Thematic program "L²-invariants and their analogues in positive characteristic"  

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We are planning to have a weekly seminar, where the visitors of the program will present their work.

Monday, June 11th, Sala Naranja ICMAT , 12:00

• Dawid Kielak (University of Bielefeld)

  L2 invariants and fibring of 3-manifolds.

  Abstract: We will discuss how the L2 torsion polytope of Friedl-Lück can be used to reprove and generalise Thurston's theorem on the structure of fibrings of a given 3-manifold.

Tuesday, May 29th, Sala 520 UAM , 10:00

• Benjamin Wasserman (Karlsruher Institut für Technologie)

  Analytic L2 -invariants and ζ -Regulization

  Abstract: link text

Tuesday, May 22th, Sala 520 UAM , 12:00

• Spencer Dowdall (Vanderbilt University)

  Splittings, suspension flows, and polynomials for free-by-cyclic groups

  Abstract: The semi-direct product of a finite-rank free group with the integers can often be expressed as such a product in infinitely many ways. This talk will explore this phenomenon and work towards 1) describing the structure of the family of such splittings of a given group, and 2) looking for for relationships between the splittings themselves. After establishing some background, I will describe a dynamical approach to studying splittings via suspension flows and a polynomial invariant that ties these perspectives together. Only limited knowledge of free groups and semi-direct products will be assumed. Joint work with Ilya Kapovich and Chris Leininger.

Thursday, May 10th, Sala 520 UAM , 12:00

• Javier Sanchez Serda (Universidade de Sao Paulo)

  On Hughes-freeness

  Abstract: In this talk it will be review an important concept for embedding group rings in division rings called Hughes-freeness. Some ideas and develomentes will be discussed.

Tuesday, April 24th, Aula Naranja, 15:00

• Juan Claramunt (Universitat Autonoma de Barcelona)

  Sylvester matrix rank functions on crossed product algebras

  Abstract: See poster here

Thursday, April 19th, Aula Naranja, 15:00

• Arnaud Brothier (University of Rome Tor Vergata)

  Jones representations of Thompson groups

  Abstract: Thompson group F is the group of homeomorphisms of [0; 1] that are piecewise linear with slopes equal to a power of 2 and breakpoints a dyadic rational. While being one of the most studied discrete groups, it still remains allusive. By investigating constructions of conformal field theories, Jones recently discovered a large class of unitary representations of Thompson group F and other related groups (such as Thompson groups T, V, etc.). Those representations are constructed via a very flexible category/functor method. Many examples are given by Jones planar algebra framework as well as certain couple of operators. I will describe concrete examples of such representations and present some ongoing research. This is a joint work with Jones.

Tuesday, April 17th, Aula Naranja, 15:00

• Mehrzad Monzavi (Texas A&M University)

  Sofic Dimension.

  Abstract: In this talk, the notion of Sofic dimension for equivalence relations and groups, introduced by Dykema, Kerr, Pichot will be defined. Some properties of this OE invariant notion as well as its relation to cost will be discussed.

Thursday, March 12th, Aula 420 (UAM), 15:00

• Maria Cumplido (Université Rennes)

  Parabolic subgroups of Artin-Tits groups of Spherical type

  Abstract: with Volker Gebhardt, Juan González-Meneses and Bert Wiest) Artin-Tits groups are a natural generalisation of braid groups from the algebraic point of view: In the same way that the braid group can be obtained from the presentation of the symmetric group with transpositions as generators by dropping the order relations for the generators, other Coxeter groups give rise to more general Artin-Tits groups. If the underlying Coxeter group is finite, the resulting Artin-Tits group is said to be of spherical type. Artin-Tits groups of spherical type share many properties with braid groups. However, some of these properties for the braid group are proved using topological or geometrical techniques, since a braid group can be seen as the fundamental group of a configuration space, and also as a mapping class group of a punctured disc. As one cannot replicate these topological or geometrical techniques in other Artin-Tits groups, they must be replaced by algebraic arguments when trying to extend these properties to all Artin-Tits groups of spherical type. That is why we are interested in parabolic subgroups of Artin-Tits groups, which are defined as conjugates of a subgroups generated by a subset of the standard generators. They are the analogue of isotopy classes of simple closed curves in the puncture disk, which are the building blocks that form the well-known complex of curves. Then, it is logical to believe that improving our understanding about parabolic subgroups will allow us to prove similar results for Artin-Tits groups of spherical type in general. In this seminar we present the new "complex of irreducible parabolic subgroups" and two new results, namely that the intersection of parabolic subgroups is a parabolic subgroup and that the set of parabolic subgroups is a lattice.

Thursday, March 22th, Aula Naranja, 15:00

• Mikolaj Fraczyk (Université Paris-Sud)

  Growth of mod-2 homology groups in higher rank locally symmetric spaces.

  Abstract: By exploiting the presence of large 2-dimensional local flats I showed that every mod -2 homology class in a higher rank locally symmetric space M is represented by a cycle of total length o(vol(M)). In the talk I will sketch the proof of this result in a simple case and explain how one can deduce that the first mod-2 betti number of M is o(vol(M)). If time permits I will discuss the difficulties encountered in adapting these methods to the mod-p homology groups for p odd.

Tuesday, March 20th, Aula Naranja, 15:00

• Henry Bradford (Mathematisches Institut of Georg-August-Universität Göttingen)

  On LEF Growth in Groups .

  Abstract: In this talk I will introduce a quantitative version of the LEF (local embeddability into finite groups) property. I will estimate this new invariant in some examples, and compare it with existing invariants which quantify residual finiteness and soficity. I will give a construction of a family of finitely generated residually finite groups which nevertheless admit many more local embeddings to finite groups than they do finite quotients. It would be of great interest to identify more groups satisfying this last property since, as I shall show, they are never finitely presentable. .

Thursday, March 15th, Aula Naranja, 14:30

• Sam Mellick (Central European University, Budapest)

  Point processes on nondiscrete groups as pmp actions.

  Abstract: A point process is a random discrete subset of a space. Invariant point processes on nondiscrete groups are then examples of probability measure preserving (pmp) actions. In fact, it turns out that all free pmp actions are isomorphic to point processes. In this talk I will introduce point processes in a way suitable for non-probabilists and discuss their relationship with amenability, cost, and weak containment. Joint work with Miklós Abért.

Tuesday, March 13th, Aula Naranja, 15:00

• Wiktor Mogilski (Indiana University South Bend)

  A Survey of Recent Computations of $L^2$-Betti Numbers

  Abstract: In this talk I will survey some recent progress on the computation of $L^2$-Betti numbers. Time permitting and depending mood, this talk will be a potpourri of many things, but overall I hope to hit the following notes. First I plan survey the current status on the Singer Conjecture and the Strong Atiyah Conjecture. Then I will present a funny trick that, in many cases, improves the Strong Atiyah Conjecture prediction of the denominators of the $L^2$-Betti numbers. In the setting of Coxeter groups, this improvement allows us to make complete computations of the $L^2$-Betti numbers for many examples, in turn leading to new higher(er) dimensional examples of manifolds satisfying the Singer Conjecture. Last, I will say a few words and give an advertisement for the weighted analogue of $L^2$-Betti numbers.

Thursday, February 22th, Aula GRIS 1, 12:00

• Lukasz Grabowski (Lancaster)

  Almost commuting matrices with respect to the rank metric

  Abstract: I will present my recent result with Gabor Elek, which deals with the variant of the "Halmos problem" for the rank metric. A particular case is as follows: if A and B are either unitary or self-adjoint matrices that almost commute with respect to the rank metric, then one can find commuting matrices C and D which are close in the rank metric to the matrices A and B respectively.

We are planning to have a weekly seminar, dedicated to the study of the newest developments in the subject of the program.