# Seminars in geometry/topology

## Spring 2015

## Monday January 19, 14:15 - 15:15, room 734, Sentralbygg 2

Alexander Schmeding (NTNU): *Character groups of Hopf algebras as infinite-dimensional Lie groups*

**Abstract:** Hopf algebras and their character groups occur naturally in diverse fields such as numerical analysis, noncommutative geometry and quantum field theory. For example, the famous Butcher group from numerical analysis turns out to be the character group of a Hopf algebra of rooted trees. In this talk we study the character group of a Hopf algebra as an infinite-dimensional Lie group. This is joint work with G. Bogfjellmo (Trondheim) and R. Dahmen (Darmstadt).

## Monday January 26, 14:15 - 15:15, room 734, Sentralbygg 2

Chad Giusti (University of Pennsylvania): *Topological analysis of neural population activity*

**Abstract:** Locating meaningful structure in neural activity and connectivity data is often complicated by the presence of (monotone) nonlinearities in both the biological system and in the measurements taken. Such nonlinearities can cause traditional eigenvalue-based methods to become
unreliable or difficult to interpret. To address this problem, we introduce a topological approach to matrix analysis that uncovers features of the data that are invariant under arbitrary monotone transformations. This method allows us to distinguish, for example, random matrices from those whose entries arise from a geometric structure. As an example, we analyze hippocampal place cell activity in rats during navigation, which we expect to exhibit such a geometric structure. Surprisingly, however, we also see such structure in the network during non-spatial behaviors, suggesting that this is a physical network phenomenon. This is joint work with Carina Curto and Vladimir Itskov.

## Wednesday January 28, 10:15 - 11:15, room 734, Sentralbygg 2

Chad Giusti (University of Pennsylvania): *Convexity and coding in neural networks*

**Abstract:** Theoretical or computational models of brain networks often involve the notion of each neuron having a "receptive field". Mathematically, this corresponds to the firing rate function being supported on some (usually convex) region in a "stimulus space". If such a model is to be valid,
the observed population activity must be consistent with the geometry induced by such convex receptive fields. The simplest manifestation of this geometry is in the pattern of intersections of the receptive fields: which neurons cofire? Here, we discuss two natural questions that arise from this observation: what cofiring patterns can be realized by such convex regions, and what role does the structure of the network play in shaping such patterns? This is joint work with Vladimir Itskov
and William Kronholm.

## Friday January 30, 10:15 - 12:00 (with break), room 734, Sentalbygg 2

Frank Neumann (University of Leicester): *Weil Conjectures and Moduli of Vector Bundles over Algebraic Curves*

**Abstract:** In 1949, Weil conjectured deep connections between the topology, geometry and arithmetic of projective algebraic varieties over a field in characteristic *p*, including an analogue of the celebrated Riemann hypothesis. These conjectures led to the development of *l*-adic etale cohomology in algebraic geometry as an analog of singular cohomology in algebraic topology by Grothendieck and his school and culminated with the proof of the Weil conjectures by Deligne in the 70s. In this talk I will outline how analogs of these Weil conjectures can be formulated and proved for the moduli stack of vector bundles over a projective algebraic curve in characteristic *p*. This basically comes down to counting correctly vector bundles up to isomorphisms via groupoid cardinalities using Behrend's trace formula for algebraic stacks. On the way I will recall the classical Weil conjectures and Grothendieck's ideas for using stacks in moduli problems. An interesting new feature in the stacky language here is the existence and interplay of different Frobenius morphisms, whose geometry is still rather mysterious.

## Monday February 9, 14:45 - 15:45, room 734, Sentralbygg 2

Gereon Quick (NTNU): *An introduction to etale homotopy theory: Motivation*

**Abstract:** Etale homotopy theory has been invented by Grothendieck, Artin, Mazur, and Friedlander in the 1970s and 1980s to make invariants from algebraic topology available for algebraic varieties over arbitrary fields. Though surprisingly, the first (but not only) spectacular application was the proof of the Adams conjecture by Quillen and Friedlander, a statement about vector bundles on topological spaces which does not at all seem to related to algebraic geometry.

My plan is to give a series of three talks (if the audience is interested). In the first talk, I will discuss the motivation and main idea of etale homotopy and then give an outline of the Quillen-Friedlander proof of the Adams conjecture. In this talk, I will not assume any specific background in algebraic geometry. So everybody who always wanted to know what etale homotopy is about is very welcome to join.

## Monday February 16, 14:15 - 15:15, room 734, Sentralbygg 2

Gereon Quick (NTNU): *An introduction to etale homotopy theory, Part 2: Constructions*

**Abstract:** This is the second part of a trilogy on etale homotopy theory. Etale homotopy theory has been invented by Grothendieck, Artin, Mazur, and Friedlander in the 1970s and 1980s to make invariants from algebraic topology available for algebraic varieties over arbitrary fields. In the first part, I discussed some motivation for defining an etale homotopy type for algebraic varieties. In the second talk, I will present the construction of the etale homotopy type. It involves several important ideas that also play a role in other fields. So there is something to learn even if you do not intend to use the etale homotopy type in your mathematical life.

Everybody who always wondered what this is all about is invited to join, independently of whether you did or did not attend the first talk. I will not assume any knowledge from part one, just that everybody is motivated.

## Friday February 27, 10:15 - 11:15, room 734, Sentralbygg 2

Gereon Quick (NTNU): *An introduction to etale homotopy theory, Part 3: Applications*

**Abstract:** This is the third part of a trilogy on etale homotopy theory. Etale homotopy theory has been invented by Grothendieck, Artin, Mazur, and Friedlander in the 1970s and 1980s to make invariants from algebraic topology available for algebraic varieties over arbitrary fields. In the third part, I will present a potpourri of applications of the etale homotopy type. In particular, I will discuss etale K-theory and the idea of Sullivan’s influential proof of the Adams conjecture using Galois symmetries in topology.

## Friday March 20, 10:15 - 11:15, room 734, Sentralbygg 2

Ehud Meir (Københavns Universitet): *Finite dimensional Hopf algebras and invariant theory*

**Abstract:**
In this talk I will present an approach to study finite dimensional semisimple Hopf algebras, based on invariant theory. I will explain why the isomorphism type of such an algebra can be determined by a (very big) collection of invariant scalars, and will discuss the intuitive meaning of these scalars.
While in some cases they can be very easily interpreted, in some other cases their meaning is much less clear.
I will explain the connection between these invariants and questions about fields of definition, some open problems in Hopf algebra theory (about the dimension of irreducible representations) and how one can use them in order to prove that every such algebra satisfies a certain finiteness condition (the existence of at most finitely many Hopf orders). I will also explain the connection with invariants of 3-manifolds one receives in Topological Quantum Field Theory.

## Friday April 10, 10:15 - 12:00, room 734, Sentralbygg 2

Truls Bakkejord Ræder (NTNU): *TR*

**Abstract:** In this talk, I will use defining TR, topological restriction homology, as an excuse to talk about some equivariant stable homotopy theory.

## Friday April 17, 10:15 - 12:00, room 734, Sentralbygg 2

Gard Spreemann (NTNU): *Using persistent homology to reveal hidden information in place cells*

**Abstract:** As we all learned from the 2014 Nobel prize in medicine, mammalian navigation is aided by place cells, a kind of neuron whose firing fields make up a cover of space. It has been demonstrated that persistent homology can provide information about the space explored by the animal using *only* the cells' firing data. Neuroscientists suspect that place cells code for more than purely spatial information, and we therefore propose a method whereby correlation of firing time series acts as a proxy for firing field intersection, allowing persistent homology to reveal hidden "non-spatial" information encoded by the cells after the effects of known encodings have been inferred away.

This is joint work with Benjamin Adric Dunn, Magnus Botnan, Nils Baas and Yasser Roudi.

## Friday April 24, 10:15 - 12:00, room 734, Sentralbygg 2

Andreas Holmström (IHES): *An elementary approach to motives*

**Abstract:** We develop a completely elementary formalism called "e-motives", which can be seen as a "decategorified" version of Grothendieck's mysterious theory of motives. This formalism can serve as a good first stepping stone to learning about motives in general, and also makes it possible to explain many interesting problems in arithmetic geometry in an elementary fashion, such as the Sato-Tate conjecture and some questions about special values of L-functions. If time allows, I will discuss how this formalism can also be used to better understand some deeper questions on motives and zeta functions of stacks.

The talk will be divided into two parts, where the first should be accessible to anyone who has heard about commutative rings, while the second may assume some more familiarity with algebraic geometry and simplicial techniques.

## Monday April 27, 14:15 - 16:00, room 734, Sentralbygg 2

Magnus Bakke Botnan (NTNU): *Algebraic stability for zigzag persistence*

**Abstract:** Arguably, the central result in the theory of persistent homology is the stability theorem, originally introduced by Cohen-Steiner et al., and later presented in a more general algebraic form by Chazal et al. Carlsson et al. proved a stability theorem for zigzag persistent homology, but no analogue of the algebraic stability theorem has been available for zigzags. In this talk I will discuss recent work with Mike Lesnick where we prove such an analogue theorem. The focus of the talk will be not so much on the proof itself but rather on its implications.