Related papers: Synthetic Differential Geometry of Chen's Iterated…
Sheaf theoretically based Abstract Differential Geometry incorporates and generalizes all the classical differential geometry. Here, we undertake to partially explore the implications of Abstract Differential Geometry to classical…
The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are…
The de Rham complex arises naturally when studying problems in electromagnetism and fluid mechanics. Stable numerical methods to solve these problems can be obtained by using a discrete de Rham complex that preserves the structure of the…
This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks.…
The Feynman Path Integral is extended in order to capture all solutions of a quantum field theory. This is done via a choice of appropriate integration cycles, parametrized by M in SL(2,C), i.e., the space of allowed integration cycles is…
A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial…
We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring…
This paper introduces a new construction of subcomplexes associated with a truncated multicomplex. Inspired by the machinery of spectral sequences, this construction yields a collection of interrelated subcomplexes whose differentials…
We introduce an algebro-geometric version of the free loop space for any scheme X of finite type. This is an ind-scheme of ind-infinite type containing the scheme of formal germs of curves on X. Then, we give a direct geometric…
The volume of a cyclic polytope can be obtained by forming an iterated integral along a suitable piecewise linear path running through its edges. Different choices of such a path are related by the action of a subgroup of the combinatorial…
On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic…
The radically synthetic foundation for smooth geometry formulated in [Law11] postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is…
This paper studies geometric properties of the Iterated Matrix Multiplication polynomial and the hypersurface that it defines. We focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the…
The chapter contains a detailed presentation of the surface integral theory for modelling light diffraction by surface-relief diffraction gratings having a one-dimensional periodicity. Several different approaches are presented, leading…
Path integrals for particles in curved spaces can be used to compute trace anomalies in quantum field theories, and more generally to study properties of quantum fields coupled to gravity in first quantization. While their construction in…
This paper studies silted algebras, namely, endomorphism algebras of 2-term silting complexes, over path algebras of Dynkin quivers. We will describe an algorithm to produce all basic 2-term silting complexes over the path algebra of a…
We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve…
We use Chen iterated line integrals to construct a topological algebra ${\cal A}_p$ of separating functions on the {\it Group of Loops} ${\bf L}{\cal M}_p$. ${\cal A}_p$ has an Hopf algebra structure which allows the construction of a group…
We give a new proof of the trace formula for regular graphs. Our approach is inspired by path integral approach in quantum mechanics, and calculations are mostly combinatorial.
Topologies of large deformation Contact-aided Compliant Mechanisms (CCMs), with self and mutual contact, exemplified via path generation applications, are designed using the continuum synthesis approach. Design domains are parameterized…