1,721,067 research outputs found
Unexpected Connections
Full text linkFor Matilde Marcolli, physics and mathematics don't intersect so much as form a two-way street—and she travels in both directions
BUILDING COSMOLOGICAL MODELS VIA NONCOMMUTATIVE GEOMETRY
Full text link This is an overview of new and ongoing research developments aimed at constructing cosmological models based on noncommutative geometry, via the spectral action functional, thought of as a modified gravity action, which includes the coupling with matter when computed on an almost commutative geometry. This survey is mostly based on recent results obtained in collaboration with Elena Pierpaoli and Kevin Teh. We describe various aspects of cosmological models of the very early universe, developed by the author and Pierpaoli, based on the asymptotic expansion of the spectral action functional and on renormalization group analysis of the associated particle physics model (an extension of the standard model with right-handed neutrinos and Majorana mass terms previously developed in collaboration with Chamseddine and Connes). We also describe nonperturbative results, more recently obtained by Pierpaoli, Teh, and the author, which extend to the more modern universe, which show that, for different candidate cosmic topologies, the form of the slow-roll inflation potentials obtained from the nonperturbative calculation of the spectral action are strongly coupled to the underlying geometry and topology. We discuss some ongoing directions of research and open questions in this new field of "noncommutative cosmology". The paper is based on the talk given by the author at the conference "Geometry and Quantum Field Theory" at the MPI, in honor of Alan Carey. </jats:p
Spectral action gravity and cosmological models
Full text linkThis paper surveys recent work of the author and collaborators on cosmological models based on the spectral action functional of gravity. A more detailed presentation of the topics surveyed here will be available in a forthcoming book [1]
Gabor frames from contact geometry in models of the primary visual cortex
Full text linkWe analyze the interplay between contact geometry and Gabor filters signal
analysis in geometric models of the primary visual cortex. We show in
particular that a specific framed lattice and an associated Gabor system is
determined by the Legendrian circle bundle structure of the -manifold of
contact elements on a surface (which models the V1-cortex), together with the
presence of an almost-complex structure on the tangent bundle of the surface
(which models the retinal surface). We identify a scaling of the lattice, also
dictated by the manifold geometry, that ensures the frame condition is
satisfied. We then consider a -dimensional model where receptor profiles
also involve a dependence on frequency and scale variables, in addition to the
dependence on position and orientation. In this case we show that a proposed
profile window function does not give rise to frames (even in a distributional
sense), while a natural modification of the same generates Gabor frames with
respect to the appropriate lattice determined by the contact geometry.Comment: LaTeX, 33 pages (v2: expanded introduction, v3: typos correction and
short explanatory comments added; v4: journal overlay published version
Information Algebras and Their Applications
No full textIn this lecture we will present joint work with Ryan Thorngren on thermodynamic semirings and entropy operads, with Nicolas Tedeschi on Birkhoff factorization in thermodynamic semirings, ongoing work with Marcus Bintz on tropicalization of Feynman graph hypersurfaces and Potts model hypersurfaces, and their thermodynamic deformations, and ongoing work by the author on applications of thermodynamic semirings to models of morphology and syntax in Computational Linguistics
Arithmetic noncommutative geometry
No full textArithmetic noncommutative geometry denotes the use of ideas and tools from the field of noncommutative geometry, to address questions and reinterpret in a new perspective results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at archimedean places of arithmetic surfaces and varieties. The main reason why noncommutative geometry can be expected to say something about topics of arithmetic interest lies in the fact that it provides the right framework in which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry, such as moduli spaces (of which modular curves are the simplest case) or arithmetic varieties (completed by suitable "fibers at infinity"), by adding boundaries that are invisible to algebraic geometry, such as degenerations of elliptic curves to noncommutative tori. The text of the book is organized around series of invited lectures delivered by the author at various universities, and the results presented are based on work of the author in collaboration with Alain Connes, Katia Consani, Yuri Manin, and Niranjan Ramachandran
Motives: an introductory survey for physicists
Full text linkWe survey certain accessible aspects of Grothendieck’s theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have
recently found applications in quantum field theory. An appendix (by Matilde Marcolli) sketches further connections between motivic theory and theoretical physics
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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