367,702 research outputs found
Reflection using the derivability conditions
Reflection principles are a way to build non-conservative
true extensions of a theory. However the application of a
reflection principle needs a proof predicate, and the effort
needed to provide this is so great as to be not really practical.
We look at a possible way to avoid this effort by using, instead
of a proof predicate, a predicate defined using only necessary
`modal' properties. Surprisingly, we can produce powerful
non-conservative extensions this way. But a reflection principle
based on such a predicate is essentially weaker, and we also
consider its limitations
Baby Verma modules for rational Cherednik algebras
Symplectic reflection algebras arise in many different mathematical
disciplines: integrable systems, Lie theory, representation theory,
differential operators, symplectic geometry. In this paper, we introduce baby
Verma modules for symplectic reflection algebras of complex reflection groups
at parameter t=0 (the so--called rational Cherednik algebras at parameter t=0)
and present their most basic properties. By analogy with the representation
theory of reductive Lie algebras in positive characteristic, we believe these
modules are fundamental to the understanding of the representation theory and
associated geometry of the rational Cherednik algebras at parameter t=0. As an
example, we use baby Verma modules to answer several problems posed by Etingof
and Ginzburg, and give an elementary proof of a theorem of Finkelberg and
Ginzburg.Comment: Relationship between baby Vermas for symmetric group and the Springer
correspondence discussed; introduction adde
Takeuti's proof theory in the context of the Kyoto School
Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project
"Building" exact confidence nets
Confidence nets, that is, collections of confidence intervals that fill out
the parameter space and whose exact parameter coverage can be computed, are
familiar in nonparametric statistics. Here, the distributional assumptions are
based on invariance under the action of a finite reflection group. Exact
confidence nets are exhibited for a single parameter, based on the root system
of the group. The main result is a formula for the generating function of the
coverage interval probabilities. The proof makes use of the theory of
"buildings" and the Chevalley factorization theorem for the length distribution
on Cayley graphs of finite reflection groups.Comment: 20 pages. To appear in Bernoull
On Representations of Conformal Field Theories and the Construction of Orbifolds
We consider representations of meromorphic bosonic chiral conformal field
theories, and demonstrate that such a representation is completely specified by
a state within the theory. The necessary and sufficient conditions upon this
state are derived, and, because of their form, we show that we may extend the
representation to a representation of a suitable larger conformal field theory.
In particular, we apply this procedure to the lattice (FKS) conformal field
theories, and deduce that Dong's proof of the uniqueness of the twisted
representation for the reflection-twisted projection of the Leech lattice
conformal field theory generalises to an arbitrary even (self-dual) lattice. As
a consequence, we see that the reflection-twisted lattice theories of Dolan et
al are truly self-dual, extending the analogies with the theories of lattices
and codes which were being pursued. Some comments are also made on the general
concept of the definition of an orbifold of a conformal field theory in
relation to this point of view.Comment: 11 pages, LaTeX. Updated references and added preprint n
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