Following the Humboldt Kolleg "Measuring America", members of the research group "Grupo de Investigación Interinstitucional en Geometría y Topología" and their students will get together with german researchers from Greifswald and Hamburg to share their latest results in Higher Structures in Algebra, Geometry and Topology.

Organizers: Camilo Arias, Georg Biedermann, Adriana Mejía, Bernardo Uribe.

PROGRAM (Abstracts at the end of the page)

Friday 22nd of March (Room CEC 1, Business School Building (G), basement)

8:30 - 9:10 Camilo Arias (Universidad Nacional de Colombia) Singular chains on Lie groups, the Cartan relations and local systems on classifying spaces.

9:20 - 10:00 Adriana Mejia (Universidad del Norte) Module categories over graded tensor categories

10:00 - 10:30 Break

10:30 - 11:10 Yiby Morales (Universidad de los Andes) A five-term exact sequence for abelian extensions of Hopf algebras.

11:20 -12:00 Augusto Stoffel (Universität Greifswald) Geometric TQFTs and parallel transport

12:00 -2:00 Lunch

Friday 22nd of March (Room 37G2, Business School Building (G), third floor)

2:00 - 2:40 Mauricio Cepeda (Universidad Nacional de Colombia) Classifying Spaces and Characteristic Classes for Tansitionally Commutative Bundles

2:50 - 3:30 Mario Velásquez (Pontificia Universidad Javeriana) The topological K-theory of the Hilbert modular group

3:30 - 4:00 Break

4:00 - 4:40 Andrés Angel (Universidad de los Andes) Representable and Non-representable classes

4:50 - 5:30 Peter Kristel (Universität Greifswald) Generalizing spinor bundles to loop spaces

Saturday 23rd of March (Room 34G1, Business School Building (G), third floor)

9:00 - 9:40 Georg Biedermann (Universidad del Norte) A generalized Blakers-Massey Theorem

9:50 - 10:30 César Fernando Venegas (Universidad de los Andes) Minimal modular extensions of super-Tannakian categories

10:30 - 11:00 Break

11:00 - 11:40 Malte Kunath (Universität Greifswald) Topological T-duality and non-abelian bundle gerbes

11:50 -12:30 Jaider Blanco (Universidad del Norte) 2-groups and multiplicative bundle gerbes

Preliminary list of Speakers

Andrés Ángel, Universidad de los Andes

Camilo Arias, Universidad Nacional de Colombia

Georg Biedermann, Universidad del Norte

Jaider Blanco, Universidad del Norte

Mauricio Cepeda, Universidad Nacional de Colombia

Peter Kristel, Universität Greifswald

Malte Kunath, Universität Greifswald

Adriana Mejía, Universidad del Norte

Yiby Morales, Universidad de los Andes

August Stoffel, Universität Greifswald

César Fernando Venegas, Universidad de los Andes

Mario Velásquez, Pontificia Universidad Javeriana

Preliminary list of participants.

Abstracts

Andrés Ángel, Representable and Non-representable classes

A natural problem in algebraic topology is Steenrod's representability problem: Is every homology class of a space represented by the continuous image of the fundamental class of a manifold?. This question was solved by Rene Thom: Every homology class with Z/2Z-coefficients is represented, but there are homology classes with integer coefficients that are not represented by manifolds.

In this talk I will give a geometric description of homology classes with integer coefficients of BZ_p x BZ_p that are not representable by manifolds and explain how the singularites look like and why the cannot removed. These non-representable classes appear as "higher products" of lens spaces.

(Joint work with Carlos Segovia and Fernando Arley Torres)

Georg Biedermann, A generalized Blakers-Massey Theorem

We repeat the classical Blakers-Massey Theorem (Homotopy Excision) and some consequences like the Freudenthal suspension theorem.

Then we give a short introduction to higher topoi. Then we introduce modalities, factorization systems whose left class is closed under base change. Then we describe our generalized Blakers-Massey Theorem. Unfortunately, due to time constraints, the main application to Goodwillie calculus will not be addressed.

(joint with Anel, Finster, Joyal)

Augusto Stoffel, Geometric TQFTs and parallel transport

I will discuss a definition of bordism categories where bordisms are equipped with general geometric structures. This is motivated by applications such as the representation of cohomology theories through field theories. Then I will discuss the case of 1-dimensional TQFTs over a manifold X. It turns out, as one would hope, that these are nothing but vector bundles with connection over X. I'll explain how the difficulties entailed in this statement and their resolution are related to the problem of representing cohomology theories. This is joint work with Matthias Ludewig and Stephan Stolz.

César Fernando Venegas, Minimal modular extensions of super-Tannakian categories.

In this talk, we parametrize the minimal modular extensions for super-Tannakian fusion categories in terms of certain cohomology group in degree 2 and 3. For this, we use equivariantization and de-equivariantization functors in the particular case of fermionic actions, extensions of braided fusion categories, and 3-homomorphisms in the Picard 3-group.

Mario Velasquez, The topological K-theory of the Hilbert modular group

Using the p-chain spectral sequence we compute the topological K-theory of the reduced C*-algebra with rational coefficients of the Hilbert modular group corresponding to the integer ring of a totally real extension of a rational numbers. This is a joint work with Luis Jorge Sánchez.