Related papers: An Introduction to $\mathbf{A}^1$-Enumerative Geom…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
This introductory text arises from a lecture given in G\"oteborg, Sweden, given by the first author and is intended for undergraduate students, as well as for any mathematically inclined reader wishing to explore a synthesis of ideas…
A standard combinatorial construction, due to Kontsevich, associates to any A-infinity algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We…
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two…
In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a…
Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a…
We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes.
This book is an account of certain topics in general and algebraic topology. Instead of laying out a synopsis of each chapter, here is a sample of some of what is taken up: 1) Nilpotency and its role in homotopy theory. 2) Bousfield's…
We develop various aspects of classical enumerative geometry, including Euler characteristics and formulas for counting degenerate fibres in a pencil, with the classical numerical formulas being replaced by identitites in the…
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an…
This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories,…
This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a…
We present an overall picture of the advances in the description of black hole physics from the perspective of loop quantum gravity. After an introduction that discusses the main conceptual issues we present some details about the classical…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
Lecture notes of an algebraic geometry graduate course. The topics covered are as follows. Cohomology: ext sheaves and groups, cohomology with support, local cohomology, local duality. Duality: relative duality, Cohen-Macaulay schemes.…
These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
Oriented cohomology theories provide a general framework to perform intersection-theory-type calculus. The Chow ring, algebraic $K$-theory, and Levine--Morel's algebraic cobordism are all instances of such theories satisfying $\mathbb…
The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory…