223 research outputs found
A Non-Commutative Extension of MELL
We extend multiplicative exponential linear logic (MELL) by a non-commutative, self-dual logical operator. The extended system, called NEL, is defined in the formalism of the calculus of structures, which is a generalisation of the sequent calculus and provides a more refined analysis of proofs. We should then be able to extend the range of applications of MELL, by modelling a broad notion of sequentiality and providing new properties of proofs. We show some proof theoretical results: decomposition and cut elimination. The new operator represents a significant challenge: to get our results we use here for the first time some novel techniques, which constitute a uniform and modular approach to cut elimination, contrary to what is possible in the sequent calculus
Topological modular forms with level structure
The cohomology theory known as Tmf, for "topological modular forms," is a
universal object mapping out to elliptic cohomology theories, and its
coefficient ring is closely connected to the classical ring of modular forms.
We extend this to a functorial family of objects corresponding to elliptic
curves with level structure and modular forms on them. Along the way, we
produce a natural way to restrict to the cusps, providing multiplicative maps
from Tmf with level structure to forms of K-theory. In particular, this allows
us to construct a connective spectrum tmf_0(3) consistent with properties
suggested by Mahowald and Rezk.
This is accomplished using the machinery of logarithmic structures. We
construct a sheaf of locally even-periodic elliptic cohomology theories,
equipped with highly structured multiplication, on the log-\'etale site of the
moduli of elliptic curves. Evaluating this sheaf on modular curves produces Tmf
with level structure.Comment: 53 pages. Heavily revised, including the addition of a new section on
background tools from homotopy theor
"Del Pezzo surfaces as Springer fibres for exceptional groups"
We show that simultaneous log resolutions of simply elliptic singularities
can be constructed inside suitable stacks of principal bundles over elliptic
curves. In particular, we give a direct geometrical construction of del Pezzo
surfaces from the corresponding exceptional simple algebraic groups.Comment: This is a re-written version of "From exceptional groups to del Pezzo
surfaces and simultaneous log resolutions via principal bundles over elliptic
curves". It contains 2 figures. This version corrects one of the figure
Additive decompositions for rings of modular forms
We study rings of integral modular forms for congruence subgroups as modules
over the ring of integral modular forms for the full modular group. In many
cases these modules are free or decompose at least into well-understood pieces.
We apply this to characterize which rings of modular forms are Cohen--Macaulay
and to prove finite generation results. These theorems are based on
decomposition results about vector bundles on the compactified moduli stack of
elliptic curves.Comment: Rewritten introduction, updated references. This article supersedes
the algebraic part of arXiv:1609.0926
Additive decompositions for rings of modular forms
We study rings of integral modular forms for congruence subgroups as modules
over the ring of integral modular forms for the full modular group. In many
cases these modules are free or decompose at least into well-understood pieces.
We apply this to characterize which rings of modular forms are Cohen--Macaulay
and to prove finite generation results. These theorems are based on
decomposition results about vector bundles on the compactified moduli stack of
elliptic curves.Comment: Complete revision. Comments welcome. arXiv admin note: text overlap
with arXiv:1609.0926
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