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Timothy Hosgood

Bio: Timothy Hosgood is an academic researcher from Aix-Marseille University. The author has contributed to research in topics: Mathematics & Abstract simplicial complex. The author has an hindex of 2, co-authored 7 publications receiving 16 citations.

Papers
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TL;DR: Weighted projective spaces as discussed by the authors allow for non-trivial weights and allow for more than enough interesting phenomena, but it is the fact that weighted projective space arises naturally in the context of classical algebraic geometry that can be surprising.
Abstract: Weighted projective space arises when we consider the usual geometric definition for projective space and allow for non-trivial weights. On its own, this extra freedom gives rise to more than enough interesting phenomena, but it is the fact that weighted projective space arises naturally in the context of classical algebraic geometry that can be surprising. Using the Riemann-Roch theorem to calculate l(E,nD) where E is a non-singular cubic curve inside projective 2-space and D=p is a point in E we obtain a non-negatively graded ring R(E) by taking the direct sum of the L(E,nD) for n greater than or equal to 0. This gives rise to an embedding of E inside the weighted projective space P(1,2,3). To understand a space it is always a good idea to look at the things inside it. The main content of this paper is the introduction and explanation of many basic concepts of weighted projective space and its varieties. There are already many brilliant texts on the topic but none of them are aimed at an audience with only an undergraduate's knowledge of mathematics. This paper hopes to partially fill this gap whilst maintaining a good balance between 'interesting' and 'simple'. The main result of this paper is a reasonably simple degree-genus formula for non-singular 'sufficiently general' plane curves, proved using not much more than the Riemann-Hurwitz formula.

15 citations

Posted Content
TL;DR: In this paper, it was shown that Green's barycentric simplicial connection is indeed admissible, and that this condition is exactly what we need in order to be able to apply Chern-Weil theory and construct characteristic classes.
Abstract: In the previous part of this diptych, we defined the notion of an admissible simplicial connection, as well as explaining how H.I. Green constructed a resolution of coherent analytic sheaves by locally free sheaves on the Cech nerve. This paper seeks to apply these abstract formalisms, by showing that Green's barycentric simplicial connection is indeed admissible, and that this condition is exactly what we need in order to be able to apply Chern-Weil theory and construct characteristic classes. We show that, in the case of (global) vector bundles, the simplicial construction agrees with what one might construct manually: the explicit Cech representatives of the exponential Atiyah classes of a vector bundle agree. Finally, we summarise how all the preceding theory fits together to allow us to define Chern classes of coherent analytic sheaves, as well as showing uniqueness in the compact case.

6 citations

Journal ArticleDOI
TL;DR: In this article , the authors put the use of diagrams on a firm mathematical footing, while also systematizing a broadly applicable framework to reason formally about systems of equations and their solutions.
Abstract: Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm mathematical footing, while also systematizing a broadly applicable framework to reason formally about systems of equations and their solutions. Our main mathematical tools are category-theoretic diagrams, which are well known, and morphisms between diagrams, which have been less appreciated. As an application of the diagrammatic framework, we show how complex, multiphysical systems can be modularly constructed from basic physical principles. A wealth of examples, drawn from electromagnetism, transport phenomena, fluid mechanics, and other fields, is included.

4 citations

Dissertation
25 Jun 2020
TL;DR: In this article, O'Brian, Toledo, and Tong constructions of classes de Chern for faisceaux analytiques coherents in the cohomologie de De Rham, in respect to the filtration de Hodge.
Abstract: L’objet de cette these est de reinterpreter et de poursuivre un travail non publie de Green, dont l’objet est de construire des classes de Chern pour les faisceaux analytiques coherents a valeurs dans la cohomologie de De Rham en respectant la filtration de Hodge. La seconde partie de la these est consacree a la construction d’un enrichissement categorique de la categorie derivee bornee des complexes de faisceaux coherents sur une variete complexe arbitraire: les objets consideres sont des fibres vectoriels « simpliciaux » munis d’un type special de connexions simpliciales. Cette construction repose sur la theorie des cochaines tordues, developpee dans ce cadre par O’Brian, Toledo, et Tong. La premiere partie est consacree a definir un relevement categorique via le modele precedent d’un caractere de Chern en cohomologie de De Rham respectant la filtration de Hodge. Cette construction peut etre realisee en adaptant la theorie de Chern-Weil classique au cadre simplicial, via la theorie de l’integration fibree de Dupont

3 citations

Posted Content
TL;DR: In this article, the authors make the case that the analogy between deep neural networks and actual brains is structurally flawed, since the wires in neural networks are more like nerve cells, in that they are what cause information to flow.
Abstract: There is an analogy that is often made between deep neural networks and actual brains, suggested by the nomenclature itself: the "neurons" in deep neural networks should correspond to neurons (or nerve cells, to avoid confusion) in the brain. We claim, however, that this analogy doesn't even type check: it is structurally flawed. In agreement with the slightly glib summary of Hebbian learning as "cells that fire together wire together", this article makes the case that the analogy should be different. Since the "neurons" in deep neural networks are managing the changing weights, they are more akin to the synapses in the brain; instead, it is the wires in deep neural networks that are more like nerve cells, in that they are what cause the information to flow. An intuition that nerve cells seem like more than mere wires is exactly right, and is justified by a precise category-theoretic analogy which we will explore in this article. Throughout, we will continue to highlight the error in equating artificial neurons with nerve cells by leaving "neuron" in quotes or by calling them artificial neurons. We will first explain how to view deep neural networks as nested dynamical systems with a very restricted sort of interaction pattern, and then explain a more general sort of interaction for dynamical systems that is useful throughout engineering, but which fails to adapt to changing circumstances. As mentioned, an analogy is then forced upon us by the mathematical formalism in which they are both embedded. We call the resulting encompassing generalization deeply interacting learning systems: they have complex interaction as in control theory, but adaptation to circumstances as in deep neural networks.

Cited by
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Book ChapterDOI
01 Jan 2013
TL;DR: In this paper, the authors define a subset of the complex plane ℂ2 of dimension two, where the real dimension of a 2-dimensional complex plane is equal to 4, and equating to zero the complex number f (x,y) is equatable to zero both its real part and its imaginary part; that is, we have two equations in the four real variables (Re x, Imx, Rey, Imy).
Abstract: The equation f (x,y) = 0, where x and y are complex variables and f is a polynomial (with complex coefficients in general), defines a subset of the complex plane ℂ2 (with coordinates x and y) of dimension two, since the real dimension of a 2-dimensional complex plane is equal to 4, and equating to zero the complex number f (x,y) means equating to zero both its real part and its imaginary part; that is, we have two equations in the four real variables (Re x, Imx, Rey, Imy).

27 citations

Journal ArticleDOI
TL;DR: In this paper, a generalization of the classical one-parameter Grobner degeneration associated to a weight has been proposed, which is the pull-back of a toric family defined by a Rees algebra along the universal torsor.
Abstract: Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Grobner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Grobner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Grobner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base $X_C$ (the toric variety associated to $C$) along the universal torsor $\mathbb A^m \to X_C$. We apply this construction to the Grassmannians ${\rm Gr}(2,\mathbb C^n)$ with their Plucker embeddings and the Grassmannian ${\rm Gr}\big(3,\mathbb C^6\big)$ with its cluster embedding. In each case, there exists a unique maximal Grobner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for ${\rm Gr}(2,\mathbb C^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.

23 citations

Posted Content
TL;DR: In this paper, it was shown that Green's barycentric simplicial connection is indeed admissible, and that this condition is exactly what we need in order to be able to apply Chern-Weil theory and construct characteristic classes.
Abstract: In the previous part of this diptych, we defined the notion of an admissible simplicial connection, as well as explaining how H.I. Green constructed a resolution of coherent analytic sheaves by locally free sheaves on the Cech nerve. This paper seeks to apply these abstract formalisms, by showing that Green's barycentric simplicial connection is indeed admissible, and that this condition is exactly what we need in order to be able to apply Chern-Weil theory and construct characteristic classes. We show that, in the case of (global) vector bundles, the simplicial construction agrees with what one might construct manually: the explicit Cech representatives of the exponential Atiyah classes of a vector bundle agree. Finally, we summarise how all the preceding theory fits together to allow us to define Chern classes of coherent analytic sheaves, as well as showing uniqueness in the compact case.

6 citations

Posted Content
TL;DR: The space of Lame functions of order m is isomorphic to the space of pairs (elliptic curve, Abelian differential) where the differential has a single zero of order 2m at the origin and m double poles with vanishing residues as mentioned in this paper.
Abstract: The space of Lame functions of order m is isomorphic to the space of pairs (elliptic curve, Abelian differential) where the differential has a single zero of order 2m at the origin and m double poles with vanishing residues. We describe the topology of this space: it is a Riemann surface of finite type; we find the number of components and the genus and Euler characteristic of each component. As an application we find the degrees of Cohn's polynomials confirming a conjecture by Robert Maier. As another application we partially describe the degeneration locus of the space of spherical metrics on tori with one conic singularity where the conic angle is an odd multiple of 2$\pi$.

4 citations