scholarly journals Hairer’s reconstruction theorem without regularity structures

2021 ◽  
Author(s):  
Francesco Caravenna ◽  
Lorenzo Zambotti
Keyword(s):  
Regularity Structures ◽  
Reconstruction Theorem

scholarly journals Adelic descent theory

2017 ◽  
Vol 153 (8) ◽  
pp. 1706-1746
Author(s):  
Michael Groechenig
Keyword(s):  
Vector Bundles ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Reductive Algebraic Group ◽  
Perfect Complexes ◽  
Descent Theory ◽  
Closed Fields ◽  
Ring Of Continuous Functions ◽  
Reconstruction Theorem ◽  
Algebraically Closed Fields

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

scholarly journals Projective Reconstruction in Algebraic Vision

2019 ◽  
Vol 63 (3) ◽  
pp. 592-609
Author(s):  
Atsushi Ito ◽  
Makoto Miura ◽  
Kazushi Ueda
Keyword(s):  
Projective Space ◽  
Arbitrary Dimension ◽  
Projective Spaces ◽  
Rational Maps ◽  
Projective Reconstruction ◽  
Reconstruction Theorem

AbstractWe discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.

scholarly journals Algebraic renormalisation of regularity structures

2018 ◽  
Vol 215 (3) ◽  
pp. 1039-1156 ◽  
Author(s):  
Y. Bruned ◽  
M. Hairer ◽  
L. Zambotti
Keyword(s):  
Regularity Structures

scholarly journals Quantum reconstruction for Fano bundles on projective space

2015 ◽  
Vol 218 ◽  
pp. 1-28
Author(s):  
Andrew Strangeway
Keyword(s):  
Projective Space ◽  
Vector Bundles ◽  
Quantum Cohomology ◽  
Lefschetz Theorem ◽  
Small Quantum ◽  
Low Degree ◽  
Reconstruction Theorem ◽  
Rank 2

AbstractWe present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivization of the bundle from a small number of low-degree Gromov-Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch program.

scholarly journals Rosenberg’s reconstruction theorem

2018 ◽  
Vol 36 (1) ◽  
pp. 98-117 ◽  
Author(s):  
Martin Brandenburg
Keyword(s):  
Reconstruction Theorem

RECONSTRUCTION THEOREMS AND RATIONAL CONFORMAL FIELD THEORIES

1991 ◽  
Vol 06 (24) ◽  
pp. 4359-4374 ◽  
Author(s):  
SHAHN MAJID
Keyword(s):  
Fourier Transform ◽  
Quantum Group ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Generalized Fourier Transform ◽  
Conformal Field ◽  
Reconstruction Theorem

We obtain an explicit reconstruction theorem for rational conformal field theories and other situations where we are presented with a braided or quasitensor category [Formula: see text]. It takes the form of a generalized Fourier transform. The reconstructed object turns out to be a quantum group in a generalized sense. Our results include both the Tannaka-Krein case where there is a functor [Formula: see text], and the case where there is no functor at all.

A reconstruction theorem for complex polynomials

2015 ◽  
Vol 26 (09) ◽  
pp. 1550073 ◽  
Author(s):  
Luka Boc Thaler
Keyword(s):  
Real Axis ◽  
Ergodic Theory ◽  
Complex Dynamics ◽  
Julia Set ◽  
Complex Polynomials ◽  
Analogous Statement ◽  
The Real ◽  
Ergodic Properties ◽  
Reconstruction Theorem ◽  
Exceptional Polynomials

Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.]. They studied the real orbits of complex polynomials, and proved that for non-exceptional polynomials ergodic properties such as measure theoretic entropy are carried over to the real orbits mapping. Here we show that the result from [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.] also holds for exceptional polynomials, unless the Julia set is entirely contained in an invariant vertical line, in which case the entropy is 0. In [The reconstruction theorem for endomorphisms, Bull. Braz. Math. Soc. (N.S.) 33(2) (2002) 231–262.] Takens proved a reconstruction theorem for endomorphisms. In this case the reconstruction map is not necessarily an embedding, but the information of the reconstruction map is sufficient to recover the (2m + 1) th image of the original map. Our main result shows an analogous statement for the iteration of generic complex polynomials and the projection onto the real axis.

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