Tautological bundles on Hilbert schemes of points often enter into enumerative and physical computations. I will explain how to use the Donaldson-Thomas theory of toric threefolds to produce combinatorial identities that are expressed geometrically using tautological bundles on the Hilbert scheme of points on a surface. I'll also explain how these identities can be used to study Euler characteristics of tautological bundles over Hilbert schemes of points on general surfaces.

The sequence of so-called Signature moments describes the laws of many stochastic processes in analogy with how the sequence of moments describes the laws of vector-valued random variables. However, even for vector-valued random variables, the sequence of cumulants is much better suited for many tasks than the sequence of moments. This motivates the study of so-called Signature cumulants. To do so, an elementary combinatorial approach is developed and used to show that in the same way that cumulants relate to the lattice of partitions, Signature cumulants relate to the lattice of so-called "ordered partitions". This is used to give a new characterisation of independence of multivariate stochastic processes.

The homotopy bordism category hCob_d has as objects closed (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. This is a simplified version of the topologically enriched bordism category Cob_d whose classifying space B(Cob_d) been completely determined by Galatius-Madsen-Tillmann-Weiss in 2006. In comparison, little is known about the classifying space B(hCob_d).

In the first part of the talk I will give an introduction to bordism categories and their classifying spaces. In the second part I will identify B(hCob_1) showing, in particular, that the rational cohomology ring of hCob_1 is polynomial on classes \kappa_i in degrees 2i+2 for all i>=1. The seemingly simpler category hCob_1 hence has a more complicated classifying space than Cob_1.

Cities are now central to addressing global changes, ranging from climate change to economic resilience. There is a growing concern of how to measure and quantify urban phenomena, and one of the biggest challenges in quantifying different aspects of cities and creating meaningful indicators lie in our ability to extract relevant features that characterize the topological and spatial patterns of urban form. Many different models that can reproduce large-scale statistical properties observed in systems of streets have been proposed, from spatial random graphs to economical models of network growth. However, existing models fail to capture the diversity observed in street networks around the world. The increased availability of street network datasets and advancements in deep learning models present a new opportunity to create more accurate and flexible models of urban street networks, as well as capture important characteristics that could be used in downstream tasks.  We propose a simple approach called Convolutional-PCA (ConvPCA) for both creating low-dimensional representations of street networks that can be used for street network classification and other downstream tasks, as well as a generating new street networks that preserve visual and statistical similarity to observed street networks.

Link to the preprint