63 research outputs found
Sard Property for the endpoint map on some Carnot groups
In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open
problems is whether the conclusions of Sard's theorem holds for the endpoint
map, a canonical map from an infinite-dimensional path space to the underlying
finite-dimensional manifold. The set of critical values for the endpoint map is
also known as abnormal set, being the set of endpoints of abnormal extremals
leaving the base point. We prove that a strong version of Sard's property holds
for all step-2 Carnot groups and several other classes of Lie groups endowed
with left-invariant distributions. Namely, we prove that the abnormal set lies
in a proper analytic subvariety. In doing so we examine several
characterizations of the abnormal set in the case of Lie groups.Comment: 39 page
Feynman integral relations from parametric annihilators
We study shift relations between Feynman integrals via the Mellin transform
through parametric annihilation operators. These contain the momentum space IBP
relations, which are well-known in the physics literature. Applying a result of
Loeser and Sabbah, we conclude that the number of master integrals is computed
by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate
techniques to compute this Euler characteristic in various examples and compare
it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional
remark
Chevalley's restriction theorem for reductive symmetric superpairs
Let (g,k) be a reductive symmetric superpair of even type, i.e. so that there
exists an even Cartan subspace a in p. The restriction map S(p^*)^k->S(a^*)^W
where W=W(g_0:a) is the Weyl group, is injective. We determine its image
explicitly.
In particular, our theorem applies to the case of a symmetric superpair of
group type, i.e. (k+k,k) with the flip involution where k is a classical Lie
superalgebra with a non-degenerate invariant even form (equivalently, a
finite-dimensional contragredient Lie superalgebra). Thus, we obtain a new
proof of the generalisation of Chevalley's restriction theorem due to Sergeev
and Kac, Gorelik.
For general symmetric superpairs, the invariants exhibit a new and surprising
behaviour. We illustrate this phenomenon by a detailed discussion in the
example g=C(q+1)=osp(2|2q,C), endowed with a special involution. In this case,
the invariant algebra defines a singular algebraic curve.Comment: 35 pages; v4: revised submission to J.Alg., accepted for publication
under the proviso of revisio
On the uniqueness of higher-spin symmetries in AdS and CFT
We study the uniqueness of higher-spin algebras which are at the core of
higher-spin theories in AdS and of CFTs with exact higher-spin symmetry, i.e.
conserved tensors of rank greater than two. The Jacobi identity for the gauge
algebra is the simplest consistency test that appears at the quartic order for
a gauge theory. Similarly, the algebra of charges in a CFT must also obey the
Jacobi identity. These algebras are essentially the same. Solving the Jacobi
identity under some simplifying assumptions spelled out, we obtain that the
Eastwood-Vasiliev algebra is the unique solution for d=4 and d>6. In 5d there
is a one-parameter family of algebras that was known before. In particular, we
show that the introduction of a single higher-spin gauge field/current
automatically requires the infinite tower of higher-spin gauge fields/currents.
The result implies that from all the admissible non-Abelian cubic vertices in
AdS(d), that have been recently classified for totally symmetric higher-spin
gauge fields, only one vertex can pass the Jacobi consistency test. This cubic
vertex is associated with a gauge deformation that is the germ of the
Eastwood-Vasiliev's higher-spin algebra.Comment: 37 pages; refs added, proof of uniquiness was improve
Topics in arithmetic combinatorics
E-thesis pagination differs from approved hard bound copy, Cambridge University Library classmark: PhD.30726This thesis is chiefly concerned with a classical conjecture of Littlewood's regarding the L^1-norm of the Fourier transform, and the closely related idempotent theorem. The vast majority of the results regarding these problems are, in some sense, qualitative or at the very least infinitary and it has become increasingly apparent that a quantitative state of affairs is desirable.
Broadly speaking, the first part of the thesis develops three new tools for tackling the problems above: We prove a new structural theorem for the spectrum of functions in A(G); we extend the notion of local Fourier analysis, pioneered by Bourgain, to a much more general structure, and localize Chang's classic structure theorem as well as our own spectral structure theorem; and we refine some aspects of Freiman's celebrated theorem regarding the structure of sets with small doubling. These tools lead to improvements in a number of existing additive results which we indicate, but for us the main purpose is in application to the analytic problems mentioned above.
The second part of the thesis discusses a natural version of Littlewood's problem for finite abelian groups. Here the situation varies wildly with the underlying group and we pay special attention first to the finite field case (where we use Chang's Theorem) and then to the case of residues modulo a prime where we require our new local structure theorem for A(G). We complete the consideration of Littlewood's problem for finite abelian groups by using the local version of Chang's Theorem we have developed. Finally we deploy the Freiman tools along with the extended Fourier analytic techniques to yield a fully quantitative version of the idempotent theorem
A PDE Approach to the Combinatorics of the Full Map Enumeration Problem: Exact Solutions and their Universal Character
Maps are polygonal cellular networks on Riemann surfaces. This paper
completes a program of constructing closed form general representations for the
enumerative generating functions associated to maps of fixed but arbitrary
genus. These closed form expressions have a universal character in the sense
that they are independent of the explicit valence distribution of the tiling
polygons. Nevertheless the valence distributions may be recovered from the
closed form generating functions by a remarkable {\it unwinding identity} in
terms of the Appell polynomials generated by Bessel functions. Our treatment,
based on random matrix theory and Riemann-Hilbert problems for orthogonal
polynomials reveals the generating functions to be solutions of nonlinear
conservation laws and their prolongations. This characterization enables one to
gain insights that go beyond more traditional methods that are purely
combinatorial. Universality results are connected to stability results for
characteristic singularities of conservation laws that were studied by
Caflisch, Ercolani, Hou and Landis as well as directly related to universality
results for random matrix spectra as described by Deift, Kriecherbauer,
McLaughlin, Venakides and Zhou
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