EPFL Applied Topology Seminar 2018/19

Please see the schedule below for the time and place of each talk.

Some seminars will take place on the Lausanne campus and others at Campus Biotech.

Program

Date and place Title Speaker
14.11.2018
15h
B1.05_Videoconference room
Campus Biotech
Rips magnitude Dejan Govc
Aberdeen
26.11.2018
11h
MA 10
Evolution of triangulations: Hausdorff and spectral dimensions José Fernando Mendes
Aveiro
05.12.2018
15h
B1.03_NeuroTopOffice
Campus Biotech
Bayesian nonparametric modeling of populations of brain networks Daniel Durante
Bocconi
18.01.2019
10h
B1.03_NeuroTopOffice
Campus Biotech
Decomposition of persistence modules Magnus Bakke Botnan
VU Amsterdam
26.02.2019
14h15
TBA
TBA Dimos Gkountaroulis
UNIL

Abstracts

Govc: Magnitude is a numerical invariant of metric spaces (and more generally, enriched categories) introduced by Tom Leinster which has been shown to arise as the graded Euler characteristic of a certain homology theory. Richard Hepworth has recently suggested to examine an analogous invariant for persistent homology, called Rips magnitude, which arises as a graded Euler characteristic of persistent homology. In the talk I will describe some of its basic properties and examine its asymptotic behaviour in the case of finite subsets of the
circle. This is joint work with Richard Hepworth.

Mendes:  How complex networks formed by triangulations and higher-dimensional simplicial complexes can represent closed evolving manifolds [1]. In particular, for triangulations, the set of possible transformations of these networks is restricted by the condition that at each step, all the faces must be triangles, which is the key constraint in this theory. Stochastic application of these operations leads to random networks with different architectures. I will show how geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties can be described. This characterisation includes the Hausdorff and spectral dimensions of the resulting networks, their degree distributions, and various structural correlations. The results reveal a rich zoo of architectures and geometries of these networks, some of which appear to be small worlds while others are finite-dimensional with a wide spectrum of Hausdorff and spectral dimensions.

[1] D. C. da Silva, G. Bianconi, R. A. da Costa, S. N. Dorogovtsev, and J. F. F. Mendes, Complex network view of evolving manifolds, Phys. Rev. E 97, 032316 (2018).

Durante: In neuroscience there is increasing interest in relating the structural connectivity network of white matter fibers in the human brain to cognitive traits and neuropsychiatric disorders. In fact, there is evidence that the structural brain network is an important driver of variability in cognitive traits and disorders. Recent connectomics pipelines can estimate the brain network based on diffusion tensor imaging and structural MRI. This produces a realization from a network-valued random variable for each individual in a study. I will present recent nonparametric Bayes advances for analyzing network-valued data, and for performing inference on the relationship between brain networks and cognitive traits. These methods are provably flexible, reduce dimension adaptively and can be used for formal inferences on group differences adjusting for multiple comparisons automatically. I will discuss key improvements relative to current approaches and illustrate the methods through application to creative reasoning and Alzheimer’s disease data.

Botnan:  In this talk I will sketch an elementary proof of the fact that any pointwise finite-dimensional persistence module over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. This result will then be applied to give a short proof of the well-known fact that a pointwise finite-dimensional persistence module over a totally ordered set is interval decomposable. A similar structure theorem for middle exact persistence module over a product of totally ordered sets will also be discussed. This is joint work with W. Crawley-Boevey.

Gkountaroulis: