Related papers: From Sparks to Grundles--Differential Characters

This is the second of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern--Simons theory. We provide a definition of trace over a crossed module…

For a second-order linear differential equation with two irregular singular points of rank three, multiple Laplace-type contour integral solutions are considered. An explicit formula in terms of the Stokes multipliers is derived for the…

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally…

We provide a natural interpretation of the secondary Euler characteristic and introduce higher Euler characteristics. For a compact oriented manifold of odd dimension, the secondary Euler characteristic recovers the Kervaire…

We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used…

We construct a new family of homomorphisms from Specht modules into Foulkes modules for the symmetric group. These homomorphisms are used to give a combinatorial description of the minimal partitions (in the dominance order) which label…

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links…

The rich spectral information of the graph Laplacian has been instrumental in graph theory, machine learning, and graph signal processing for applications such as graph classification, clustering, or eigenmode analysis. Recently, the Hodge…

This is a series of lecture notes, with embedded problems, aimed at students studying differential topology. Many revered texts, such as Spivak's "Calculus on Manifolds" and Guillemin and Pollack's "Differential Topology" introduce forms by…

We investigate the second-order correlation function of the characteristic polynomial of a sample covariance matrix. Starting from an explicit formula for the generating function, we re-obtain several well-known kernels from random matrix…

Graphs are used to represent and analyze data in domains as diverse as physics, biology, chemistry, planetary science, and the social sciences. Across domains, random graph models relate generative processes to expected graph properties,…

The Chern character maps are one of the most important working tools in mathematics. Although they admit numerous different constructions, they are not yet fully understood at the conceptual level. In this note we eliminate this gap by…

We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the…

This is a note on my mini-course in the International Workshop on Real and Complex Singularities held at ICMC-USP (Sao Carlos, Brazil) in July 2012. Here we introduce a new branch of the Thom polynomial theory for singularities of…

We describe algorithms based on invariant theory to solve problems on the geometry of curves, mainly those of genus 2, 3 and 4. New theoretical results building on the first author's PhD thesis are also included.

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding…

We present the construction of a Chern character in cyclic cohomology, involving an arbitrary number of associative algebras in contravariant or covariant position. This is a generalization of the bivariant Chern character for bornological…

We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show…

We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either…

Graph invariants provide a powerful analytical tool for investigation of abstract structures of graphs. They, combined in convenient relations, carry global and general information about a graph and its various substructures such as cycle…