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On the geometry of regular maps from a quasi-projective surface to a curve
We explore consequences of the triviality of the monodromy group, using the condition of purity of the mixed Hodge structure on the cohomology of the surface X.
Infeasibility certificates for linear matrix inequalities
Farkas' lemma is a fundamental result from linear programming providing linear certi cates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry. More precisely, we show that a linear matrix inequality L(x)⪰0 is infeasible if and only if −1 lies in the quadratic module associated to L. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certi cate, we use a more involved algebraic certificate motivated by the real radical and Prestel's theory of semiorderings. Completely different methods, namely complete positivity from operator algebras, are employed to consider linear matrix inequality domination.
G-complete reducibility in non-connected groups
In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G0 is G0-cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.
On the autonomous metric on the groups of Hamiltonian diffeomorphisms of closed hyperbolic surfaces
Let g be a closed hyperbolic surface of genus g and let Ham g be the group of Hamiltonian diffeomorphisms of g. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that Ham g is unbounded with respect to this metric
On the Markov inequality in the L2-norm with the Gegenbauer weight
Let wλ(t):=(1−t2)λ−1/2, where λ>−12, be the Gegenbauer weight function, let ∥⋅∥wλ be the associated L2-norm, |f∥wλ={∫1−1|f(x)|2wλ(x)dx}1/2, and denote by Pn the space of algebraic polynomials of degree ≤n. We study the best constant cn(λ) in the Markov inequality in this norm ∥p′n∥wλ≤cn(λ)∥pn∥wλ,pn∈Pn, namely the constant cn(λ):=suppn∈Pn∥p′n∥wλ∥pn∥wλ. We derive explicit lower and upper bounds for the Markov constant cn(λ), which are valid for all n and λ.
Mehler-Heine asymptotics of a class of generalized hypergeometric polynomials
We obtain a Mehler–Heine type formula for a class of generalized hypergeometric polynomials. This type of formula describes the asymptotics of polynomials scale conveniently. As a consequence of this formula, we obtain the asymptotic behavior of the corresponding zeros. We illustrate these results with numerical experiments and some figures.
An extension problem and trace Hardy inequality for the sublaplacian on H-type groups
In this paper we study the extension problem for the sublaplacian on a H-type group and use the solutions to prove trace Hardy and Hardy inequalities for fractional powers of the sublaplacian.
Noncompact harmonic manifolds
The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szab´o [Sz] for harmonic manifolds with compact universal cover. E. Damek and F. Ricci [DR] provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient. The classification of all noncompact harmonic spaces is still a very difficult open problem. In this paper we provide a survey on recent results on noncompact simply connected harmonic manifolds, and we also prove many new results, both for general noncompact harmonic manifolds and for noncompact harmonic manifolds with purely exponential volume growth.
On the derived category of Grassmannians in arbitrary characteristic
In this paper we consider Grassmannians in arbitrary characteristic. Generalizing Kapranov's well-known characteristic-zero results we construct dual exceptional collections on them (which are however not strong) as well as a tilting bundle. We show that this tilting bundle has a quasi-hereditary endomorphism ring and we identify the standard, costandard, projective and simple modules of the latter.