Seminars in geometry/topology

September 11, 13:15 - 14:15, room 734, Sentralbygg 2

Glen Wilson (University of Oslo): The eta-inverted sphere over the rationals

Abstract: For any field \(\mathbf{F}\) Morel and and Voevodsky associate to it a stable homotopy category \(SH(\mathbf{F})\) that encodes topologicalvinformation and information about algebraic varieties over \(\mathbf{F}\). Just as with the topological stable homotopy category, we would like to calculate the motivic stable homotopy groups of spheres \(\pi_{s,w}(1) = SH(\mathbf{F})(\Sigma^{s,w}1,1)\). To gain some relevant information about these groups, Guillou and Isaksen—with input from Andrews and Miller—calculated the eta-inverted stable homotopy groups of the \(2\)-complete sphere spectrum over the complex numbers and the real numbers. In this talk we will review the eta-inverted calculations of Guillou and Isaksen and present new results about the structure of the stable homotopy groups of the eta-inverted \(2\)-complete sphere spectrum over fields of cohomological dimension at most \(2\) (characteristic not \(2\)) and the field of rational numbers.

September 25, 13:15 - 14:15, room 734, Sentralbygg 2

Ana Romero Ibáñez (Universidad de La Rioja): Kenzo, a Symbolic Computation system for Algebraic Topology: introduction and applications for the computation of persistent homology and generalized spectral sequences

Abstract: The Kenzo system, developed by Francis Sergeraert and some coworkers, is a Symbolic Computation system for Algebraic Topology. It works with complex algebraic structures: chain complexes, differential graded algebras, simplicial sets, simplicial groups, morphisms between these objects, etc. In particular, it implements the effective homology method and can be used to compute homology and homotopy groups of complicated spaces.

The talk will consist in an introductory description of the program with some examples of calculations. Moreover, we will explain some new modules that we have depeloped computing persistent homology and generalized spectral sequences.

Joint work with A. Guidolin, J. Rubio and F. Sergeraert.

October 9, 13:15 - 14:15, room 734, Sentralbygg 2

Bjørn Ian Dundas (University of Bergen): Homotopy Type Theory

Abstract: Mathematics is vital to modern society in increasingly sophisticated ways; but can we certify our results or may it be that the ATM at the corner goes berserk because you made a stupid mistake? What are the nuts and bolts needed for automated proof assistants?

Surprisingly, at the core of a revolutionary approach spearheaded by Fields medalist Vladimir Voevodsky, a mix of computer science and algebraic topology takes center stage. I aim at giving a lighthearted overview of what it is all about, both practically (what life with computer scientists is like) and mathematically. The central question is “What does it mean that two things are equal?” The answer depends on the context, and we need tools to handle it constructively. Surprisingly, the answer comes from understanding how spaces behave.

October 16, 13:15 - 14:15, room 734, Sentralbygg 2

David Sprehn (Københavns Universitet): Stable Homology of Classical Groups

Abstract: Quillen computed the homology of the general linear groups over a finite field of characteristic \(p\), with coefficients away from characteristic \(p\). He wasn't able to compute the \(p\)-torsion, but he gave a beautiful argument showing that, in the stable limit, the \(p\)-torsion vanishes. For \(p\) odd, Nathalie Wahl and I were able to prove the analogous result for the other classical groups (symplectic, orthogonal, unitary) in the course of improving the known homological stability ranges.

October 25, 12:30 - 13:30, room 656, Sentralbygg 2

Rune Haugseng (Københavns Universitet): Higher categories of higher categories

Abstract: I will discuss ongoing work aimed at constructing higher categories of (enriched) higher categories. This should give the appropriate targets for many interesting examples of extended topological quantum field theories, including extended versions of the classical examples of TQFTs due to Turaev-Viro, Reshetikhin-Turaev, etc.

November 1, 13:00 - 14:00, room 656, Sentralbygg 2

Claudia Scheimbauer (Oxford University): Constructing extended AKSZ topological field theories in derived algebraic geometry using an explicit model for the higher category of bordisms

Abstract: I will first explain an explicit model of the higher category of bordisms which allows to explicitly construct extended topological field theories in the sense of Lurie. Then I will sketch examples thereof which describe ``semi-classical TFTs". The main tool is derived algebraic geometry and derived symplectic geometry in the sense of Pantev-Toen-Vaquié-Vezzosi, which allows for a reinterpretation/analog of the classical AKSZ construction for certain \(\sigma\)-models. The target of the TFT is a higher category whose objects are \(n\)-shifted symplectic derived stacks and (higher) morphisms are (higher) Lagrangian correspondences. The TFT itself is given by taking mapping stacks with a fixed target building. This is joint work in progress with Damien Calaque and Rune Haugseng.

November 13, 13:15 - 14:15, room 734, Sentralbygg 2

Andreas Holmström (Stockholms Universitet): Multiplicative functions and Tannakian symbols

Abstract: Multiplicative functions appear everywhere in number theory. Elementary examples include the Euler phi function and the Möbius function, while more sophisticated examples arise from Dirichlet coefficients of zeta functions and L-functions.

There are many operations on multiplicative functions defined in the literature, some of an elementary nature and others related to algebraic geometry and the theory of L-functions. I will present a unified framework for understanding all these operations, based on the language of lambda-rings. This framework has a number of applications, as well as interesting connections to problems in K-theory and representation theory.

This is joint work with Ane Espeseth and Torstein Vik.

November 30 (Thursday!!!), 14:15 - 15:15, room 734, Sentralbygg 2

Richard Williamson (SINTEF Ocean) and Reidun Persdatter Ødegaard (NTNU): Elementary writhe-like and parity-like invariants of knots

Abstract: The self-linking number, introduced by Kauffman in 2004, is a very useful and powerful invariant of virtual knots. It relies on the parity of certain numbers (i.e. whether they are odd or even), and investigation of further invariants involving parity has been a fruitful theme in virtual knot theory. The self-linking number is, however, always zero for classical knots. The writhe of a knot, though invariant only under the R2 and R3 moves, is also a very useful and powerful tool, playing for example a crucial role in the Kauffman bracket approach to the Jones polynomial. In this talk, we will introduce a family of new, elementary knot invariants, inspired by and loosely related to the self-linking number and writhe. Despite their simplicity, these invariants are extremely powerful: we have carried out quite extensive computer calculations, and all knots which we have checked are distinguished by at least one of the new invariants from the unknot.