanding of the general algebraic deformation pattern to various geometric settings (including the algebraic geometry context in which we obtained our earlier results). (b) we will investigate concrete (as opposed to philosophical - see the discussion in x1) links between the various geometric settings both on the undeformed, the deformed, and the general non-commutative level.#### Researcher(s)

- Promotor: Lowen Wendy

#### Research team(s)

## Non-commutative deformations of saturated spaces.

#### Abstract

The project belongs to the field of noncommutative algebraic geometry, following the philosophy of Kontsevich, Van den Bergh and others. In analogy to derived categories of quasi coherent sheaves, one considers derived categories of modules over a noncommutative ring in order to study this ring geometrically. More general noncommutative spaces are represented by abelian categories, their derived categories and their algebraic models: dg categories and A-infinity-categories ( B. Keller). These methods have been shown to be very useful to study subjects like deformation quantization (M. Kontsevich), homological mirror symmetry (Kontsevich) and Hodge theory (D.Kaledin). W. Lowen and M. Van den Bergh constructed of a deformation theory, with associated Hochschild cohomology for abelian categories. To further develop this theory we have to study the following related aspects: - To obtain structure theorems for deformations of known spaces by means of their associated sheaf categories. In our project we describe deformations of projective varieties by means of Z-algebras. In doing so we obtain a general setting in which results of Van den Bergh, Bondal and Polischuk can fit. - To investigate properties under deformation. For sufficiently nice schemes the categories of quasi coherent sheaves are Grothendieck categories, which is a property preserved under deformation. There are other geometric properties of which the behaviour under deformation still has to be investigated.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: De Deken Olivier

#### Research team(s)

## Hochschild cohomology, non-commutative deformations and mirror symmetry.

#### Abstract

Our research programme addresses several interesting current issues in non-commutative algebraic geometry, and important links with symplectic geometry and algebraic topology. Non-commutative algebraic geometry is concerned with the study of algebraic objects in geometric ways. One of the basic philosophies is that, in analogy with (derived) categories of (quasi-)coherent sheaves over schemes and (derived) module categories, non-commutative spaces can be represented by suitable abelian or triangulated categories. This point of view has proven extremely useful in non-commutative algebra, algebraic geometry and more recently in string theory thanks to the Homological Mirror Symmetry conjecture. One of our main aims is to set up a deformation framework for non-commutative spaces represented by "enhanced" triangulated categories, encompassing both the non-commutative schemes represented by derived abelian categories and the derived-affine spaces, represented by dg algebras. This framework should clarify and resolve some of the important problems known to exist in the deformation theory of derived-affine spaces. It should moreover be applicable to Fukaya-type categories, and yield a new way of proving and interpreting instances of "deformed mirror symmetry". This theory will be developed in interaction with concrete applications of the abelian deformation theory developed in our earlier work, and with the development of new decomposition and comparison techniques for Hochschild cohomology. By understanding the links between the different theories and fields of application, we aim to achieve an interdisciplinary understanding of non-commutative spaces using abelian and triangulated structures.#### Researcher(s)

- Promotor: Lowen Wendy
- Fellow: Lowen Wendy

#### Research team(s)

## HHNcdMir - Hochschild cohomology, non-commutative deformations and mirror symmetry.

#### Abstract

The research project addresses some interesting current issues in non-commutative algebraic geometry, and some important links with sympletic geometry and algebraic topology. The main focus will be on non-commutative deformations, which can be considered as the first step on the road to general non-commutative spaces.#### Researcher(s)

- Promotor: Lowen Wendy

#### Research team(s)

#### Project website

## Non-commutative deformations and mirror symmetry

#### Abstract

The project addresses current issues in non-commutative geometry, and links with symplectic geometry and topology. Non-commutative spaces can be represented by suitable abelian or triangulated categories. This point of view has proven extremely useful i.p. in string theory, thanks to the Homological Mirror Symmetry conjecture. We will set up a deformation framework, applicable to Fukaya-type categories, yielding a new way of proving and interpreting instances of ``deformed mirror symmetry''.#### Researcher(s)

- Promotor: Lowen Wendy

#### Research team(s)

#### Project website

## Deformations and cohomology in non-commutative derived geometry.

#### Abstract

My research project is at the crossroads of non-commutative geometry (in the sense of Kontsevich, Van den Bergh, . . . ) and homotopical derived geometry (in the sense of Toën, . . . ). An important inspiration is the fact [6] that a smooth proper scheme is equivalent in the derived sense to a differential graded (dg) algebra [28], and smoothness and properness boil down to properties of this dg algebra. Hence, dg algebras become models of "noncommutative schemes" [37], [60]. This approach has proven useful in topics ranging from deformation quantization to homological mirror symmetry. In this spirit, we study dg algebras [28], their twins, A1-algebras [27], stacks, and in particular deformations and Hochschild cohomology of these objects.#### Researcher(s)

- Promotor: Van Oystaeyen Fred
- Fellow: Lowen Wendy