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## Stringy K-theory and the Chern character

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Geometry#Mathematics - Differential Geometry#Mathematics - K-Theory and Homology#Mathematics - Quantum Algebra#19L47#53D45#55N15#14N35#55R65#57R20

For a finite group G acting on a smooth projective variety X, we construct
two new G-equivariant rings: first the stringy K-theory of X, and second the
stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct
a new ring called the full orbifold K-theory of Y. For a global quotient
Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra
of the full orbifold K-theory of the the stack Y and is linearly isomorphic to
the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a
different, ``quantum,'' product, which respects the natural group grading. We
prove there is a ring isomorphism, the stringy Chern character, from stringy
K-theory to stringy cohomology, and a ring homomorphism from full orbifold
K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy
Grothendieck-Riemann-Roch for etale maps.
We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's
construction. Since our constructions do not use complex curves, stable maps,
admissible covers, or moduli spaces, our results simplify the definitions of
Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of
Abramovich-Graber-Vistoli's orbifold Chow.
We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler
Resolution Conjecture holds for symmetric products.
Our results hold both in the algebro-geometric category and in the
topological category for equivariant almost complex manifolds.; Comment: Exposition improved and additional details provided. To appear in
Inventiones Mathematicae

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## Symmetric homotopy theory for operads

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/03/2015
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#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - K-Theory and Homology#Mathematics - Quantum Algebra#Mathematics - Representation Theory#18D50, 18G55

The purpose of this foundational paper is to introduce various notions and
constructions in order to develop the homotopy theory for differential graded
operads over any ring. The main new idea is to consider the action of the
symmetric groups as part of the defining structure of an operad and not as the
underlying category. We introduce a new dual category of higher cooperads, a
new higher bar-cobar adjunction with the category of operads, and a new higher
notion of homotopy operads, for which we establish the relevant homotopy
properties. For instance, the higher bar-cobar construction provides us with a
cofibrant replacement functor for operads over any ring. All these
constructions are produced conceptually by applying the curved Koszul duality
for colored operads. This paper is a first step toward a new Koszul duality
theory for operads, where the action of the symmetric groups is properly taken
into account.; Comment: 40 pages. Comments are welcome

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## Idempotent (Asymptotic) Mathematics and the Representation Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/06/2002
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A brief survey of some basic ideas of the so-called Idempotent Mathematics is
presented; an "idempotent" version of the representation theory is discussed.
The Idempotent Mathematics can be treated as a result of a dequantization of
the traditional mathematics over numerical fields in the limit of the vanishing
"imaginary Planck constant"; there is a correspondence, in the spirit of N.
Bohr's correspondence principle, between constructions and results in
traditional mathematics over the fields of real and complex numbers and similar
constructions and results over idempotent semirings. In particular, there is an
"idempotent" version of the theory of linear representations of groups. Some
basic concepts and results of the "idempotent" representation theory are
presented. In the framework of this theory the well-known Legendre transform
can be treated as an idempotent version of the traditional Fourier transform.
Some unexpected versions of the Engel theorem are given.; Comment: 10 pages

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## Parametrized K-Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/04/2013
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#Mathematics - K-Theory and Homology#Mathematics - Commutative Algebra#Mathematics - Algebraic Geometry#Mathematics - Category Theory#18F25, 19D99 (Primary) 13D15, 14F05, 14F20, 18D10, 18D30, 18D99,
18E10, 18F10, 19E08 (Secondary)

In nature, one observes that a K-theory of an object is defined in two steps.
First a "structured" category is associated to the object. Second, a K-theory
machine is applied to the latter category to produce an infinite loop space. We
develop a general framework that deals with the first step of this process. The
K-theory of an object is defined via a category of "locally trivial" objects
with respect to a pretopology. We study conditions ensuring an exact structure
on such categories. We also consider morphisms in K-theory that such contexts
naturally provide. We end by defining various K-theories of schemes and
morphisms between them.; Comment: 31 pages

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## Knot Homology from Refined Chern-Simons Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#High Energy Physics - Theory#Mathematics - Algebraic Geometry#Mathematics - Geometric Topology#Mathematics - Representation Theory

We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold
via the refined topological string and the (2,0) theory on N M5 branes. The
refined Chern-Simons theory is defined on any three-manifold with a semi-free
circle action. We give an explicit solution of the theory, in terms of a
one-parameter refinement of the S and T matrices of Chern-Simons theory,
related to the theory of Macdonald polynomials. The ordinary and refined
Chern-Simons theory are similar in many ways; for example, the Verlinde formula
holds in both. We obtain new topological invariants of Seifert three-manifolds
and torus knots inside them. We conjecture that the knot invariants we compute
are the Poincare polynomials of the sl(n) knot homology theory. The latter
includes the Khovanov-Rozansky knot homology, as a special case. The conjecture
passes a number of nontrivial checks. We show that, for a large number of torus
knots colored with the fundamental representation of SU(N), our knot invariants
agree with the Poincare polynomials of Khovanov-Rozansky homology. As a
byproduct, we show that our theory on S^3 has a large-N dual which is the
refined topological string on X=O(-1)+O(-1)->P^1; this supports the conjecture
by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to sl(n)
knot homology. We also provide a matrix model description of some amplitudes of
the refined Chern-Simons theory on S^3.; Comment: 73 pages...

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## Twisted and untwisted K-theory quantization, and symplectic topology

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/08/2015
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#Mathematics - Symplectic Geometry#Mathematical Physics#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - K-Theory and Homology

A prequantization space $(P,\alpha)$ is a principal $S^1$-bundle with a
connection one-form $\alpha$ over a symplectic manifold $(M,\omega),$ with
curvature given by the symplectic form. In particular $\alpha$ is a contact
form. Using the theory of $Spin ^{c} $ Dirac quantization, we set up natural
K-theory invariants of structure group $Cont _{0} (P, \alpha) $ fibrations of
prequantization spaces. We further construct twisted K-theory invariants of
Hamiltonian $(M, \omega)$ fibrations. As an application we prove that the
natural map $BU(r) \to BU$ of classifying spaces factors as $BU(r) \to B
\mathcal{Q}(r \to BU,$ where $\mathcal{Q}(r)=Cont_0(S^{2r-1},\alpha_{std})$ for
the standard contact form on the odd-dimensional sphere. As a corollary we show
that the natural map $BU(r) \rightarrow B\mathcal{Q}(r)$ induces a surjection
on complex K-theory and on integral cohomology, strengthening a theorem of
Spacil \cite{SpacilThesis,CasalsSpacil} for rational cohomology, and that it
induces an injection on integral homology. Furthermore, we improve a theorem of
Reznikov, showing the injectivity on homotopy groups of the natural map $BU (r)
\to B \text{Ham} (\mathbb{CP} ^{r-1}, \omega )$, in the stable range. Finally,
we produce examples of non-trivial $\mathcal{Q}(r)$ and $\text{Ham}
(\mathbb{CP} ^{r-1}...

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## A1-homotopy invariance of algebraic K-theory with coefficients and Kleinian singularities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Rings and Algebras#Mathematics - Representation Theory#13F35, 14A22, 14H20, 19D25, 19D35, 19E08, 30F50

C. Weibel and Thomason-Trobaugh proved (under some assumptions) that
algebraic K-theory with coefficients is A1-homotopy invariant. In this article
we generalize this result from schemes to the broad setting of dg categories.
Along the way, we extend Bass-Quillen's fundamental theorem as well as
Stienstra's foundational work on module structures over the big Witt ring to
the setting of dg categories. Among other cases, the above A1-homotopy
invariance result can now be applied to sheaves of (not necessarily
commutative) dg algebras over stacks. As an application, we compute the
algebraic K-theory with coefficients of dg cluster categories using solely the
kernel and cokernel of the Coxeter matrix. This leads to a complete computation
of the algebraic K-theory with coefficients of the Kleinian singularities
parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain
some vanishing and divisibility properties of algebraic K-theory (without
coefficients).; Comment: 20 pages. Revised version

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## Algebraic K-theory of strict ring spectra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/03/2014
Português

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#Mathematics - Algebraic Topology#Mathematics - Geometric Topology#Mathematics - K-Theory and Homology#Mathematics - Number Theory#19D10, 55P43, 19F27, 57R50

We view strict ring spectra as generalized rings. The study of their
algebraic K-theory is motivated by its applications to the automorphism groups
of compact manifolds. Partial calculations of algebraic K-theory for the sphere
spectrum are available at regular primes, but we seek more conceptual answers
in terms of localization and descent properties. Calculations for ring spectra
related to topological K-theory suggest the existence of a motivic cohomology
theory for strictly commutative ring spectra, and we present evidence for
arithmetic duality in this theory. To tie motivic cohomology to Galois
cohomology we wish to spectrally realize ramified extensions, which is only
possible after mild forms of localization. One such mild localization is
provided by the theory of logarithmic ring spectra, and we outline recent
developments in this area.; Comment: Contribution to the proceedings of the ICM 2014 in Seoul

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## A Theory of Adjoint Functors--with some Thoughts about their Philosophical Significance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/11/2005
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The question "What is category theory" is approached by focusing on universal
mapping properties and adjoint functors. Category theory organizes mathematics
using morphisms that transmit structure and determination. Structures of
mathematical interest are usually characterized by some universal mapping
property so the general thesis is that category theory is about determination
through universals. In recent decades, the notion of adjoint functors has moved
to center-stage as category theory's primary tool to characterize what is
important and universal in mathematics. Hence our focus here is to present a
theory of adjoint functors, a theory which shows that all adjunctions arise
from the birepresentations of "chimeras" or "heteromorphisms" between the
objects of different categories. Since representations provide universal
mapping properties, this theory places adjoints within the framework of
determination through universals. The conclusion considers some unreasonably
effective analogies between these mathematical concepts and some central
philosophical themes.; Comment: 58 pages. Forthcoming in: What is Category Theory? Giandomenico Sica
ed., Milan: Polimetrica

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## On determinant functors and $K$-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - Number Theory#19A99, 19B99, 18F25, 18G50, 18G55, 18E10, 18E30

In this paper we introduce a new approach to determinant functors which
allows us to extend Deligne's determinant functors for exact categories to
Waldhausen categories, (strongly) triangulated categories, and derivators. We
construct universal determinant functors in all cases by original methods which
are interesting even for the known cases. Moreover, we show that the target of
each universal determinant functor computes the corresponding $K$-theory in
dimensions 0 and 1. As applications, we answer open questions by Maltsiniotis
and Neeman on the $K$-theory of (strongly) triangulated categories and a
question of Grothendieck to Knudsen on determinant functors. We also prove
additivity theorems for low-dimensional $K$-theory and obtain generators and
(some) relations for various $K_{1}$-groups.; Comment: 73 pages. We have deeply revised the paper to make it more
accessible, it contains now explicit examples and computations. We have
removed the part on localization, it was correct but we didn't want to make
the paper longer and we thought this part was the less interesting one.
Nevertheless it will remain here in the arXiv, in version 1. If you need it
in your research, please let us know

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## The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Number Theory

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible
in its ring of regular functions, an ample family of line bundles, and a global
bound on the virtual mod-2 cohomological dimensions of its residue fields. We
prove that the comparison map from the hermitian K-theory of X to the homotopy
fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence
in general, and an integral equivalence when X has no formally real residue
field. We also show that the comparison map between the higher
Grothendieck-Witt (hermitian K-) theory of X and its \'etale version is an
isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum
conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of
complex algebraic varieties and rings of 2-integers in number fields, and hence
values of Dedekind zeta-functions.; Comment: 17 pages, to appear in Adv. Math

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## K-Theory in Quantum Field Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/06/2002
Português

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#Mathematical Physics#High Energy Physics - Theory#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - K-Theory and Homology#81T30, 81T45, 81T50, 19L99

We survey three different ways in which K-theory in all its forms enters
quantum field theory. In Part 1 we give a general argument which relates
topological field theory in codimension two with twisted K-theory, and we
illustrate with some finite models. Part 2 is a review of pfaffians of Dirac
operators, anomalies, and the relationship to differential K-theory. Part 3 is
a geometric exposition of Dirac charge quantization, which in superstring
theories also involves differential K-theory. Parts 2 and 3 are related by the
Green-Schwarz anomaly cancellation mechanism. An appendix, joint with Jerry
Jenquin, treats the partition function of Rarita-Schwinger fields.; Comment: 56 pages, expanded version of lectures at "Current Developments in
Mathematics"

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## Comparison between algebraic and topological K-theory of locally convex algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - K-Theory and Homology#Mathematics - Rings and Algebras#18G, 19K, 46H, 46L80, 46M, 58B34

This paper is concerned with the algebraic K-theory of locally convex
algebras stabilized by operator ideals, and its comparison with topological
K-theory. We show that the obstruction for the comparison map between algebraic
and topological K-theory to be an isomorphism is (absolute) algebraic cyclic
homology and prove the existence of an 6-term exact sequence.
We show that cyclic homology vanishes in the case when J is the ideal of
compact operators and L is a Frechet algebra with bounded app. unit. This
proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki
and announced in his paper "Algebraic K-theory and functional analysis", First
European Congress of Mathematics, Vol. II (Paris, 1992), 485--496, Progr.
Math., 120, Birkh\"auser, Basel, 1994.
We also consider stabilization with respect to a wider class of operator
ideals, called sub-harmonic. We study the algebraic K-theory of the tensor
product of a sub-harmonic ideal with an arbitrary complex algebra and prove
that the obstruction for the periodicity of algebraic K-theory is again cyclic
homology.
The main technical tools we use are the diffeotopy invariance theorem of
Cuntz and the second author (which we generalize), and the excision theorem for
infinitesimal K-theory...

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## Unified Foundations for Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/03/2004
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There are different meanings of foundation of mathematics: philosophical,
logical, and mathematical. Here foundations are considered as a theory that
provides means (concepts, structures, methods etc.) for the development of
whole mathematics. Set theory has been for a long time the most popular
foundation. However, it was not been able to win completely over its rivals:
logic, the theory of algorithms, and theory of categories. Moreover, practical
applications of mathematics and its inner problems caused creation of different
generalization of sets: multisets, fuzzy sets, rough sets etc. Thus, we
encounter a problem: Is it possible to find the most fundamental structure in
mathematics? The situation is similar to the quest of physics for the most
fundamental "brick" of nature and for a grand unified theory of nature. It is
demonstrated that in contrast to physics, which is still in search for a
unified theory, in mathematics such a theory exists. It is the theory of named
sets.

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## Tilting theory for trees via stable homotopy theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Topology#Mathematics - Algebraic Geometry#Mathematics - Category Theory#Mathematics - Representation Theory#55U35 (Primary) 16E35, 18E30, 55U40 (Secondary)

We show that variants of the classical reflection functors from quiver
representation theory exist in any abstract stable homotopy theory, making them
available for example over arbitrary ground rings, for quasi-coherent modules
on schemes, in the differential-graded context, in stable homotopy theory as
well as in the equivariant, motivic, and parametrized variant thereof. As an
application of these equivalences we obtain abstract tilting results for trees
valid in all these situations, hence generalizing a result of Happel.
The main tools introduced for the construction of these reflection functors
are homotopical epimorphisms of small categories and one-point extensions of
small categories, both of which are inspired by similar concepts in homological
algebra.; Comment: To appear in J. Pure Appl. Algebra, it is a sequel to arXiv:1401.6451
and continues the development of abstract tilting theory. Version 2: various
improvements in the presentation. Version 3: a detailed explanation added (in
Construction 9.13 and Lemma 9.15) for the key fact that both the branches of
Figure 2 lead to the same category

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## Representation theory and homological stability

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Representation Theory#Mathematics - Algebraic Topology#Mathematics - Group Theory#Mathematics - Geometric Topology

We introduce the idea of *representation stability* (and several variations)
for a sequence of representations V_n of groups G_n. A central application of
the new viewpoint we introduce here is the importation of representation theory
into the study of homological stability. This makes it possible to extend
classical theorems of homological stability to a much broader variety of
examples. Representation stability also provides a framework in which to find
and to predict patterns, from classical representation theory
(Littlewood--Richardson and Murnaghan rules, stability of Schur functors), to
cohomology of groups (pure braid, Torelli and congruence groups), to Lie
algebras and their homology, to the (equivariant) cohomology of flag and
Schubert varieties, to combinatorics (the (n+1)^(n-1) conjecture). The majority
of this paper is devoted to exposing this phenomenon through examples. In doing
this we obtain applications, theorems and conjectures.
Beyond the discovery of new phenomena, the viewpoint of representation
stability can be useful in solving problems outside the theory. In addition to
the applications given in this paper, it is applied in [CEF] to counting
problems in number theory and finite group theory. Representation stability is
also used in [C] to give broad generalizations and new proofs of classical
homological stability theorems for configuration spaces on oriented manifolds.; Comment: 91 pages. v2: minor revisions throughout. v3: final version...

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## Almost ring theory - sixth release

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Geometry#Mathematics - Commutative Algebra#Mathematics - Number Theory#Mathematics - Rings and Algebras#13D03, 12J20, 14A99, 14F99, 18D10, 14G22

We develop almost ring theory, which is a domain of mathematics somewhere
halfway between ring theory and category theory (whence the difficulty of
finding appropriate MSC-class numbers). We apply this theory to valuation
theory and to p-adic analytic geometry. You should really have a look at the
introductions (each chapter has one).; Comment: This is the sixth - and assuredly final - release of "Almost ring
theory". It is about 230 page long; it is written in AMSLaTeX and uses XYPic
and a few not so standard fonts. Any future corrections (mainly typos, I
expect) will be found on my personal web page:
http://www.math.u-bordeaux.fr/~ramero/

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## Homotopy Type Theory: Univalent Foundations of Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/08/2013
Português

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#Mathematics - Logic#Computer Science - Programming Languages#Mathematics - Algebraic Topology#Mathematics - Category Theory

Homotopy type theory is a new branch of mathematics, based on a recently
discovered connection between homotopy theory and type theory, which brings new
ideas into the very foundation of mathematics. On the one hand, Voevodsky's
subtle and beautiful "univalence axiom" implies that isomorphic structures can
be identified. On the other hand, "higher inductive types" provide direct,
logical descriptions of some of the basic spaces and constructions of homotopy
theory. Both are impossible to capture directly in classical set-theoretic
foundations, but when combined in homotopy type theory, they permit an entirely
new kind of "logic of homotopy types". This suggests a new conception of
foundations of mathematics, with intrinsic homotopical content, an "invariant"
conception of the objects of mathematics -- and convenient machine
implementations, which can serve as a practical aid to the working
mathematician. This book is intended as a first systematic exposition of the
basics of the resulting "Univalent Foundations" program, and a collection of
examples of this new style of reasoning -- but without requiring the reader to
know or learn any formal logic, or to use any computer proof assistant.; Comment: 465 pages. arXiv v1: first-edition-257-g5561b73...

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## Ghosts in modular representation theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Representation Theory#Mathematics - Algebraic Topology#Mathematics - Group Theory#20C20, 20J06, 55P42

A ghost over a finite p-group G is a map between modular representations of G
which is invisible in Tate cohomology. Motivated by the failure of the
generating hypothesis---the statement that ghosts between finite-dimensional
G-representations factor through a projective---we define the ghost number of
kG to be the smallest integer l such that the composition of any l ghosts
between finite-dimensional G-representations factors through a projective. In
this paper we study ghosts and the ghost numbers of p-groups. We begin by
showing that a weaker version of the generating hypothesis, where the target of
the ghost is fixed to be the trivial representation k, holds for all p-groups.
We then compute the ghost numbers of all cyclic p-groups and all abelian
2-groups with C_2 as a summand. We obtain bounds on the ghost numbers for
abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2.
Using these bounds we determine the finite abelian groups which have ghost
number at most 2. Our methods involve techniques from group theory,
representation theory, triangulated category theory, and constructions
motivated from homotopy theory.; Comment: 15 pages, final version, to appear in Advances in Mathematics. v4
only makes changes to arxiv meta-data...

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## Categorical Foundations for K-Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/11/2011
Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Category Theory#19-02, 18F25 (Primary) 18D05, 18D10, 18D30, 18D35, 18F10, 19E08,
19L99, 55N15 (Secondary)

Recall that the definition of the $K$-theory of an object C (e.g., a ring or
a space) has the following pattern. One first associates to the object C a
category A_C that has a suitable structure (exact, Waldhausen, symmetric
monoidal, ...). One then applies to the category A_C a "$K$-theory machine",
which provides an infinite loop space that is the $K$-theory K(C) of the object
C.
We study the first step of this process. What are the kinds of objects to be
studied via $K$-theory? Given these types of objects, what structured
categories should one associate to an object to obtain $K$-theoretic
information about it? And how should the morphisms of these objects interact
with this correspondence?
We propose a unified, conceptual framework for a number of important examples
of objects studied in $K$-theory. The structured categories associated to an
object C are typically categories of modules in a monoidal (op-)fibred
category. The modules considered are "locally trivial" with respect to a given
class of trivial modules and a given Grothendieck topology on the object C's
category.; Comment: 176 + xi pages. This monograph is a revised and augmented version of
my PhD thesis. The official thesis is available at
http://library.epfl.ch/en/theses/?nr=4861

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