1,605 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Algebraic solutions of linear differential equations: an arithmetic approach
Given a linear differential equation with coefficients in , an
important question is to know whether its full space of solutions consists of
algebraic functions, or at least if one of its specific solutions is algebraic.
After presenting motivating examples coming from various branches of
mathematics, we advertise in an elementary way a beautiful local-global
arithmetic approach to these questions, initiated by Grothendieck in the late
sixties. This approach has deep ramifications and leads to the still unsolved
Grothendieck-Katz -curvature conjecture.Comment: 47 page
Equivariant toric geometry and Euler-Maclaurin formulae
We consider equivariant versions of the motivic Chern and Hirzebruch
characteristic classes of a quasi-projective toric variety, and extend many
known results from non-equivariant to the equivariant setting. The
corresponding generalized equivariant Hirzebruch genus of a torus-invariant
Cartier divisor is also calculated. Further global formulae for equivariant
Hirzebruch classes are obtained in the simplicial context by using the Cox
construction and the equivariant Lefschetz-Riemann-Roch theorem. Alternative
proofs of all these results are given via localization at the torus fixed
points in equivariant - and homology theories. In localized equivariant
-theory, we prove a weighted version of a classical formula of Brion for a
full-dimensional lattice polytope. We also generalize to the context of motivic
Chern classes the Molien formula of Brion-Vergne. Similarly, we compute the
localized Hirzebruch class, extending results of Brylinski-Zhang for the
localized Todd class.
We also elaborate on the relation between the equivariant toric geometry via
the equivariant Hirzebruch-Riemann-Roch and Euler-Maclaurin type formulae for
full-dimensional simple lattice polytopes. Our results provide generalizations
to arbitrary coherent sheaf coefficients, and algebraic geometric proofs of
(weighted versions of) the Euler-Maclaurin formulae of Cappell-Shaneson,
Brion-Vergne, Guillemin, etc., via the equivariant Hirzebruch-Riemann-Roch
formalism. Our approach, based on motivic characteristic classes, allows us to
obtain such Euler-Maclaurin formulae also for (the interior of) a face, or for
the polytope with several facets removed. We also prove such results in the
weighted context, and for Minkovski summands of the given full-dimensional
lattice polytope. Some of these results are extended to local Euler-Maclaurin
formulas for the tangent cones at the vertices of the given lattice polytope.Comment: 93 pages, comments are very welcom
Mirror symmetry for Dubrovin-Zhang Frobenius manifolds
Frobenius manifolds were formally defined by Boris Dubrovin in the early 1990s, and serve as a bridge between a priori very different fields of mathematics such as integrable systems theory, enumerative geometry, singularity theory, and mathematical physics. This thesis concerns, in particular, a specific class of Frobenius manifolds constructed on the orbit space of an extension of the affine Weyl group defined by Dubrovin together with Youjin Zhang. Here, we find Landau-Ginzburg superpotentials, or B-model mirrors, for these Frobenius structures by considering the characteristic equation for Lax operators of relativistic Toda chains as proposed by Andrea Brini. As a bonus, the results open up various applications in topology, integrable hierarchies, and Gromov-Witten theory, making interesting research questions in these areas more accessible. Some such applications are considered in this thesis. The form of the determinant of the Saito metric on discriminant strata is investigated, applications to the combinatorics of Lyashko-Looijenga maps are given, and investigations into the integrable systems theoretic and enumerative geometric applications are commenced
Non-perturbative topological string theory on compact Calabi-Yau 3-folds
We obtain analytic and numerical results for the non-perturbative amplitudes
of topological string theory on arbitrary, compact Calabi-Yau manifolds. Our
approach is based on the theory of resurgence and extends previous special
results to the more general case. In particular, we obtain explicit
trans-series solutions of the holomorphic anomaly equations. Our results
predict the all orders, large genus asymptotics of the topological string free
energies, which we test in detail against high genus perturbative series
obtained recently in the compact case. We also provide additional evidence that
the Stokes constants appearing in the resurgent structure are closely related
to integer BPS invariants.Comment: 85 pages, 16 figures, 15 table
Enumerative invariants in self-dual categories. I. Motivic invariants
In this series of papers, we propose a theory of enumerative invariants
counting self-dual objects in self-dual categories. Ordinary enumerative
invariants in abelian categories can be seen as invariants for the structure
group , and our theory is an extension of this to structure
groups and . Examples of our invariants
include invariants counting principal orthogonal or symplectic bundles, and
invariants counting self-dual quiver representations.
In the present paper, we take the motivic approach, and define our invariants
as elements in a ring of motives. One can also extract numerical invariants
from these invariants. We prove wall-crossing formulae relating our invariants
for different stability conditions. We also provide an explicit algorithm
computing invariants for quiver representations, and we present some numerical
results.Comment: 122 page
A Unifying Framework for Differentially Private Sums under Continual Observation
We study the problem of maintaining a differentially private decaying sum
under continual observation. We give a unifying framework and an efficient
algorithm for this problem for \emph{any sufficiently smooth} function. Our
algorithm is the first differentially private algorithm that does not have a
multiplicative error for polynomially-decaying weights. Our algorithm improves
on all prior works on differentially private decaying sums under continual
observation and recovers exactly the additive error for the special case of
continual counting from Henzinger et al. (SODA 2023) as a corollary.
Our algorithm is a variant of the factorization mechanism whose error depends
on the and norm of the underlying matrix. We give a
constructive proof for an almost exact upper bound on the and
norm and an almost tight lower bound on the norm for a
large class of lower-triangular matrices. This is the first non-trivial lower
bound for lower-triangular matrices whose non-zero entries are not all the
same. It includes matrices for all continual decaying sums problems, resulting
in an upper bound on the additive error of any differentially private decaying
sums algorithm under continual observation.
We also explore some implications of our result in discrepancy theory and
operator algebra. Given the importance of the norm in computer
science and the extensive work in mathematics, we believe our result will have
further applications.Comment: 32 page
Discrete signature varieties
Discrete signatures are invariants computed from time series that correspond
to the discretised version of the signature of paths. We study the algebraic
varieties arising from their images, the discrete signature variety. We
introduce them and compute their dimension in many cases. From the analysis of
a particular subclass of these varieties, we also derive a partial solution to
the Chen-Chow theorem in an algebraically closed setting.Comment: 24 pages, one diagra
Map enumeration on surfaces: from bijective techniques to asymptotic counting
L'objectiu principal del treball és l'enumeració de mapes a superfÃcies orientables amb l'ús majoritari de tècniques bijectives. Deduirem la fórmula exacta per mapes planars amb n arestes amb 3 mètodes diferents. Un d'ells explica en detall la relació que hi ha entre mapes i arbres planars, a més d'oferir una base per l'enumeració de mapes en gèneres majors. En aquestes superfÃcies, enumerarem asimptòticament els mapes a partir d'una intel·ligent bijecció entre mapes i mapes amb una sola cara, anomenats g-trees.El objetivo principal del trabajo es la enumeración de mapas en superficies orientables con el uso mayoritario de técnicas biyectivas. Deduciremos la fórmula exacta para mapas planos con n aristas con tres métodos diferentes. Uno de ellos explica en detalle la relación que hay entre mapas y árboles planos, a más de dar una base para la enumeración de mapas en superficies de género mayor. En dichas superficies, enumeraremos asintóticamente los mapas a partir de una biyección entre mapas y mapas con una sola cara, llamados g-trees.The main objective of this work is the enumeration of maps on orientable surfaces with the use of mostly bijective techniques. We are going to deduce the exact formula for planar maps with n edges by three different methods. One of them explains in detail the relation that planar maps and trees have, in addition to giving the foundations for the enumeration of maps in higher genus surfaces. In said surfaces, we are going to asymptotically enumerate maps via a clever bijection between maps and maps with one face, which are called g-trees
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