Algebraic topology seminar

Adélie Garin, EPFL, May 30

Title: Topological Data Analysis meets Geometric Group Theory: Stratifying the space of barcodes using Coxeter complexes

At the intersection of data science and algebraic topology, topological data analysis (TDA) is a recent field of study, which provides robust mathematical, statistical and algorithmic methods to analyze the topological and geometric structures underlying complex data. TDA has proved its utility in many applications, including biology, material science and climate science, and it is still rapidly evolving. Barcodes are frequently used invariants in TDA. They provide topological summaries of the persistent homology of a filtered space. Understanding the structure and geometry of the space of barcodes is hence crucial for applications. In this talk, we use Coxeter complexes to define new coordinates on the space of barcodes.These coordinates define a stratification of the space of barcodes with n bars where the highest dimensional strata are indexed by the symmetric group. This creates a bridge between the fields of TDA, geometric group theory and permutation statistics, which could be exploited by researchers from each field. No prerequisite on TDA or Coxeter complexes are required.

 

Renaud Detcherry, Université de Bourgogne, Apr 28

Title: On the kernel of SO(3) quantum representations

The Witten-Reshetikhin-Turaev quantum representations are families of representations of mapping class groups of surfaces, indexed by an odd integer r. They have infinite image, are asymptotically faithful but not faithful. We will present partial results on the open question whether the kernel of the r-th quantum representations is generated by r-th powers of Dehn twists.

 

Alexandru Suciu, Northeastern University, Boston, Mar 20

Title: Tropical bounds for the BNSR invariants

There are several topological invariants associated to a finite-type CW-complex, which keep track of various finiteness properties of its covering spaces. These invariants, which include the cohomology jump loci and the Bieri–Neumann–Strebel-Renz invariants, are interconnected in ways that makes them both more computable and more informative. I will describe one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to low-dimensional topology and complex geometry.

 

Owen Rouillé, INRIA Paris / IMJ-PRG, Mar 8

Title: Computation of Large Asymptotics of 3-Manifold Quantum Invariants

Quantum topological invariants have played an important role in computational topology, and they are at the heart of major modern mathematical conjectures. In the first part of this talk, I will present the experimental problem of computing large r values of Turaev-Viro invariants TVr. The algorithm we consider uses backtracking and consists in the enumeration of a set of admissible colorings. We provide an easily computable parameter to estimate the complexity of the enumeration space and look at the volume conjecture proposed by Chen and Yang on a census of closed 3-manifolds. The second part of this talk will include general considerations about costly computations in mathematics and computer sciences. (joint work with Clément Maria)

 

Paul Wedrich, Hamburg University, Feb 17

Title: A Kirby color for Khovanov homology

The Jones polynomial is a famous knot invariant that has been categorified by Khovanov to a knot homology theory. Jones polynomials of knots can be computed relatively straightforwardly using the Temperley-Lieb algebras, which admit a diagrammatic presentation. Surprisingly, closely related diagrammatic algebras, the dotted Temperley-Lieb algebras (a.k.a. nil-blob algebras) appear when extending Khovanov homology to an invariant of smooth 4-dimensional manifolds. I will define these algebras, assemble them into a monoidal category, and then construct a special ind-object that we call the Kirby color for Khovanov homology. This object, discovered in joint work with Hogancamp and Rose, allows a concrete description of a certain 2-handle gluing rule for the invariants of smooth 4-manifold constructed in joint work with Morrison and Walker.

 

Marco de Renzi, University of Zurich, Feb 14

Title: Algebraic presentation of cobordisms and quantum invariants in dimension 3 and 4

The category 2Cob of 2-dimensional cobordisms is freely generated by a commutative Frobenius algebra: the circle. This yields a classification of 2-dimensional TQFTs (Topological Quantum Field Theories). In this talk, I will discuss some consequences of an analogous algebraic presentation in dimension 3 and 4, due to Bobtcheva and Piergallini. In both cases, the fundamental algebraic structures are provided by certain Hopf algebras called BPH algebras. In dimension 3, I will consider the category 3Cob of connected cobordisms between connected surfaces with one boundary component. I will explain that an algebraic presentation conjectured (or rather announced without proof) by Habiro is in fact equivalent to the one established by Bobtcheva and Piergallini. In dimension 4, I will focus on a category denoted 4HB, whose morphisms are 2-deformation classes of 4-dimensional 2-handlebodies. I will show that any unimodular ribbon category contains a BPH algebra, which can be characterized very explicitly. This result proves the existence of a very large family of TQFT functors on 4HB. Finally, I will explain that a unimodular ribbon category has the potential to detect exotic phenomena in dimension 4 only if it is neither semisimple nor factorizable. This is a joint work with A. Beliakova, I. Bobtcheva, and R. Piergallini.

 

Ilaria Colazzo, University of Exeter, Jan 17

Title: Combinatorial solutions to the Yang-Baxter equation and applications

The first part of the talk will connect the topological problem of distinguishing mathematical knots with combinatorial solutions to the Yang-Baxter equation. The second one will be devoted to a new algebraic structure (skew braces) that provides the perfect tool to study and classify such solutions.