3,710 research outputs found
Normalisation Control in Deep Inference via Atomic Flows
We introduce `atomic flows': they are graphs obtained from derivations by
tracing atom occurrences and forgetting the logical structure. We study simple
manipulations of atomic flows that correspond to complex reductions on
derivations. This allows us to prove, for propositional logic, a new and very
general normalisation theorem, which contains cut elimination as a special
case. We operate in deep inference, which is more general than other syntactic
paradigms, and where normalisation is more difficult to control. We argue that
atomic flows are a significant technical advance for normalisation theory,
because 1) the technique they support is largely independent of syntax; 2)
indeed, it is largely independent of logical inference rules; 3) they
constitute a powerful geometric formalism, which is more intuitive than syntax
Goodwillie's Calculus of Functors and Higher Topos Theory
We develop an approach to Goodwillie's calculus of functors using the
techniques of higher topos theory. Central to our method is the introduction of
the notion of fiberwise orthogonality, a strengthening of ordinary
orthogonality which allows us to give a number of useful characterizations of
the class of -excisive maps. We use these results to show that the pushout
product of a -equivalence with a -equivalence is a
-equivalence. Then, building on our previous work, we prove a
Blakers-Massey type theorem for the Goodwillie tower. We show how to use the
resulting techniques to rederive some foundational theorems in the subject,
such as delooping of homogeneous functors.Comment: 40 pages, (a slightly modified version of) this paper is accepted for
publication by the Journal of Topolog
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Chern class identities from tadpole matching in type IIB and F-theory
In light of Sen's weak coupling limit of F-theory as a type IIB orientifold,
the compatibility of the tadpole conditions leads to a non-trivial identity
relating the Euler characteristics of an elliptically fibered Calabi-Yau
fourfold and of certain related surfaces. We present the physical argument
leading to the identity, and a mathematical derivation of a Chern class
identity which confirms it, after taking into account singularities of the
relevant loci. This identity of Chern classes holds in arbitrary dimension, and
for varieties that are not necessarily Calabi-Yau. Singularities are essential
in both the physics and the mathematics arguments: the tadpole relation may be
interpreted as an identity involving stringy invariants of a singular
hypersurface, and corrections for the presence of pinch-points. The
mathematical discussion is streamlined by the use of Chern-Schwartz-MacPherson
classes of singular varieties. We also show how the main identity may be
obtained by applying `Verdier specialization' to suitable constructible
functions.Comment: 26 pages, 1 figure, references added, typos correcte
Billey's formula in combinatorics, geometry, and topology
In this expository paper we describe a powerful combinatorial formula and its
implications in geometry, topology, and algebra. This formula first appeared in
the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey
discovered it independently five years later, and it played a prominent role in
her work to evaluate certain polynomials closely related to Schubert
polynomials.
Billey's formula relates many pieces of Schubert calculus: the geometry of
Schubert varieties, the action of the torus on the flag variety, combinatorial
data about permutations, the cohomology of the flag variety and of the Schubert
varieties, and the combinatorics of root systems (generalizing inversions of a
permutation). Combinatorially, Billey's formula describes an invariant of pairs
of elements of a Weyl group. On its face, this formula is a combination of
roots built from subwords of a fixed word. As we will see, it has deeper
geometric and topological meaning as well: (1) It tells us about the tangent
spaces at each permutation flag in each Schubert variety. (2) It tells us about
singular points in Schubert varieties. (3) It tells us about the values of
Kostant polynomials. Billey's formula also reflects an aspect of GKM theory,
which is a way of describing the torus-equivariant cohomology of a variety just
from information about the torus-fixed points in the variety.
This paper will also describe some applications of Billey's formula,
including concrete combinatorial descriptions of Billey's formula in special
cases, and ways to bootstrap Billey's formula to describe the equivariant
cohomology of subvarieties of the flag variety to which GKM theory does not
apply.Comment: 14 pages, presented at the International Summer School and Workshop
on Schubert Calculus in Osaka, Japan, 201
Reversible Multiparty Sessions with Checkpoints
Reversible interactions model different scenarios, like biochemical systems
and human as well as automatic negotiations. We abstract interactions via
multiparty sessions enriched with named checkpoints. Computations can either go
forward or roll back to some checkpoints, where possibly different choices may
be taken. In this way communications can be undone and different conversations
may be tried. Interactions are typed with global types, which control also
rollbacks. Typeability of session participants in agreement with global types
ensures session fidelity and progress of reversible communications.Comment: In Proceedings EXPRESS/SOS 2016, arXiv:1608.0269
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