CGTA Inaugural Event Event

For the schedule of talks, titles and abstracts , please see this page .

The distinguished speakers include:

• Prof Kathryn Hess (École Polytechnique Fédérale de Lausanne (EPFL), Switzerland)
• Prof Nigel Higson (Pennsylvania State University, USA)
• Prof Sayan Mukherjee (Duke University, USA)
• Prof Ulrike Tillmann (Oxford University, UK)
• Prof Karen Vogtmann (Warwick University, UK)
• Prof Guoliang Yu (Texas A&M University, USA)

Titles and abstracts

Kathryn Hess: Topological characterization of neuron morphologies

Abstract:

The Topological Morphology Descriptor (TMD) encodes complex branching patterns, such as those of neuronal trees, as barcodes. It provides an unbiased benchmark test for the categorization of neuronal morphologies, enabling us to quantify and characterize the structural differences between distinct morphological classes and thus to increase our understanding of the anatomy and diversity of branching morphologies. For example, a consensus on the number of morphologically different types of pyramidal cells (PCs) in the neocortex has yet to be reached, despite over a century of anatomical studies, due to a lack of agreement on the subjective classifications of neuron types based on expert analyses of the shapes of somata, dendrites, and axons. We showed that applying the TMD to dendritic arbors of rat PCs provides an objective, reliable classification into 17 types and insight into the challenging problem of determining when neurons belong to different types or to a continuum of the same type. Our topological classification does not require expert input, is stable, and helps settle the long-standing debate on whether cell-types are discrete or form a morphological continuum. Similarly, there have been few quantitative characterizations of the morphological and biophysical properties of neurons in the human neocortex. Applying the TMD to the apical dendrites of 60 3D reconstructed pyramidal neurons from layers 2 and 3 in the human temporal cortex revealed the existence of two morphologically distinct classes that also had distinct electrical behavior.

References:

• L. Kanari, P. Dlotko, M. Scolamiero, R. Levi, J. C. Shillcock, K. Hess, and H. Markram, A topological representation of branching morphologies, Neuroinformatics, 2017, DOI: 10.1007/s12021-017-9341-1 .
• L. Kanari, S. Ramaswamy, Y. Shi, S. Morand, J. Meystre, R. Perin, M. Abdellah, Y. Wang, K. Hess, and H. Markram, Objective classification of neocortical pyramidal cells, available on bioRXiv, DOI: 10.1101/349977
• Y. Deitcher, G. Eyal, L. Kanari, M. B. Verhoog, G. A. Atenekeng, K. H. D. Mansvelder, C. P. J. de Kock, I. Segev, Comprehensive morpho-electrotonic analysis shows 2 distinct classes of L2 and L3 pyramidal neurons in human temporal cortex, Cerebral Cortex, 2017, DOI: 10.1093/cercor/bhx226

Nigel Higson: Noncommutative geometry and group representations

Abstract:

Alain Connes' noncommutative geometry involves the application of Hilbert space ideas and techniques to the study of geometric spaces, both old and new. I shall try to illustrate some of Connes' ideas by describing parts of Harish-Chandra's tempered representation theory from the noncommutative-geometric point of view. Geometric spaces of the traditional sort, for example homogeneous spaces, play a role here, but so do spaces that are not usually considered from a geometric point of view, for example Harish-Chandra's tempered dual. Noncommutative geometry allows them to be treated together, with interesting consequences.

Sayan Mukherjee: How Many Directions Determine a Shape

Abstract:

In this paper we consider topological transforms based on Euler calculus. These transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. The transforms take a shape, viewed as a tame subset M of R^d, and associates to each direction v in S^{d-1} a shape summary obtained by scanning M in the direction v. By using an inversion theorem of Schapira, we show these transforms are injective on the space of shapes---each shape has a unique transform. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, our main result provides the first (to our knowledge) finite bound required to specify any shape in certain uncountable families of shapes, bounded below by curvature. This result is perhaps best appreciated in terms of shattering number or the perspective that any point in these particular moduli spaces of shapes is indexed using a tree of finite depth.

Ulrike Tillmann: Classifying commutativity

Abstract:

One of the fundamental objects in geometry and algebra is the representation space of a group. When one considers the special case of the free abelian group, this becomes the study of commuting tuples and quotients there of. This seemingly simple case has already a rich structure. We will review some of the recent progress. Taking the study of commutativity a step further, we are led to consider commutative bundles. A commutative structure on a vector bundle is given by a choice of local trivialisations such that the transition functions commute pointwise. Similarly we define a commutative  structure on a smooth manifold bundle by a choice of trivialisations such that its transition functions commute locally up to homotopy. Like the former, also the latter gives rise to a generalised cohomology theory. We explore these theories and describe a general recipe of how to construct the associated infinite loop spaces.
This is work in progress with Simon Gritschacher.

Karen Vogtmann: Spaces of finite metric graphs

Abstract:

Finite metric graphs are used to model phenomena in various branches of mathematics and science. The set of all graphs modeling a particular phenomenon forms a topological space in a natural way. I will describe examples of this from biology, physics and geometric group theory, and indicate how studying the topology and geometry of these spaces has proved useful in each case.

Guoliang Yu: The Novikov conjecture and groups of volume preserving diffeomorphisms

Abstract:

In this talk, I will give an introduction to the Novikov conjecture and give an outline why the Novikov conjecture holds for discrete subgroups of the group of volume preserving diffeomorphisms. An interesting ingredient is isometry rigidity of a certain infinite dimensional symmetric space.
This is joint work with Sherry Gong and Jianchao Wu.