"Algebraic K-theory via higher categories" (Thomas Nikolaus)
Abstract: The groups K_i(R) for a ring R and natural numbers i are abelian groups which we want to discuss in this lecture series. Our goal is to give an introduction into K-theory from the point of view of higher category theory.
The zeroth K-theory group K_0(R) is defined as the group completion of the monoid of isomorphism classes of finitely generated projective R modules, which is a monoid under direct sum. The higher K-groups are defined as the homotopy groups of a K-theory space/spectrum K(R). Roughly speaking, K(R) is defined as the group completion of the category of finitely generated, projective R-modules. This can best be made precise using a higher categorical version of group completion (as we are dealing with categories instead of monoids). We will explain this point of view and how to compute the groups K_*(R) in specific cases.
We will explain how K(R) is functorial in R and which additional structure the K-groups carry (i.e. they form a graded ring if R is commutative). If time permits, we will also explain variants of K-theory such as cyclic K-theory and Grothendieck-Witt theory. I will generally work under the assumption that the audience is familiar with the basic ideas of higher categories (e.g. infinity-categories) and stable homotopy theory but doesn’t know anything about K-theory. Most of the higher categorical results can be treated as a black-box, but it is important to understand the rough idea of what a higher category is and what a structure "up to coherence" is. Here are some recommendations to have a look if you want to get familiar to that circle of ideas (all sources have way more details than we need and technical details can safely be skipped):
Chapter 1 of "Higher Topos Theory" by J. Lurie
"A short course on infty categories" by M. Groth
The YouTube Lecture series about "Higher algebra" by myself and A. Krause, can be found here.
Many more excellent sources...
"Topological adventures in neuroscience" (Kathryn Hess)
Slides: Lecture 1 Lecture 2 Lecture 3
Abstract: Over the past decade, research at the interface of topology and neuroscience has grown remarkably fast. Topology has, for example, been successfully applied to objective classification of neuron morphologies, to automatic detection of network dynamics, to understanding the neural representation of natural auditory signals, and to demonstrating that the population activity of grid cells exhibits toroidal structure, as well as to describing brain structure and function and analyzing the relationship between them in a novel and effective manner. In this series of lectures, I’ll provide an overview of various promising recent applications of topology in neuroscience.
Lecture 1: Topology and neuron morphologies. Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD), that assigns a barcode to any geometric tree (i.e, any finite binary tree embedded in R3). We showed that the TMD effectively determines the reliability of clusterings of random and neuronal trees. Moreover, using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites. We have also reverse-engineered the TMD, in order to digitally synthesize dendrites, to compensate for the dearth of available biological reconstructions. The algorithm we developed, called Topological Neuron Synthesis (TNS), stochastically generates a geometric tree from a barcode, in a biologically grounded manner. The synthesized neurons are statistically indistinguishable from real neurons of the same type, in terms of morpho-electrical properties and connectivity. We synthesized networks of structurally altered neurons, revealing principles linking branching properties to the structure of large-scale networks. In this talk I will provide an overview of the TMD and the TNS and then describe the results of our theoretical and computational analysis of their behavior and properties, in which symmetric groups play a key role. In particular, I will specify the extent to which the TNS provides an inverse to the TMD. This is joint work with Adélie Garin and Lida Kanari, building on earlier collaborations led by Lida Kanari.
Lectures 2 and 3: Topological analysis of network structure and function. Graph theory has proven to be extremely useful for analyzing network structure and function, for networks arising in brain imaging, power grids, social networks, and more. More recently, the tools of algebraic topology have been successfully applied to characterizing and quantifying network structure and function, particularly in networks of neurons and brain regions.
In these two lectures I will first present the general framework for such topological analyses, then give a brief introduction to the necessary elements of neuroscience, before presenting particular applications. In particular I’ll survey the results my collaborators and I have obtained when applying topological tools to studying the relationship between brain structure and function, primarily in collaboration with the Blue Brain Project, as well as more recent work of Ran Levi and collaborators using “tribes” in networks and of Skander Moalla and Jacob Bamberger integrating information theory into our topological analyses.
I will also describe a novel approach to classifying network dynamics that employs tools from topological data analysis. The efficacy of this method was established by studying simulated activity in three small artificial neural networks in which we varied certain “biological” parameters, giving rise to dynamics that could be classified into four regimes, and showing that a machine learning classifier trained on features extracted from persistent homology could accurately predict the regime of the network it was trained on and also generalize to other networks not presented during training. I will also discuss the application of these methods to networks of actual neurons. This is joint work with Jean-Baptiste Bardin and Gard Spreemann, which was further extended by Tâm Nguyen.
"Graph Complexes and spaces of graphs" (Karen Vogtman)
Abstract: Graph complexes are chain complexes with very simple descriptions in terms of finite graphs. They were introduced by Kontsevich in his work on deformation quantization, but have proved to have applications in a wide variety of other areas. These include the study of groups important in low-dimensional topology such as automorphism groups of free groups and surface mapping class groups, which act naturally on spaces of graphs. I will describe several flavors of graph complexes and how they are related to each other and to the relevant groups.