Agricola, Ilka (University of Marburg):
Homogeneous Einstein metrics and their geometric properties
The scientific focus of this project is on the relation between geometric structures on homogeneous Riemannian manifolds and special metrics that they may carry, in particular, Einstein metrics. We will classify those homogeneous Einstein manifolds that are spin, determine the underlying G-structure and its characteristic connection, and, finally, compute the spectrum of Kostant's cubic Dirac operator (this is precisely the Dirac operator of the characteristic connection).
Burban, Igor (University of Cologne):
Classical Yang-Baxter equation and sheaves on degenerations of elliptic curves
The goal of this project is to study solutions of Yang-Baxter equations (associative, classical and quantum) arising from the geometry of simple vector bundles on curves of genus one. In particular, using this geometric approach, we expect to find new rational solutions of the classical Yang-Baxter equation. This project turns out to be closely related with representation theory of finite dimensional algebras, in particular with matrix problems and representations of bocses. On the other hand, it leads to new algebra-geometric problems about Fourier-Mukai transforms on elliptic fibrations.
Geometry and representation theory in computational complexity
Can one compute the permanent of an n by n matrix with a number of arithmetic operations bounded by a polynomial in n? How many arithmetic operations are sufficient for calculating the product of two matrices? These two questions are undoubtedly the most important open problems of algebraic complexity theory. The first question is closely related with the famous P versus NP problem. Surprisingly, both problems allow a natural formulation in terms of geometric invariant theory and representation theory. For the matrix multiplication problem, this has been realized long ago by Strassen. For the algebraic P versus NP problem, this connection was discovered and considerably pushed further by Mulmuley and Sohoni in the last years. In both cases, questions of computational complexity are first reduced to geometric questions, and then to specific questions about the splitting of irreducible representations when restricting to the subgroup of symmetries of the problem under investigation. One ends up with questions about special instances of Kronecker products and plethysms. Unfortunately, those classical problems concerning the representations of symmetric groups and general linear groups are not well understood. The goal of this project is to prove better lower complexity bounds for the permanent and matrix multiplication based on the Mulmuley Sohoni approach, in collaboration with specialists of representation theory.
Cerulli Irelli, Giovanni (University of Bonn):
Categorification of Positivity in Cluster Algebras
In every cluster algebra there is a natural notion of positivity, first noticed by Sherman and Zelevinsky. This project has two main objectives: on one hand investigate this notion in relation with the wellâ€“known notion of total positivity in semisimple algebraic groups, in the case when the cluster algebra is the coordinate ring of the corresponding group. On the other hand, our aim is to interpret the positive elements in the language of categorification of the cluster algebra theory. The final goal is to be able to give an explicit, satisfactory answer to the following question: are cluster algebras and their categorifications the right framework to study total positivity?
Affine Nichols algebras of diagonal type and modular tensor categories
Root systems and crystallographic Coxeter groups are central tools in the study of semisimple Lie algebras. In the structure theory of pointed Hopf algebras a similar role is expected to be played by Weyl groupoids and their root systems. Finite universal Weyl groupoids correspond to the so-called crystallographic arrangements. These are arrangements of hyperplane satisfying a certain global axiom of integrality. In a series of papers, Heckenberger and Cuntz achieved a complete classification of crystallographic arrangements up to isomorphisms.
We propose to extend the results on Nichols algebras of diagonal type and Weyl groupoids to the affine case. We expect similar results as in the classical theory. In particular, there should also exist an exotic Fourier transform matrix connecting the combinatorics of crystallographic arrangements to certain modular tensor categories. Our second objective is to find for each finite Weyl groupoid $W$ an associated algebra $H$ having $W$ as symmetry structure. We propose to obtain this result by the study of the sheaf of (skew) differential operators on the toric variety of the corresponding crystallographic arrangement.
Investigations into the Abelian Defect Group Conjecture
One of the central questions in modular representation theory of ﬁnite groups is to what extent the ‘global’ representation theory of a ﬁnite group G over a ﬁeld k of characteristic p > 0 is already controlled by ‘local’ data, that is, representations of p-subgroups of G and their normalizers. This area has been extremely active and rich in fascinating development in recent years. In his 1990 landmark paper, M. Brou´e conjectured that every block of kG with an abelian defect group is derived equivalent to its Brauer correspondent, and has thus essentially the ‘same’ representation theory as a block of a much smaller group algebra. Whilst the conjecture has been proved in a number of special cases, it remains open in its general form. The aim of this project is to make further progress on this long-standing conjecture, by verifying it for substantial series of ﬁnite groups, to give new evidence for the conjecture to hold true, and to improve on the methods to prove the conjecture in general. To achieve this, we will combine theoretical methods with powerful techniques from computational representation theory.
Spectral theory of Green functors and other commutative 2-rings
Let G be a finite group. Every Green functor for G - such as the Burnside, the cohomology, or the representation Green functor - captures some aspect of the representation theory of G. If the Green functor is commutative, as in the previous examples, then it admits a symmetric tensor product of its modules, and its derived category is a tensor triangulated category. Our first goal is to understand the extent to which this is like the derived category of a commutative ring. In particular we aim at proving a classification theorem for its thick subcategories of perfect complexes, in terms of subsets of a suitable topological space, the spectrum of the Green functor; this result should generalize the classification for commutative rings (the case G=1) and Quillen stratification in modular representation theory. Our second goal lies in developing the proof methods themselves, i.e. new techniques of tensor triangular geometry adapted to Mackey and Green functors. We also wish to explore the natural context for this enterprise, namely presheaves over small symmetric tensor categories, or "commutative 2-rings". Our third goal is to provide applications to G-equivariant stable homotopy (via the Burnside ring Green functor)and to G-equivariant KK-theory (via the representation ring Green functor).
Ebeling,Wolfgang (Leibniz University Hannover):
Homological mirror symmetry for singularities
The primary objective of the project is to study homological mirror symmetry for singularities in order to gain, by means of representation theory, a better understanding of some mysterious phenomena discovered in singularity theory. A two-year postdoc position (David Ploog, University of Toronto) will be financed by the DFG and funds for travel and research visits are approved.
Farnsteiner, Rolf (University of Kiel):
Extensions of Tame algebras and finite group schemes of domestic representation type
Let k be an algebraically closed ﬁeld. Given an extension A : B of ﬁnite-dimensional kalgebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is preserved under ascent and descent. These results are then used to show that principal blocks of ﬁnite group schemes of odd characteristic are of polynomial growth if and only if they are Morita equivalent to trivial extensions of radical square zero tame hereditary algebras. In that case, the blocks are of domestic representation type and the underlying group schemes are closely related to binary polyhedral group schemes.
Dualities in the representation theory and geometry of loop groups
There are two essentially different ways to link the geometry of affine Grassmannians to representation theory. The first relates D-modules to representations via the Beilinson-Bernstein localization functor. The second uses moment graph localization and produces, starting from parity sheaves, certain G_1T-modules for the Langlands dual datum. In the project we want to study the relations between these two approaches. In particular, we want to understand how the Satake-equivalence fits into the picture.
Symmetry groups of differential or difference equations and their representations on the solution spaces
The aim of the project is to study symmetry groups of differential or difference equations by means of their representations on the solution spaces. To a differential or difference equation one associates an algebraic group which describes the symmetries and encodes a lot of information about the solutions of the equation. By Tannakian theory, this group is determined by the structure of its representations on objects of the tensor category generated by the solution space. The aim of this project is to apply methods from the structure and representation theory of algebraic groups to study differential (or difference) equations. The first application is the computation of the symmetry group (Galois group) of a given equation. A second goal is the construction of objects with interesting symmetry groups.
Heckenberger, István (University of Marburg):
Spherical subalgebras of quantized enveloping algebras --- structure theory and classification problems
Spherical subgroups of Lie or algebraic groups have been investigated since the 1970ies because of their interesting geometric, algebraic, combinatorial and representation theoretical properties. Nowadays generalizations like spherical varieties are in the focus of interest. For another generalization towards quantum groups one first has to find the proper setting: an obvious description via Hopf subalgebras of Hopf algebras fails due to the lack of sufficiently many Hopf subalgebras. Based on the case-by-case construction of quantum symmetric spaces in the 1990ies and on recent developments on right coideal subalgebras of quantized enveloping algebras, now we are in the position to develop a substantial structure theory of spherical subalgebras of quantized enveloping algebras using right coideal subalgebras and to initiate classification projects. (In the classical, cocommutative setting right coideal subalgebras are automatically Hopf subalgebras.) It is a very interesting question, to which extent the classical and the quantum theories and examples are analogous and whether one can observe significant new aspects.
Complex geometry of actions and related representations
The interplay between actions of a given group on geometric objects having rich structure and linear representations on associated function spaces is studied. The groups being considered are complex Lie groups and their real forms. The geometric objects on which they act are of a complex analytic nature, e.g., complex manifolds. The representations are on spaces of functions which respect the complex structure, often just vector spaces of special holomorphic functions. A typical goal is to attempt to understand attributes of the function algebra in terms of the group structure and its action. In many aspects of the project the information of interest comes from restricting a given action of a complex Lie group to a real form. In that setting the Hamiltonian method is applied in order to identify special regions of representation theoretic relevance. This is often an orbit of the real form. In that case a nonlinear transform is introduced in order to realize the representation space as a function space where the introduction of invariant linear geometry,unitarization, is one particular goal of the project.
Investigations on Alperin's weight conjecture through Hecke algebras
In its original version, the conjecture postulates that two seemingly unrelated sets of objects, constructed from the representation theory of a group, have the same number of elements. One of these sets is defined by "local data", i.e. by data constructed from proper subgroups, the other set only by "global data" of the group itself. Thus the philosophy underlying the conjecture agrees with that behind so many other famous conjectures of group theory and representation theory: global data should be determined by local data. This project follows a suggestion of Alperin to study his conjecture through the modular Hecke algebra. (If G is a finite group and p a prime, the modular Hecke algebra (with respect to p) is the endomorphism ring of the G-permutation module on the cosets of a Sylow p-subgroup of G.) This could serve as a mediator between local and global theory. To date there are only few attempts to investigate the modular Hecke algebra systematically, in particular its role in Alperin's conjecture. The experimental results of Naehrig's 2008 PhD thesis suggest that a deeper investigation of the representation theory of this object might well be worthwhile. This is what we pursue in our project.
Investigations on the conjectures of McKay and Alperin-McKay
This research project is located in the representation theory of finite groups. Our aim is the investigation of two prominent and long standing open conjectures, the McKay conjecture and its refinement by Alperin, the Alperin-McKay conjecture, respectively. These conjectures were formulated in the mid 70th of the last century. In its original formulation, the McKay conjecture postulates that two seemingly unrelated sets of objects, constructed from the representation theory of a group, have the same number of elements. One of these sets is defined by "local data", i.e. by data constructed from proper subgroups, the other set only by "global data" of the group itself. That global data should be determined by local data, is the philosophy behind the McKay conjecture and other famous conjectures of representation theory. Two recent developments have inspired this project. Firstly, in 2007, Isaacs, Malle and Navarro published a powerful reduction theorem for the McKay conjecture. This leaves one to verify some rather complicated conditions for the finite simple groups. Secondly, the results of Späth's 2007 PhD thesis provide a means to verify these complicated conditions in the local situation, at least in special cases. The main objectives of this project are to construct suitable bijections between the two sets of objects of the McKay conjecture in local configurations of classical groups and to prove the McKay conjecture for finite groups of Lie type in non-defining characteristic. A further objective is to prove the Alperin-McKay conjecture for unipotent blocks of finite groups of Lie type in non-defining characteristic.
Actions of Algebraic Groups, Fans and Tilting Modules
To any finite dimensional algebra of finite global dimension one can associate a fan of tilting modules of projective dimension at most one. This fan is connected for representation finite and tame hereditary algebras. Moreover, the support of the fan classifies the actions of the corresponding group on the representation space with a dense orbit. In particular, the action of a parabolic subgroup in a General Linear Group on a subideal in the Lie algebra of the unipotent radical can be described in this way.The principal aim of this project is to understand the fan for hereditary algebras, in particular its connected components. Moreover, for actions of parabolic groups the fan might be connected as well, which in turn would classify all such actions with a dense orbit. Finally, we would like to generalize the results to actions for parabolic subgroups in the classical linear algebraic groups.
Cluster categories and torsion theory
Cluster categories form a categorification of Fomin and Zelevinsky's cluster algebras, a topic linking diverse mathematical areas in unexpected ways. In the proposed project we plan to study the structure of cluster categories and their higher generalisations, the d-cluster categories. Torsion theory is fundamental in the representation theory of algebras and recently also for triangulated categories. A particular focus of the project will be on classifications of torsion pairs in cluster categories, in particular for cluster categories attached to cluster algebras coming from triangulations of surfaces. Along the way, interesting special cases, e.g. Dynkin and extended Dynkin types, might already present challenging problems. Moreover, we plan to examine cluster behaviour in triangulated categories with Auslander-Reiten quivers of infinite Dynkin type, and to classify their (weak) cluster tilting subcategories. To this end, topological/geometric models for such categories would be useful which allow explicit calculations. We also plan to find geometric realisations of such categories 'in nature'.
Knop, Friedrich (University of Erlangen-Nürnberg):
Multiplicity Free Actions
In this project, we study spaces, called spherical varieties, which have in a precise sense the highest possible degree of symmetry. They are generalizations of the round 2-sphere. In recent years, extensive research was devoted towards the goal of finding a complete list of spherical varieties. One aim of this project is to finalize this classification. Another goal is to study the geometric properties of spherical varieties. These are controlled by their set of spherical functions, which are generalizations of spherical harmonics on the 2-sphere. We study, in particular, the multiplicative properties of spherical functions. Finally, certain generalizations of spherical varieties are considered where functions are also allowed to anticommute. These structures are encountered, for example, in the geometry of compact Hermitian symmetric spaces.
Homological structures at the interface of abstract representation theory and algebraic Lie theory
This project investigates and uses homological structures of finite dimensional algebras, in particular homological dimensions such as representation dimension or dominant dimension. Results are to be used in other branches of representation theory, in particular in algebraic Lie theory.
Quiver representations, singularity categories, and monoidal structures
Representations of algebras are studied from various directions, involving monoidal and triangulated structures. One goal is to describe in terms of generators and relations representation rings and monoids of projections functors for representations of quivers. Another goal is to classify thick and localising subcategories of triangulated singularity categories. This involves continuous and discrete parametrisations, reflecting the existing monoidal or combinatorial structures. The choice of singularity categories is motivated by connections with the representation theory of finite groups, the study of matrix factorisations, and applications in algebraic geometry, including weighted projective lines.
Ladkani, Sefi (University of Bonn):
Derived categories of sheaves over finite partially ordered sets and their homological properties
Triangulated and derived categories have been successfully used to relate objects of different mathematical origins (e.g. Kontsevich's Homological mirror symmetry conjecture) as well as objects of the same nature (e.g. Rickard's Morita theory, Broue's conjecture). In this project we will investigate derived categories arising from combinatorial objects, such as certain quivers with relations and finite partially ordered sets (posets), leading to categories which can be described in algebraic ("modules") as well as topological ("sheaves") terms. We will study both abstract posets, as well as specific ones arising from combinatorial contexts such as cluster algebras and cluster categories. The main questions are: (a) When two such objects lead to equivalent derived categories? (b) Is there an algorithm which decides on this question? (c) Are there basic combinatorial operations ("mutations") taking an object to a derived equivalent one, with the property that any two derived equivalent objects are related by a sequence of such operations? To address them, we will: 1. Develop new constructions of derived equivalences; 2. Investigate the role of combinatorial and numerical invariants, especially the Euler bilinear form, in determining derived equivalence. For these investigations, the process of interpreting certain linear algebra statements as categorical ones (known as "categorification") will play an important role. Outside mathematics, one can find applications to chemistry concerning structure discriminators of molecules.
PBW-filtration of representations, degenerate flag varieties and polytopes
The project is part of a long term program to better understand the geometric, algebraic and combinatorial aspects of the PBW-filtration of a representation. Fix a simple Lie algebra and the nilpotent radical n of an opposite Borel subalgebra. The PBW-filtration of the enveloping algebra of n induces a filtration on any finite dimensional irreducible representation V, let V' be the associated graded space. This construction leads naturally to an algebraic group G' acting on the spaces V' and a degenerate version of the flag variety.
Geometric, algebraic and combinatorial aspects of this construction have been investigated for the special linear group and the symplectic group in a series of papers by E. Feigin, M. Finkelberg, G. Fourier and the PI. The aim of this project is to generalize the results to other types, to find type independent constructions and to link the newly developed methods and tools to other known constructions in representation theory like crystal bases and generalized Gelfand-Tsetlin patterns.
Structure and representations of cyclotomic Hecke algebras
The project concerns the structure and representation theory of cyclotomic Hecke algebras. These deformations of group algebras of complex reflection groups arise as analogues of the Iwahori-Hecke algebras which in turn play a central role in the representation theory of finite groups of Lie type. It has become apparent in recent years that many properties of Iwahori-Hecke algebras have counterparts in the theory of cyclotomic Hecke algebras. Often, though, new approaches have to be found for their proof. The project aims at investigating analogues of several recent constructions for Weyl groups. The first is the new construction by M. Geck of Lusztig's algebra $J$ in the case of Coxeter groups, which seems to lend itself to a generalization to cyclotomic Hecke algebras. Together with a suitably modified concept of $W$-graph for the explicit construction of irreducible representations this should lead to a better understanding of these algebras. A further topic concerns representations of rational Cherednik algebras attached to complex reflection groups.
Semibounded unitary representations of double extensions of pre-Hilbert--Lie groups
The goal of this project is to develop a geometric approach to the important class of semibounded unitary representations for groups which are so-called double extensions of pre-Hilbert--Lie groups (groups whose Lie algebra carries an invariant scalar product). Typical examples of such groups are oscillator groups, double extensions of Hilbert--Lie groups and affine Kac--Moody groups. Semiboundedness of a unitary representation is a stable version of the ``positive energy'' condition which characterizes many representations arising in mathe\-matical physics, resp., field theories. For a unitary representation of a Lie group it means that the selfadjoint operators from the derived representation are uniformly bounded below on some open subset of the Lie algebra. Our long term goal is to understand the decomposition theory and the irreducible representations for this class.
Perling, Markus (University of Bochum):
Toric structures of exceptional sequences and representations of algebras
In this project we want to construct non-commutative coordinatizations of algebraic varieties by construction of exceptional sequences. On a given variety X, an exceptional sequence is a sequence of coherent sheaves satisfying certain conditions of cohomology vanishing. The endomorphism ring of the direct sum of such a sequence forms a noncommutative coordinate system for X in the sense that the derived category of X, i.e. the category of complexes of coherent sheaves on X, is equivalent to the derived category of modules over the endomorphism algebra. This connection serves as a bridge between geometry and the representation theory of algebras and has attracted much interest in recent years, not only in mathematics but also in the context of superstring theory. Recent results indicate that this correspondence is linked in a deep and unexpected way to the geometry of toric varieties. In this project we want to improve our understanding of this link and we want to use toric geometry to give more general constructions of exceptional sequences and their associated endomorphism rings.
Structures and representations of infinite- and finite-dimensional Lie algebras
This project concentrates on collaborations with three colleagues: Gregg Zuckermann(Yale University), Vera Serganova (University of California, Berke- lley), Alexander Tikhomirov(Yaroslavl University). With Zuckermann we study the structure of (g; k)-modules of finite type, where g is a finite-dimensional semisimple Lie algebra and k is a reductive (not necessarily symmetric!) subalgebra. The current challenge is to get an understanding (as detailed as possible) of (g; sl(2))-modules. With Serganova we study integrable modules of llocally finite Lie algebras. One of the challenges is to prove that the categories of tensor modules over the classical finite-dimensional Lie algebras so(&infin) and sp(&infin) are equivalent. May be this is infinite Howe duality? With Tikhomirov we address the geometry of homogeneous ind-spaces. The conjecture we are currently working on is to prove that any vector bundle on G=B for G = GL(&infin) has a subbundle of rank 1. This is part of a more general conjecture about finite rank vector bundles on flag ind-varieties.
Quiver moduli and quantized Donaldson-Thomas type invariants
The central aims of the project are the explicit computation of quantized Donaldson-Thomas type invariants for quivers with stability and superpotential, the continuation of the categorification programme of M. Kontsevich and Y. Soibelman for quantized Donaldson-Thomas type invariants in terms of Cohomological Hall algebras, the exploration of the relation between quantized Donaldson-Thomas type invariants and (refinements of) Kac polynomials with a view towards the Kac conjecture, and the geometrization of the GW/Kronecker correspondence between Gromov-Witten invariants of toric surfaces and DT type invariants of quivers.
Schweigert, Christoph (University of Hamburg):
Logarithmic conformal field theories
Logarithmic conformal field theories are a class of quantum field theories in which indecomposable but not irreducible representations arise. Some relevant representation categories are mathematically well-understood, e.g.\ via quantized universal envelopping algebras. We propose on the one hand side to make new classes of examples explicitly accessible. On the other hand, we propose to construct in general and in classes of examples physically relevant quantities using tools from representation theory and category theory.
We have three detailed goals for dissertation projects:
1. Construction of invariants of actions of mapping class groups which are candidates for physical correlators.
2. Calculations of these invariants in concrete examples, e.g. representation categories of Nichols algebras or group algebras in ``bad'' characteristic.
3. Description of physically relevant quantities like boundary conditions or defect types in terms of categorical quantities.
Seppänen, Henrik (University of Paderborn):
Asymptotic branching laws and Okounkov bodies
Koszul duality in representation theory
The project aims at understanding what some standard operations on representations should correspond to under the Koszul-duality equivalence, in particular the operation of tensoring with a finite dimensional representation and Induction.
Combinatorics of affine Grassmannians and affine flag varieties
The aim of the proposed research project is the translation and extension of combinatorial models for studying the complex affine Grassmannian to the complex affine flag variety, and then to both the affine flag variety and Grassmannian in positive characteristic. This should prove to be useful to help understand geometric and representation theoretic problems related to these objects. In the case of the above mentioned complex affine Grassmannian a lot is known thanks to the work of numerous authors which relates the geometry of the affine Grassmannian with the representation theory of algebraic groups, by both combinatorial and geometric approaches. While some of these techniques are already established, others are quite new or need some major modifications to be applicable. The proposed project wants to generalize these established methods and use them in an extended context (the affine flag variety) as well as developing new tools (in particular for the positive characteristic case). The proposed research project is related to the following topics in mathematics: Representation theory of complex algebraic groups, representation theory of algebraic groups/group schemes over a local/finite field, arithmetic geometry, complex algebraic geometry, theory of building over the complex numbers as well as local fields, combinatorics of galleries and alcove walks, and the structure of Iwahori- Hecke algebras.
Invariant theory of theta-representations
One of the main tasks of mathematics is to describe certain objects up to a certain equivalence relation. Often this relation is given by an algebraic group action. Then the equivalence classes are the orbits and orbits closures correspond to degenerations of our objects. Thus, describing orbits of algebraic actions, as well as deciding whether one orbit lies in the closure of another, is an important and interesting problem. However, this is possible only in a very few cases. One of this instances is provided by the theta-representations. The notion was proposed by V.G.Kac and E.B.Vinberg as a generalisation of the theory of symmetric pairs. Here we are dealing with a finite order automorphism theta of a complex reductive algebraic group G and the arising representations of the fixed-points subgroup on the theta-eigenspaces in the Lie algebra of G. The aim of this project is to get a better understanding of the orbit structure of theta-representations, in particular, to find out when orbits closures have such nice properties as normality or rationality of singularities.