The objective of this project is to study the topological entropy for Reeb flows and its relations with symplectic and contact topology. Reeb flows are a special class of dynamical systems which lie at the intersection of geometry, topology and mathematical physics. The class of Reeb flows includes the geodesic flows of Riemannian metrics and important examples of Hamiltonian dynamical systems. The dynamical properties of Reeb flows are strongly related to the topological properties of contact and symplectic manifolds. In this project we study the behaviour of the topological entropy of Reeb flows. The topological entropy is an important dynamical invariant which codifies in a single non-negative number the exponential complexity of a dynamical system. If the topological entropy of a dynamical system is positive then the system exhibits some type of chaotic behaviour. In this project we propose to: A) better understand the dynamics of Reeb flows with positive topological entropy using invariants coming from Floer theory; B) to use topological methods to construct new examples of Reeb flows with zero topological entropy; C) to use Floer theory to study how topological entropy varies under perturbations of Reeb flows.

Geometric mechanics refers to a variety of topics that lie at the intersection of differential geometry, dynamical systems, and analytical mechanics. The main idea is to identify the geometric structures underlying many classes of physical and engineering systems. These geometric structures can be useful for the qualitative study of the system and can also be used for instance in the design of control laws and geometric integrators. The property that a system can be derived from a variational principle is one of these useful structures. In this project we will use the inverse problem of the calculus of variations to find stabilizing controls for a variety of mechanical systems. One of the advantages of finding a variational structure is that we can then use energy methods to show stability, or find conditions for stability. We will also introduce more flexibility to the classical inverse problem to extend its possible applications. More precisely, for the inverse problem on a Lie algebra we will allow variable structure constants in order to have more freedom in the energy shaping step. The theory of exterior differential systems has been applied successfully to the inverse problem to identify variational cases. We will also adapt these techniques to the inverse problem for constrained systems with an eye towards the problem of Hamiltonization of nonholonomic systems. Finally we will also study geometric integrators for metriplectic and dissipative systems.

To each monoid we can naturally associate a topos, consisting of sets with a right action of this monoid. This opens the door to many geometrical invariants associated to the monoid, following the philosophy of toposes as generalized topological spaces. For example, toposes have points, and for the toposes associated to monoids, calculating the points can give surprising results. A simple example is the monoid of nonzero natural numbers under multiplication. Alain Connes and Caterina Consani showed that the points of the associated topos are up to isomorphism given by a double quotient featuring the finite adeles. They then constructed a structure sheaf on the topos, and showed that this combination of topos and structure sheaf, their Arithmetic Site, is related to the noncommutative geometry approach to the Riemann Hypothesis. In this research project, we will systematically study the toposes associated to monoids from a geometric point of view. In certain cases, we will construct structure sheaves on these toposes, leading to generalized Connes-Consani arithmetic sites.

The project is about the study of two groups which are naturally assosciated to a field and which contain different information on the arithmetic of the field, related to quadratic form theory. One is the quotient of the group of nonzero sums of squares modulo the subgroup of sums of two squares, the other one is the so-called Kaplansky radical. These two groups are studied in a context of arithmetic geometry, namely for algebraic function fields over a complete discretely valued field. Such a field can be viewed as the function field of an arithmetic surface over a discrete valuation ring. Classical reduction theory of algebraic curves relates our problem to the study of the special fiber of this arithmetic surface, which is an algebraic curve over the residue field. The reduction curve is connected, and the analysis of the genera of its irreducible components and of the intersections of these components relates our problems (on the two specific groups) to a combinatorial problem on the associated intersection graph. The aim of the PhD project of Gonzalo Manzano Flores in this context is in particular to obtain interesting examples of curves which illustrate how big the two groups can be under given conditions on the genus of the function field. We are searching for such examples over very specific complete discretely valued fields, namely Laurent series fields in one variable over the real and over the complex numbers, R((t)) and C((t)). The PhD candidate has obtained results on the first group (sums of squares modulo sums of two squares) in the case of a function field of a hyperelliptic curve over the field R((t)). These examples show that the upper bound for the size of this group obtained in Becher, Van Geel, Sums of squares in function fields of hyperelliptic curves. Math. Z. 261 (2009): 829–844, is best possible in the case of the function field of a hyperelliptic curve of any even degree with a nonreal function field. On the other hand, Gonzalo Manzano Flores also found evidence that this bound can be slightly improved in the case of a real function field, and that the improved bound then is optimal. The PhD candidate is currently writing up these results, which will cover a substantial part of his thesis. Complementary to this topic, we started in 2019 to work on the Kaplansky radical for function fields of arithmetic surfaces, and obtained first results and additionally evidence for some intriguing relations to the other problem. The Kaplansky radical was introduced in the 1970's by C. Cordes as a way to generalise a couple of results in quadratic form theory by substituting this group for the subgroup of squares in the field. However, it took a while until examples of fields with a nontrivial Kaplansky radical were discovered, in the 1980's by M. Kula. In the last two decades, a series of deep results on local-global principles have come up in quadratic form theory, which hold over function fields of arithmetic surfaces, and it has turned out that there is often in these situations a failure of the local-global principle in dimension 2. It was described in Becher, Leep, The Kaplansky radical of a quadratic field extension, Journal of Pure and Applied Algebra 218 (2014), 1577-1582 that this failure in dimension 2 is directly expressed by the Kaplansky radical of the corresponding function field. The aim of the collaboration with Gonzalo Manzano Flores and his supervisor at USACH, prof. David Grimm, during the period of funding will be to complement this observation by a variety of generic examples, in particular for hyperelliptic curves over C((t)). These results together with the topic on sums of squares shall lead in the academic year 2020/21 to a completion of the PhD of Gonzalo Manzano Flores, according the dubble degree convention between USACH and UA, with myself and prof. David Grimm as promotors.

Semitoric systems are a type of dynamical system (such as a spinning top) which satisfy certain symmetries. These systems can be well understood in terms of five invariants which can be recovered from the system. Semitoric systems lie in the field of symplectic geometry, another subfield of symplecitc geometry is Floer theory, which attempts to compute and understand certain invariants of symplectic manifolds and their Lagrangian submanifolds (a type of submanifold which arises naturally in the study of semitoric systems). We propose to (1) initiate research to better understand the invariants of semitoric systems (2) expand results and ideas from semitoric systems to more general systems (including those with so-called hyperbolic points, which are more common in nature) (3) explore the connection between semitoric systems, integrable systems, and Floer theory.

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?

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.

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.

In algebra, one studies algebraic operations (compositions) on a given vector space, obeying some compatibility laws. The simplest example is an associative algebra structure on a vector space. The famous Eckmann-Hilton argument shows that, given two compatible unital associative algebra structures on the same vector space, both structures are equal and are commutative. It can be interpreted by saying that the world of vector spaces is very rigid, and no interesting higher structures can be derived from two compatible associative algebra structures. The higher category theory provides a more relaxed environment than the category of vector spaces. Thus, two compatible associative structures in such a relaxed environment produce a homotopy 2-algebra structure on a given space. The generalized Deligne conjecture aims to derive higher structures similar to a homotopy 2-algebra, starting with a (higher n-) monoidal abelian category. For n=1 it gives precisely a homotopy 2-algebra structure, as it was proven in two recent promoter's papers. In historically the first case, considered by P.Deligne, and referred to as the classical Deligne conjecture, one has a homotopy 2-algebra structure on the Hochschild cochain complex of an associative algebra. The main goal of this project is to generalize our previous results for an arbitrary n-monoidal abelian category, for n greater than 1. The case n=2 is of special importance for deformation theory of associative bialgebras.

During the past decades, research of symplectic geometry accelerated providing lots of new tools and applications for very different topics in mathematics. This Excellence of Science (EoS) research project pushes these ideas further to areas not immediately connected to symplectic geometry.

Geometric methods in the study of dynamical systems have the advantage of providing global rather than local results. Quite often, the dynamical systems that arise in theoretical physics or other sciences can be seen to be essentially defined by a geometric structure and further auxiliary data. In this project we will mainly concentrate on aspects related to symmetries of such dynamical systems. Symmetry has the defining property that it maps solutions of the system onto solutions. Since the number of dynamical systems for which we can easily write down the solutions in closed form is extremely limited, finding symmetries is a key step in the process of solving the governing differential equations for the problem at hand. Symmetries and, possibly, their associated conserved quantities can often be used to reduce the problem to a smaller system of differential equations, which may be easier to solve. Any further attempt to integrate the reduced system will rely on how much of the geometric structure of the unreduced system is transferred to the reduced one. This project has the overall goal to investigate, mainly in the context of singular Lagrangian systems, the most appropriate conditions for the existence of such structure-preserving reduction.

Named after the Irish physicist, astronomer, and mathematician W. R. Hamilton (1805-1875), Hamiltonian systems are an important class of dynamical systems with certain conservation laws and rigidity features. A well-known classical example is the n-body problem (`movement of the planets around the sun'). In fact, Hamiltonian systems appear in many shapes throughout mathematics, physics, chemistry, biology, and engineering. Classical Hamiltonian problems are formulated as systems of ordinary differential equations on finite dimensional spaces. Nevertheless, there are also equations that can be reformulated as Hamiltonian systems, but this time on infinite dimensional spaces. Such systems are called Hamiltonian partial differential equations, in short Hamiltonian PDEs. Examples are the Korteweg-de Vries equation, the Sine-Gordon equation, the nonlinear Schrödinger equation, nonlinear sigma models etc. This project starts out from a `triholomorphic' Dirac-type equation on a so-called hyperkähler manifold. It can be transformed it into a Hamiltonian PDE on the infinite dimensional loop space of the manifold. For this new equation, we expect to show conservation laws, integrability (`extra symmetries'), features from modern symplectic geometry (`non-squeezing' properties, symplectic capacities etc.), and estimates on the number of periodic solutions (`infinite dimensional Arnold conjecture'). So far, only three people studied Hamiltonian PDEs with modern symplectic methods. Hamiltonian PDEs with hyperkähler background having symplectic features, as in this project, have never before been investigated. Due to its link to the Cauchy-Riemann-Fueter equation in the supersymmetric sigma model, our results may also be of interest to physicists.

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.

For a large number N, the Riemann Hypothesis would give a very precise estimate of the amount of prime numbers smaller than N. Because prime numbers are the foundation of number theory, many mathematical problems depend on it, which is why the Riemann Hypothesis is considered to be one of the most important unresolved problems in mathematics. In the 1940's, André Weil, a famous mathematician and brother of philosopher Simone Weil, has proved a variant of the Riemann Hypothesis regarding estimation problems for a very specific type of polynomials. His proof was related to geometry, more specifically to the study of curves. In a recent series of papers, Alain Connes and Caterina Consani have described an approach to the Riemann Hypothesis by introducing and studying the Arithmetic Site. This is a new geometric object describing the distribution of the prime numbers, constructed with contemporary mathematical techniques. It has properties similar to that of a curve in geometry, so the hope is that eventually Weil's proof can be translated to a proof of the original Riemann Hypothesis. While Connes and Consani focus on a tropical geometry point of view, we will construct alternatives that allow for a more algebraic point of view. In particular, we want to relate their approach to the study of noncommutative algebras, a subject for which the University of Antwerp is well-known.

Semitoric systems are a type of dynamical system which satisfy certain symmetries. These systems can be well understood in terms of five invariants. Semitoric systems lie in the field of symplectic geometry, another subfield of symplecitc geometry is Floer theory, which attempts to compute and understand certain invariants of symplectic manifolds and their Lagrangian submanifolds (a type of submanifold which arises naturally in the study of semitoric systems). This project will (1) initiate research to better understand the invariants of semitoric systems (2) expand results and ideas from semitoric systems to more general systems (including those with so-called hyperbolic points, which are common in nature) (3) explore the connection between semitoric systems, integrable systems, and Floer theory.

In mathematics, a ring is a collection of objects which one can add, subtract and multiply sort of like we are used to with numbers. For example, the collection of integers (…, -2, -1, 0, 1, 2, …) forms a ring, since they can be added, subtracted or multiplied to form new integers. In our research, we intend to study techniques which might establish relationships between the computational complexity of different rings. While we find this interesting in its own right, it is further motivated by questions about which parts of mathematics can be 'automatized', in the sense that one can write a computer program to solve certain types of mathematical problems. A notorious example of this is the following, stated in some form by David Hilbert in 1900: to find an algorithm which tells you whether a given equation has a solution or not. As it turned out, what such a program might look like and even whether it exists depends heavily not only on the sorts of equations one considers, but also on what kinds of solutions one allows: only integers? Also fractions of integers? What about irrational numbers like pi? For integers, it is known that such a program cannot possibly exist; the problem cannot be automatized. On the other hand, a program is known in the case one allows all real numbers. For fractions of integers, the question remains unsolved. The techniques we will consider might bring us closer to an answer, for example by relating the complexity of the fractions and the integers.

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.

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.

A lot has changed in geometry since the study of shapes (like triangles) in ancient Greece. The shapes studied nowadays very often have more than three dimensions and are curved, and can be extremely complicated. This is useful for fields like physics or engineering, which have also evolved drastically since Pythagoras and need geometrical techniques that become more and more involved. In this project we want to investigate a way to stretch the abstraction of geometry still further. To this extent, we will need concepts from algebra, like coordinates and equations, but further elaborated and more abstract. More precisely, we will study 'rings', collections of values that you can multiply or add. One particular kind of rings that will be important are 'Azumaya algebras'. We will need to solve some specific questions about these algebras to get a better understanding of the geometry in question. The study of Azumaya algebras or rings in general is also interesting on its own because they appear everywhere in mathematics, and there are still a lot of unresolved questions about them. Additionally, the proposed research will have implications in physics, more precisely string theory. From a string theorist's perspective, the smallest building blocks of the universe are vibrating strings (like guitar strings). Endpoints of these strings are called 'D-branes', and they are described accurately by the geometry we propose to explore.

To a superpotential \Phi is associated an associative algebra J_{Q,\Phi} by taking cyclic derivatives of \Phi. If C\Phi is a 1-dimensional G-subrepresentation of CQ/[CQ,CQ], then G acts on J_{Q,\Phi} as algebra automorphisms. The point of this research is to study J_{Q,\Phi} in the hope of finding Cayley-smooth orders (that is, algebras finite over their center such that their trace preserving representation varieties are smooth) on which G acts. The starting point would be to take a superpotential \Phi_0 which gives an algebra J_{Q,\Phi_0} that is PI and consider other superpotentials \Phi such that C\Phi \cong C\Phi_0 as G-representation, such that \Phi degenerates to \Phi_0 in a controlled way.

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.

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.

This research project studies aspects of interactions between integrable Hamiltonian systems and modern symplectic geometry: The symplectic classification in the sense of Pelayo & Vu Ngoc's is completed for two seminal examples of semitoric systems, namely the coupled spin oscillator and coupled angular momenta. In addition, more general families are studied and partially classified.

A lot has changed in geometry since the study of shapes (like triangles) in ancient Greece. The shapes studied nowadays very often have more than three dimensions and are curved, and can be extremely complicated. This is useful for fields like physics or engineering, which have also evolved drastically since Pythagoras and need geometrical techniques that become more and more involved. In this project we want to investigate a way to stretch the abstraction of geometry still further. To this extent, we will need concepts from algebra, like coordinates and equations, but further elaborated and more abstract. More precisely, we will study 'rings', collections of values that you can multiply or add. One particular kind of rings that will be important are 'Azumaya algebras'. We will need to solve some specific questions about these algebras to get a better understanding of the geometry in question. The study of Azumaya algebras or rings in general is also interesting on its own because they appear everywhere in mathematics, and there are still a lot of unresolved questions about them. Additionally, the proposed research will have implications in physics, more precisely string theory. From a string theorist's perspective, the smallest building blocks of the universe are vibrating strings (like guitar strings). Endpoints of these strings are called 'D-branes', and they are described accurately by the geometry we propose to explore.

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.

Quadratic forms are homogeneous polynomials of degree 2. They play an important role in modern research as special cases of various mathematical structures. Their explicit aspects make them suitable for applications. In the 1930s Ernst Witt classified quadratic forms over an arbitrary field (an algebraic structure like the rational or real numbers) by introducing the so-called Witt ring. Around the same time a similar link was drawn regarding central simple algebras, which can be classified using the Brauer group. A connection between these two algebraic structures is given by associating to a quadratic form its Clifford algebra. Merkurjev's Theorem from 1981 describes the connection. In theory, it characterizes when two forms have the same Clifford algebra and it discribes the algebras which are equivalent to Clifford algebras. The first aim of this project is to link the problem on the number of generators of the third power of the fundamental ideal in the Witt ring to Merkurjev's Theorem. Secondly, we wish to search for a more explicit proof of Merkurjev's Theorem. Thirdly, we want to make use of the fact that the central simple algebras characterized by Merkurjev's Theorem carry involutions. Further, we want to study pairs of quadratic forms in order to study the isotropy (existence of a solution) of quadratic forms over the rational function field over an arbitrary field. Finally, we wish to address the first two problems using axiomatic theory of Witt rings.

Physical systems are often described in terms of differential equations. To find solutions for these systems, one can investigate the symmetries of the system: these are transformations which generate solutions, through well-defined relations in a specific algebra. Often, this leads to special solutions which are referred to as 'special functions' (a whole mathematical subject on its own). In this project, we will start from a specific type of equations (conformally invariant equations for massless particles of arbitrary spin) and investigate the special functions appearing in this framework.

Let the density of an unknown probability distribution belong to a family of densities which are determined up to an unknown parameter vector. Based on a sample of observations, the maximum likelihood estimator (MLE) then provides an estimate for the unknown parameter vector by picking the vector under which the chance of observing the particular given sample is maximal. Statisticians value the MLE because it behaves well asymptotically. This roughly means that the estimates given by the MLE will converge to the true value of the unknown parameter vector as the sample size grows to infinity. However, doubts about the applicability of the MLE have emerged as `misspecified models', i.e. models in which the density of the unknown probability distribution fails to belong to the family of densities producing the MLE, are common in realistic settings. Many researchers have investigated under which additional regularity conditions the MLE in a misspecified model continues to behave well asymptotically. Here we want to follow a different route. More precisely, instead of adding extra regularity conditions on the model (which may again destroy the applicability), we will try to establish results in which we measure how irregular a model is and then use this information to assess how asymptotically well the MLE behaves. To this end, we will use approach theory, a mathematical theory which is designed to cope rigorously with notions such as `almost behaving well asymptotically'

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.

The project addresses problems in Quadratic Form Theory motivated by major results obtained recently in this area. It emphasizes the use of explicit and computational approaches to obtain new results as well as refinements of results, in particular on statements about the existence of a certain representation of an object such as a quadratic form or an algebra with involution with certain properties (e.g. in terms of ramification or values for invariants). It further strives for providing criteria helping to decide for special fields whether certain forms satisfy a local-global principle or to bound certain field invariants.

A general goal of the project is to study the higher algebraic structures, which make it possible to apply the Tamarkin scheme for proving the formalities similar to the Kontsevich-Tamarkin formality, for more involved deformation problems.

This project represents a formal research agreement between UA and on the other hand the client. UA provides the client research results mentioned in the title of the project under the conditions as stipulated in this contract.

Three different approaches lead recently to the solution of a longstanding open problem in algebra, namely to prove that the u-invariant of the function field of a curve over a p-adic number field is equal to eight. The u-invariant of a nonreal field is the smallest integer n such that every quadratic form in more than n varaibles over the field has a nontrivial zero. The different approaches are: (i) a combinatorial approach giving a far stronger result on systems of quadratic forms even over finitely generated extensions over any local number field; (ii) an approach using Galois cohomology and the construction of elements therein with given ramification; (iii) an approach called 'Field Patching' leading to new local-global principles for isotropy of quadratic forms over function fields of curves over a complete discrete valued field. Any of the three aproaches leads to more general results that so far are not attainable by the other two methods, for example (ii) and (iii) do currently not apply when p=2. The proposed doctoral research project strives for a comparative analysis of the three methods and a better understanding of their strengths and limitations. As an application it is expected that the u-invariant as well as related field invariants for other fields can be determined in this manner.

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.

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.

This project consists of three main topics: T1 Construction of new realizations of osp(1/2n) T2 Integral transforms associated with osp(1/2n) T3 Analysis related to the orthosymplectic transvector algebra

In this project we aim to develop a comprehensive and universally applicable theory of quantitative analysis of hitherto only topological structures (e.g. weak convergence, finite dimensional convergence, convergence in probability and in law) on spaces of probability measures and random variables (in particular continuous and cadlag stochastic processes) by replacing the topologies by canonical and intrinsically richer isometric counterparts, eventually aiming to prove quantitative versions of the fundamental results of stochastic analysis such as e.g. Prohorov's theorem and various important limit theorems.

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.

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.

We study the Brauer group of some concrete monoidal categories, with emphasis on the equivariant Brauer group of a triangular pointed Hopf algebra, the Brauer group of differential graded algebras, and the Brauer group of a coquasitriangular dual pointed Hopf algebra. We will study the groupoid of biGalois objects over a Hopf algebra. We will introduce a categorical version of the Clifford functor and the Brauer-Wall group.

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.

In this project we aim to develop a comprehensive and universally applicable theory of quantitative analysis of hitherto only topological structures (e.g. weak convergence, finite dimensional convergence, convergence in probability and in law) on spaces of probability measures and random variables (in particular continuous and cadlag stochastic processes) by replacing the topologies by canonical and intrinsically richer isometric counterparts, eventually aiming to prove quantitative versions of the fundamental results of stochastic analysis such as e.g. Prohorov's theorem and various important limit theorems.

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''.

The theory of lax algebras is a recent theory which helps us to describe certain categories in a different way. We obtain new characterisations for Top and Ap. I look closer to Ap and describe this category by using functional ideals. Now I try to adapt the technique to Lip (Lipschitz spaces), again I consider ideals of functions, but with another saturation condition. Later on the question will be: "Can we describe a category of lax algebras, starting from an arbitrary operation and saturation condition?"

The first cornerstone is the study of new quantitative convergence structures in measure theoretic context, in particular on spaces of random variables and probability measures. We will mainly be concerned with structures strongly related to the p-Wasserstein distance, a topic popular for both applications and theoretical aspects. The second cornerstone is the application of the theory to stochastic analysis (convergence of Feller processes, martingales and solutions of stochastic differential equations) and statistics (convergence of estimators in parametric and non-parametric models).

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.

Starting with the works by Nadine Bellamy (1999), Damien Lamberton (1997), Steve E. Shreve (2004) as well as that of Erhan Bayraktar and others (2003), respectïvely about stochastic processes with jumps in finance and market models with inertia, we intend to define basic differential equations (with jumps) which reflect at the same time the effect of inertia and jumps in view of evaluation and hedging of options in a financial market.

We aim to apply approach theorie, developed by the research group ATS, on spaces of measures, and thus to obtain a generalisation of the already exisiting so called weak approach structure on such spaces. Thereby, the interaction with the already existing convexity theories [Z. Semadeni, N. Pumpluen, R. Rohrl] will be of paramount importance, as it will enable us to recover the algebraic component of our theory. Thus we have strong indications to obtain a monad for describing Stochastic Processes [M. Giry] in a non metrical context.

During the previous years the candidate has studied Clifford algebra valued modular forms on arithmetic subgroups of the orthogonal group O(1,n) and that are annihilated by Dirac type operators. The aim of this project is to apply and to extend these techniques to generalizations of Clifford algebras. This shall give further insight in the study of discrete quantum groups and Hopf algebras in the framework of non-commutative geometry.

An important part of the program will consist of a study of Markov semi-groups and propagators, by employing techniques taken from functional analysis and from the theory of stochastic processes. For locally compact spaces there is a one-to-one correspondence between strongly continuous Feller-Dynkin semi-groups and strong Markov processes (with certain continuity properties. Feller-Dynkin semigroups leave the space of bounded continuous functions vanishing at infinity invariant. This correspondence gives rise to an interaction between stochastic analysis and classical operator semi-group theory. However, many interesting spaces are not locally compact. Nevertheless these more general spaces are interesting and important from the viewpoint of stochastic analysis and possibilities for applications. Examples of such spaces are: Wiener space, loop spaces, Fock space. Many of these spaces are Polish. The idea is to develop an analysis which includes these Polish spaces. In this context we also want to investigate the martingale problem. In the commutative case this leads to problems as described in the paper: J.A. van Casteren, Some problems in stochastic analysis and semigroup theory (in Proceedings of the First International Conference of Semigroups of Operators: Theory and Applications, December 1998, Newport Beach, California The main organizer/chairman is A.V. Balakrishnan). Series: Progress in Nonlinear Differential Equations and Their Applications, Vol. 42; Publisher: Birkhauser Verlag, Basel, Switzerland, 2000; pp. 43--60. These problems were for a part repeated and further elaborated in J.A. Van Casteren, Markov processes and Feller semigroups "Conferenze del Seminario di Matematica dell'Universita di Bari" Estratti Dalle Conferenze del Seminario di Matematica, Proceedings of the Summer School Operator methods for Evoluton Equations and Approximation Problems (OMEEAP) 2002, Roma 2003, pages 19-93. In the non-commutative situation some other but related problems were described in the same publication (here ``positive'' is replaced with ``completely positive'', and Feller generator is substituted by Lindblad generator). Shortly the book Jan A. Van Casteren, Markov processes, Feller semigroups and evolution equations, 650 pages will be resubmitted for publication. This book contains many of the results described above.

Approach frames are a pointfree abstraction of approach spaces. We research the properties of this category, which concepts from approach theory and frame theory form natural constructions in this new framework and how this interaction between frame theory and approach theory gives us new concepts to study approach spaces.

In this project the usability of OFDM modulation for digital acoustical underwater communication will be investigated. In addition, spectral noise characteristics will be determined, as these will enable us to specify a number of system parameters of the OFDM communications system.

The idea is to generalize and improve theorems, which exist for Markov processes with locally compact state space, to a more general topological setting. In fact the state space should be replaced with a not necessarily locally compact polish (or more general) topological space. Some new techniques and methods will be developed. These methods will have a functional analytic as well as a stochastic aspect. One of the problems will be to give an adequate definition of a generator of such a process. The corresponding martingale problem also requires a new approach. We hope to apply our results to concrete infinite-dimensional problems in mathematical analysis.

One of the main results obtained in earlier research is a proof for the Popov conjecture [1] for quiver representations, i.e. the quotient maps are not equidimensional if the quotient variety has singularities. This study is to expand to the more general situation of Brauer-Severi varieties (introduced by M. Van den Bergh) over (smooth) Cayley-Hamilton orders. The dimension of the fiber of the Brauer-Severi fibration over an arbitrary point in the variety of isomorphism classes of trace preserving representations of a smooth Cayley-Hamilton order will be computed. In the end, this should lead to determining when such a Brauer-Severi fibration is a flat morphism.

Approach frames are a pointfree abstraction of approach spaces. We research the properties of this category, which concepts from approach theory and frame theory form natural constructions in this new framework and how this interaction between frame theory and approach theory gives us new concepts to study approach spaces.

In this project, the limits of ultrasonic underwater datacommunication concerning bandwidth and distance will be investigated. Parameters will be caracterized using theoretical models and empirical measurements. Ultrasonic transducers must be selected. A simulation model of the system will be built. With this model engineering choices can be made to build a prototype communication system.

A central issue in this project is the use of stochastic methods and techniques (like Markov Processes, Brownian motion, martingale techniques) to investigate forward and backward (stochastic) differential equations. A crucial role are played by the linear (and non-linear) Feynman-Kac formula, as well as by related formulae like the Girsanov (or drift) transformation (Cameron-Martin formula). Operator semi-groups are extensively employed throughout the whole project.

In our work we look for methods to desingularise quotient varieties of the variety of n-dimensional representations of an algebra A, under the natural action of GLn. In particular in the case of isolated singularities, we hope to make progress.

The following topics wil be discussed. (1) Establishing and better understanding the intimate relationship between Euclidean Quantum Mechanics, martingale measures and generators of diffusions. (2) Extend the existing results on generators of strong Markov processes (with locally compact state spaces) to Polish and other topological spaces. The strict topology plays a central role here. Do something similar in the non-commutative context: again a notion of strict topology is available. (3) Try to improve the knowledge about stochastic partial differential equations and of backward stochastic differential equations. In particular, indicate the existing relationship between Neumann semigroups, reflected Markov processes, backward stochastic differential equations, and singular limits of quadratic forms.

Rationality problem for quotients of PFLn-varieties. Connection between ringtheoretical properties of Sklyanin algebras and arithmetic of elliptic curves.

Recent interactions between physics and noncommutative algebra gave rise to the creation of a new area in mathematics : 'Noncommutative Geometry'. The European Science Foundation selected this for a European Priority Programme that was funded by 12 member countries. The NOG-programme is involved in the organization of congresses, workshops and summerschools and also provides fellowships and travel grants for research cooperation. See web-page win-www.uia.ac.be/u/nog2000 for more information.

General Mathematical research with particular emphasis on interdisciplinary aspects.