Previous Talks

30. June 2010
Prof. Dr. Andreas Zastrow (Gdansk University)
On how descriptive set-theory could help to answer a question from geometric topology


02. June 2010
Dr. Anna Felikson (MPI)
Double pants decompositions of 2-surfaces
A pants decomposition of a 2-surface S is a maximal set of mutually non-intersecting closed curves on S whose complement is a union of "pants" (i.e. spheres with 3 holes). A double pants decomposition is a union of two pants decompositions of S. We define a remarkable class DP of double pants decompositions which we call "admissible", as well as a natural class T of transformations acting on double pants decompositions. We show that the transformations from T act transitively on the set of admissible double pants decompositions. These transformations also generate a group of automorphisms of DP, in particular, T contains the mapping class group.
This is a joint work with S. Natanzon.


12. May 2010
Thomas Lessmann
On the number of projective solutions of diagonal equations over F_p


21. April 2010
Dr. Pierre Will (Grenoble)
Discrete subgroups of isometries in complex hyperbolic space.
The goal of this talk is to present results and examples about discrete subgroups of PU(2,1), which is the automorphism group of the complex hyperbolic plane. These groups are a complex 2-dimensional analogues of Fuchsian groups in PSL(2,R), or Kleinian groups in PSL(2,C).
The complex hyperbolic space is an example of a rank one symmetric space with negative pinched curvature. It is biholomorphic to a ball, and is a natural generalisation of the usual Poincaré disk or upper half plane. I will try to illustrate on examples the differences between the "classical" cases of Fuchsian and Kleinian groups and the complex hyperbolic case.

03. February 2010
Saeid Hamzeh Zarghani
Hecke Operators on non-congruence Subgroups.
The action of Hecke operators on the modular forms associated to a congruence subgroup (of the modular group) has been used quite successfully to understand the properties of these modular forms (and cusp forms) in the last decades. But, as conjectured by Atkin and proved in a special case by Serre (in a letter to J. G. Thumpson in 1987), in case of a non-congruence subgroup G, this action is "almost" the same as the action of the Hecke operators on the modular forms (and cusp forms) associated to the congruence closure of G. In its general form, this fact was finally proved by Berger in 1994.
In my talk, I am going to review these facts quickly and then recall, without going through details, the Eichler-Shimura correspondence between the space of cusp forms and (a certain subgroup of) the first cohomology group of G (with coefficients in a certain G-module depending on the weight of the forms), which leads to a natural question: Are the actions of the Hecke operators on the first cohomology groups of G and its congruence closure (with coefficients in some G-module) related in a similar manner?
The fact is, that this is true not only for the first cohomology, but also for higher cohomology groups and for every coefficient module, and I am going to give the main ideas of its proof.

27. January 2010
Dr. Shelly Garion (MPI)
Triangle groups, finite simple groups and applications.
In this talk we will discuss the following problem:
Given a triple of integers (r,s,t), which finite simple groups are quotients of the triangle group T(r,s,t)?
This problem has many applications, especially concerning Riemann surfaces (in differential geometry) and Beauville surfaces (in algebraic geometry).
In the talk we will focus on the group theoretical aspects of this problem.

20. January 2010
Prof. Dr. Kai-Uwe Bux (Bielefeld)
On Asada's solution for the congruence subgroup problem in Aut(F_2).
(joint work with Mikhail Ershov and Andrei Rapinchuk)
Congruence subgroups in Aut(F_n) are defined to consist of those automorphism that induce the identity on a finite quotient F_n/K where K is a characteristic subgroup. Aut(F_n) has the congruence subgroup property if every finite index normal subgroup contains a congruence subgroup. Asada has shown that Aut(F_2) has the congruence subgroup property. His arguement is very high powered. We give a low-tech, down-to-earth interpretation of the main points. In return, we obtain a somewhat more explicit construction of congruence subgroups.

09. December 2009
Prof. Dr. Jan Trlifaj (Prague)
Flat Mittag-Leffler modules and Drinfeld vector bundles.

11. November 2009
Prof. Dr. Vladimir Platonov (Moscow)
Arithmetic of quadratic function fields and torsion in Jacobians.

21. October 2009
Prof. Dr. Kai-Uwe Bux (Bielefeld)
Arithmetic groups in positive characteristic
The connection of topology and group theory is via the fundamental group: every group G arises as the fundamental group of some space Y. In fact, Y can be chosen to be a CW complex with contractible universal cover. In this case, we say that Y is an Eilenberg-MacLane complex for G. It turns out that G determines its Eilenberg-MacLane complex up to homotopy equivalence. Hence, homotopy-invariants (e.g., homology and cohomology) of Y are actually invariants of G. Another source of obtaining invariants for the group G is to employ the ambiguity inherent in the Eilenber- MacLane complexes for G. The geometric dimension of G is the minimum dimension of an Eilenberg-MacLane complex for G. The finiteness length of G is the maximum m for which there is an Eilenberg-MacLane complex for G with finite m-skeleton. The finiteness length of G is >=1 if and only if G is finitely generated and the finiteness length is >=2 if and only if G is finitely presented.
Arithmetic groups, such as SL_n(Z), SL_n( Z[1/3] ), SL_n( F_q[t] ), or SL_n( F_q[t,1/t] ) where F_q is a finite field provide a good case study for finiteness properties since they depend on two parameters that can be varied independently: one has to choose the group scheme (in our case SL_2, SL_3, ...) and the coefficient ring Z, Z[t], F_q[t], ...; the interesting question is how the finiteness length depends on the choice of these parameters. I will describe what is known with respect to this problem and what is conjectured (recently there has been mounting evidence for a particular conjecture that would settle the question for semi-simple groups). I also intend to at least point toward yet uncharted territory where only scattered results are known.

22. July 2009
Nadine Hansen (Dortmund)
Commensurators of some Baumslag-Solitar Groups (Diploma Thesis)

15. July 2009
Dr. Evija Ribnere
Representations of Aut(F_n)

08. July 2009
Dr. Anna Felikson (MPI)
Quivers of finite mutation type
We discuss quivers of finite mutation type (which are in one-to-one correspondence with cluster algebras of finite mutation type with skew-symmetric exchange matrices). Besides quivers of rank 2 and ones associated with triangulations of surfaces there are exactly 11 exceptional quivers of finite mutation type. More precisely, 9 of them are associated with root systems E_6, E_7, E_8, \widetilde E_6, \widetilde E_7, \widetilde E_8, E_6^(1,1), E_7^(1,1), E_8^(1,1). We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type. The talk is aimed to general audience: no special knowledge is assumed.

24. June 2009
Prof. Dr. Sergei Tabachnikov (PennState University / MPI)
Outer billiards: results and open problems
I shall survey three problems on lesser known class of billiards, the outer (a.k.a. dual) billiards. The first is the existence of periodic trajectories for polygonal outer billiards, a theorem in affine, and an open problem in the hyperbolic plane (the problem is open and very hard for polygonal inner billiards!). The second is a version of Birkhoff conjecture for outer billiards, which is a theorem in the algebraic setting, and a conjecture in the smooth one. The third is Moser's problem whether the orbits of polygonal outer billiard may escape to infinity; this problem was recently solved (in the affirmative) by R. Schwartz.

10. June 2009
Dr. Alina Vdovina (University of Newcastle)
Fundamental groups of CW-complexes from different points of view
We are going to present methods to construct CW-complexes with specified local structure. Very often various properties of the fundamental groups are encoded in the structure of the complexes.
As one of the applications, we present new infinite family of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our family is given as Cayley graphs of finite groups with very short presentations with only 2 generators and 4 relations. These groups are particular finite quotients of a group G of infinite upper triangular matrices over the ring M(3,F_2).
The group G is constructed as a fundamental group of a 2-dimensional CW complex with links isomorphic to incidence graphs of finite projective planes.
The pro-2 completion of G satisfies the Golod-Shafavarevich inequality |R| >= (|X|^2)/4, it is infinite, not p-adic analytic, contains a free nonabelian subgroup, but not a free pro-p group. We also conjecture that the group G has finite width 3 and finite average width 8/3.

03. June 2009
Alexander Rahm (Göttingen)
The integral homology of PSL_2 of imaginary quadratic integers of class number two
Joint work with Mathias Fuchs
We calculate the integral homology of the non-Euclidean Bianchi groups PSL_2 of the ring of integers of the number field Q[\sqrt{-m}] for m=5,6,10,13, making essential use of an equivariant cellular decomposition of a retract of hyperbolic three-space due to Floege.
arXiv:0903.4517v1

27. May 2009
Prof. Dr. E. Ventura (UPC Barcelona)
Algebraic extensions and computations of closures in free groups
In this talk, I'll introduce the concept of algebraic extension of free groups, which appeared in the literature under different contexts and by independent authors (I'll follow a unified version due to Miasnikov-Ventura-Weil, and using Stallings graphs in an essential way). This is a modern development trying to mimic the theory of field extensions, among subgroups of a given free group. Some results are totally analog to the classical ones in field theory (algebraic and free extensions, algebraic closure, free factors and free complements, composition of extensions, equation conditions, etc), while some other aspects behave in a considerably more complicated way.
Surprisingly, the concept of algebraic extension was essentially considered much before by Takahasi in the 1950’s. We'll give a modern proof of Takahasi's theorem, and will see the connection with many algorithmic problems about subgroups of the free group.
As an application, I'll present an algorithm (due to S. Margolis, M. Sapir and P. Weil) to compute (a basis of) the closure of a finitely generated subgroup of a free group, with respect to the pro-p topology, for every prime p.

20. May 2009
Prof. Dr. O. Kegel (Freiburg)
On universal groups

13. May 2009
Prof. Dr. O. Bogopolski
On conjugacy separability of virtually free groups
Joint work with Prof. F. Grunewald

11. February 2009
Shelly Garion
Aut(F_n) actions on group presentations and representations
In this talk, I will describe some recent developments regarding the action of Aut(F_n) on Epi(F_n, G), where F_n is the free group on n generators, and G is either finite, compact or non-compact simple Lie group, which is n-generated.
For finite groups, the question of transitivity of this action was raised in the 50's by B.H. Neumann and H. Neumann. In recent years, there is renewed interest in this subject due to its applications in computational group theory to the Product Replacement Algorithm.
I will present several recent results concerning the transitivity of this action for finite simple groups as well as its ergodicity for compact and some non-compact simple Lie groups. This is based on works of Gelander, Glasner, Minsky and my joint works with Avni and Shalev. These results were presented at a workshop in Sde-Boker, Israel, in January 2009.

28. January 2009
Gautami Bhowmik (Lille / MPI)
Analytic Continuation of Dirichlet Series

21. January 2009
Keivan Mallahi-Karai (Bremen)
Relative growth in Arithmetic groups

Monday, 12. January 2009 at 4pm in lecture hall 5E
Prof. Dr. Oleg Bogopolski (Dortmund)
On endomorphisms of hyperbolic groups

07. January 2009
Prof. Dr. Holger Reich
Geometry of classifying spaces

17. December 2008
Prof. Dr. Gordana Stojanovic (Georgia Tech / MPI)
Embeddings of manifolds with certain non-degeneracy conditions
It is a well known theorem of Whitney that every smooth connected manifold can be embedded into a Euclidean space of large enough dimension - double that of the manifold itself. One can consider the same question, but with some extra conditions imposed on the embeddings. For instance, how many dimensions of ambient Euclidean space does one need for an embedding of a smooth loop which doesn't allow for parallel tangents? (Such loops are called skew.)
We will define some such classes of maps and explore their relations to some well studied problems of differential and algebraic topology.

19. November 2008
Dr. Marc Ensenbach (Aachen)
Cosets of unimodular groups over Dedekind domains
PDF version
In the case of the GL_2(o) for a norm-finite Dedekind domain o, a formula for the calculation of the number of right cosets contained in a given double coset of this group has been found. In this talk, the derivation of this formula is outlined. Furthermore, applications of this formula to indices of congruence subgroups and computational aspects in Hecke algebras are given.

29. October 2008
Jasmin Matz
Zassenhaus Rings
In 1963 A.L.S. Corner showed that every reduced, torsion-free ring R of finite rank is the endomorphism ring of a reduced, torsion-free abelian group G with rk G=2rk R. Corner showed in the same paper that the rank of G cannot be chosen to be less than 2rk R in general.
Over the years it was shown by different people that the rank of G can be chosen to be rk R for rings, which fulfill some additional properties.
We will introduce the notion of Zassenhaus-Rings over an integral domain A and we will show that many A-algebras are Zassenhaus-Rings over A if A is a Dedekind domain. Moreover, we will see that there does not only exist countable Zassenhaus-Rings, but that there are arbitrary large Zassenhaus-Rings.

22. October 2008
Annika Günther (Aachen)
Permutation groups of doubly-even self-dual codes
Self-dual binary codes are of particular interest in algebraic coding theory, and have many practical applications. The best error-correcting self-dual binary codes have the additional property of being doubly-even (or Type II), which means that the weight of every codeword, i.e. the number of its nonzero entries, is a multiple of 4. In constructing these codes, it is often helpful to consider their permutation groups. For a binary code C of length n, its permutation group is
P(C) := { pi in S_n | C pi = C},
where S_n is the symmetric group on n points.
This talk presents a recent result, which says that the permutation group of a binary self-dual doubly-even code of length n is always contained in the alternating group A_n. Moreover, given a subgroup G of S_n, sufficient conditions on G will be given such that G is contained in the permutation group of a binary self-dual doubly-even code.

30. July 2008
Dr. Andy Novocin (Florida State)
Factoring Univariate Polynomials over the Rationals
In this talk I will present a brief history of polynomial factorization algorithms, concluding with my new state-of-the-art algorithm. The new algorithm is of interest in the computer algebra community because it is not only the fastest algorithm in practice but also the fastest algorithm in theory. This is the first polynomial factoring algorithm in the last 25 years which can make such a claim.

23. July 2008
Dr. Natalia Iyoudu (Belfast)
On techniques from the classical algebraic K-theory and RIT-algebras
We will discuss the generalization of the local-global principle appeared as one of the main ingredients of the solution of Serre's problem on freeness of projective modules over polynomial extension of commutative rings to the case of iterated Ore extension. Next we consider the class of quadratic RIT algebras, which has close connection to the above problem since it mainly consists of algebras allowing stably free non free modules. We present some results on the representation theory of these algebras and the combinatorial characterization of their properties in terms of graphs of relations.

Dr. Nicole Raulf (Lille)
Distribution of eigenvalues of Hecke operators
In this talk we will determine the asymptotic behaviour for the eigenvalues of Hecke operators. The operators we work with act on PSL_2(O)-invariant functions on the upper half-space. Here O denotes the ring of integers of an imaginary quadratic number field. Interest in this area has increased following the work of Taylor on the Sato-Tate conjecture. We modify the setting of the Sato-Tate conjecture and determine the distribution of the eigenvalues of the operator Tp with p being fixed. This is joint work with Ozlem Imamoglu.

16. July 2008
Dr. Claus Fieker (Sydney)
Minimizing Group Representations
Given an absolutely irreducible representation of some finite group G into the GL(n, K) for some number field K, it is an important problem to find fields affording this representation that are as "small as possible". By translating this problem into a problem involving central simple algebras we can then use some relatived Brauer group to decide which fieldxs afford the representation and relaise this constructively. The key idea stems from unpublished work of Plesken.

14. July 2008
Prof. Dr. A. A. Mikhalev (Moscow)
Free differential calculus and automorphisms of free algebras

07. July 2008
Prof. Dr. V. A. Artamonov (Moscow)
On Hopf-Algebras

02. July 2008
Heiko Dietrich (Braunschweig)
Periodic patterns in the coclass graph of p-groups of maximal class
The coclass of a p-group of order p^n and nilpotency class c is defined as n-c. It seems to be a fruitful approach to classify p-groups by coclass. In the talk, I follow the philosophy of coclass theory and report on some periodic patterns in the graph corresponding to p-groups of maximal class. These graph theoretic periodicities are reflected in the structure of the groups; this supports the conjecture that the p-groups of maximal class can be classified.

18. June 2008
Kristina Schindelar (Aachen)
Linear exact modelling with variable coefficients
With respect to a signal set consisting of multivariate polynomial trajectories, the aim is to describe these by Ore modules. In the classical most powerful model, we are searching for a linear model that is invariant under partial differentiation. The algebraic methods, known from this classical case, fail for Ore modules.

11. June 2008
Dr. Michael Lönne (Bayreuth)
Discriminant knot groups generalising braid groups
We consider some natural discriminant divisors and give finite presentations of the corresponding knot groups, ie. of the fundamental groups of their complements. Both of the settings we study correpond to a geometric generalisation of a setup for the braid group of the disc, resp. sphere. We find that the presentation are also generalisations (of that of the braid group) and discuss some natural questions.

09. June 2008
Dr. Andrew Duncan (Newcastle)
Automorphisms of Partially Commutative Groups

04. June 2008
Christian Greve
Galois groups of Eisenstein polynomials over local fields
Roots of Eisenstein polynomials generate fully ramified extensions of local fields. I will describe an algorithm for calculating the galois group of an Eisenstein polynomial, if the generated extension has a certain "easy" ramification structure.

16. May 2008
Shelly Garion
Connectivity properties of the product replacement algorithm graph of finite simple groups
The Product Replacement Algorithm (PRA) is a practical algorithm for generating random elements of a finite group G. The algorithm was introduced in 1995, and it quickly became popular and was included in the two commonly used computer algebra packages GAP and MAGMA.
One can describe this algorithm as a random walk on a certain graph, called the PRA graph, whose vertices are the generating k-tuples of G. The connected components of this graph correspond to the transitivity classes of the action of the automorphism group of the free group Aut(F_k) on the set of epimorphisms from F_k onto G.
In the talk, I will discuss the connectivity properties of PRA graphs and present some new results on the graphs corresponding to finite simple groups. These are related to the well-known Wiegold's Conjecture.

14. May 2008
Prof. Dr. Gabor Wiese (Essen)
Modular Forms in Inverse Galois Theory
Modular forms which are eigenfunctions for all Hecke operators give rise to 2-dimensional mod p representations of the absolute Galois group of the rationals. In the talk we will show how these representations, and hence modular forms, can be used to derive results on the occurrence of groups of the type PSL_2(F_p^r) as Galois groups over the rationals.

07. May 2008
Daniel Appel
On the abelianization of congruence subgroups of the automorphism group of the rank two free group
For an epimorphism pi of the rank two free group F_2 onto a finite group G write Gamma(G,pi) for the group of all automorphisms f of F_2 for which pi*f = pi. This is called the standard congruence subgroup of Aut(F_2) associated to G and pi. Congruence subgroups associated to abelian groups are closely connected to certain congruence subgroups of SL(2,Z). I will explain this connection and show how to use it to determine the abelianization of Gamma(G,pi) for abelian G.

30. April 2008
Mehmet Haluk Sengun (Wisconsin)
Galois Representations of Small Quadratic Fields
I will present the result that for small quadratic fields K and primes p, there is no irreducible continuous mod p representation of the absolute Galois group of K that is unramified away from p and infinity. I will discuss Serre's conjecture and its generalizations as a motivation to this result.

23. April 2008
Prof. Dr. Elena Klimenko
The geometry of Kleinian groups with real parameters
Kleinian groups are the discrete subgroups of PSL(2,C), the full group of orientation preserving isometries of hyperbolic 3-space. The classification of hyperbolic 3-manifolds was given earlier in topological terms together with the proof of Thurston's ending lamination conjecture and Marden's tameness conjecture. However, there are still open problems in the area. For example, it is still unknown in general whether two matrices of SL(2,C) generate a discrete group, and if they do, what is the quotient orbifold. We concentrate on the two-generator Kleinian groups with real traces of the generators and their commutator, and give a complete classification of such groups and corresponding orbifolds.

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