Related papers: Lecture notes on the Fundamental Structures of Dif…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…
We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions. To train a…
The structural invariant subspaces of the discrete-time singular Hamiltonian system are used in 1] to give an analytic nonrecursive expression of all the admissible trajectories. A deeper insight into the features of these subspaces,…
Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen and…
We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of…
These are lecture notes of a course taken in Leipzig 2023, spring semester. It deals with extremal combinatorics, algebraic methods and combinatorial geometry. These are not meant to be exhaustive, and do not contain many proofs that were…
These are notes for a graduate-level introductory course on singularity categories.
These informal notes deal with some basic properties of metric spaces, especially concerning lengths of curves.
We study fundamental forms of algebraic varieties using the sheaves of principal parts of line bundles and establish a vanishing theorem for any order fundamental forms. We also give connection of fundamental forms with the higher order…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the $\alpha$-structure of frame bundles over…
These notes aim to give a first introduction to intersection cohomology and perverse sheaves with applications to representation theory or quantum groups in mind.
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other…
Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport,…
In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic…
Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to…
Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such…
The paper contains a review on the general connection theory on differentiable fibre bundles. Particular attention is paid to (linear) connections on vector bundles. The (local) representations of connections in frames adapted to holonomic…
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
In this paper, we describe a general theory of "spaces with structure sheaves." Specializations of this theory include the classical theory of schemes, the theory of Deligne-Mumford stacks, and their derived generalizations.