Seminars in geometry/topology

Monday August 31, 14:15 - 15:15, room 734, Sentralbygg 2

Alexander Schmeding (NTNU): (Re-)constructing Lie-groupoids from their groups of bisections

Abstract: In this talk we will revisit the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. Namely, we discuss two (re-)construction principles for Lie groupoids from candidates for their bisection Lie groups. If time permits we will discuss the main application of these results to the prequantisation of (pre)symplectic manifolds.

This is joint work with C. Wochel (Hamburg), cf. (Re)constructing Lie groupoids from their bisections and applications to prequantisation, arXiv:1506.05415 [math.DG].

September 7, 14:15 - 15:15, room 734, Sentralbygg 2

Gereon Quick (NTNU): Continuous homotopy fixed points of Lubin-Tate spectra

Abstract: In the chromatic approach to stable homotopy theory, the action of the Morava stabilizer group \(G_n\) on Lubin-Tate spectra \(E_n\) plays a central role. in particular, Morava’s change of rings theorem suggests that a certain localization of the sphere spectrum should be equivalent to the homotopy fixed points of \(E_n\) under the action of \(G_n\). In this talk, I will discuss different ways to make sense of such homotopy fixed points. In particular, I will present joint work with Daniel G. Davis in which we provide a new construction of iterated continuous homotopy fixed point spectra. Our approach builds on the work of Goerss, Hopkins, and Miller, and extends previous results of Devinatz-Hopkins, Davis, Behrens-Davis and Fausk.

October 5, 14:15 - 15:15, room 734, Sentralbygg 2

Truls Bakkejord Ræder (NTNU): THH and free commutative ring spectra

Abstract: In this talk, I will briefly describe classical and modern constructions of topological Hochschild homology, and we will see how the latter can be used to explicitly identify THH of certain free commutative ring spectra.

October 12, 14:15 - 15:15, room 734, Sentralbygg 2

Markus Szymik (NTNU): On the rational homology of the automorphism groups of free 2-nilpotent groups

Abstract: I will give some context and then explain some recent stable computations of mine.

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

Gard Spreemann (NTNU): Using persistent homology to reveal hidden information in neural data

Abstract: Mammalian navigation is aided by place cells, which are neurons that fire preferentially when the animal is in certain regions of space. It is known that the firing activities of these and related neurons are not governed solely by spatial position, but also by head direction, theta wave phase, sensory stimuli, etc., and probably also by further unknown influences. Such covariates are thought of as being reflected in the animal's "state space", and knowledge of its topological properties can reveal hidden information about a priori unknown covariates.

We propose a method wherein an approximation of such a state space is built from spike train recordings of neurons. Persistent homology is then used to reveal properties of the space. Through an inference process, we remove the contributions of known covariates to the spike trains, and thus to the reconstructed stace space. After all known covariates have been accounted for, persistent homology reveals properties of any potential remaining unknown ones.

See also arXiv:1510.06629.

November 9, 14:15 - 15:15, room 734, Sentralbygg 2

Pedro de M. Rios (Sao Paolo): Symbol Correspondences for Spin Systems

Abstract: In this talk, I will give an overview of the motivation and basic features of the theory of Symbol Correspondences for Spin Systems, as developed in the homonymous book by Eldar Straume and myself that was published recently (Birkhauser/Springer, 2014).

In a nutshell: a quantum spin-\(j\) system is a finite-dimensional complex Hilbert space \(C^k\) (\(k = 2j + 1\), \(j\) integer or half-integer) together with an unitary irreducible representation of \(\mathrm{SU}(2)\) on \(C^k\) and the algebra of operators on \(C^k\) (the familiar algebra of complex \(k\)-square matrices); the classical spin system is the \(2\)-sphere \(S^2\) with its standard symplectic form together with its Poisson algebra of smooth functions on \(S^2\); a spin-\(j\) symbol correspondence is a map from \(k\)-square matrices to smooth functions on \(S^2\), satisfying some natural properties, so that questions about the (semi)classical limit of quantum spin systems (\(j\) going to infinity) can be addressed in terms of the symbols (smooth functions on \(S^2\)) and their algebra (induced from matrix algebra).

November 16, 14:15 - 15:15, room 734, Sentralbygg 2

Magnus Bakke Botnan (NTNU): Algebraic Stability of Zigzag Persistence Modules

Abstract: The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of \(R\)-valued functions, the result was later cast in a more general algebraic form, in the language of persistence modules and interleavings. In this talk, we present an analogue of the algebraic stability theorem for zigzag persistence modules.

This is joint work with Michael Lesnick (Columbia University).