Fractional Delayed Control Design for Linear Periodic Systems

In this paper, the fractional Chebyshev collocation (FCC) method is proposed to design fractional delay controllers for linear systems with periodic coefficients. In our previous study, it was shown that this method can be successfully used to stabilize fractional periodic time-delay systems with the delay terms being of integer orders. In the current paper, it is shown that this method can be extended successfully to design fractional delay controllers for fractional periodic systems. For this propose, the solution of linear periodic systems with fractional delay terms is expressed in a Banach space. The short memory principle is used to show that the actual response of the system can be approximated by an approximated monodromy operator. The approximated monodromy operator yields the solution of a fixed length interval by mapping the solution of the previous interval with the same length. Usually obtaining the approximated monodromy operator is complicated or even impossible. The spectral radius of the approximated monodromy matrix indicates the asymptotic stability of the system. The efficiency of the proposed fractional delayed control is illustrated in the case of a second order system with periodic coefficients.

Monodromy Filtrations and the Topology of Tropical Varieties

We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, “multiplicity-free” parameterization of Trop(X) by a topological space ГXand give a geometric interpretation of the cohomology of ГXin terms of the action of a monodromy operator on the cohomology ofX. This gives bounds on the Betti numbers of ГXin terms of the Betti numbers ofXwhich constrain the topology of Trop(X). We also obtain a description of the top power of the monodromy operator acting on middle cohomology ofXin terms of the volume pairing on ГX.

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Opérateur de Lefschetz sur les tours de Drinfeld et Lubin–Tate

We define and study a Lefschetz operator on the equivariant cohomology complex of the Drinfeld and Lubin–Tate towers. For ℓ-adic coefficients we show how this operator induces a geometric realization of the Langlands correspondence composed with the Zelevinski involution for elliptic representations. Combined with our previous study of the monodromy operator, this suggests a possible extension of Arthur’s philosophy for unitary representations occurring in the intersection cohomology of Shimura varieties to the possibly non-unitary representations occurring in the cohomology of Rapoport–Zink spaces. However, our motivation for studying the Lefschetz operator comes from the hope that its geometric nature will enable us to realize the mod-ℓ Langlands correspondence due to Vignéras. We discuss this problem and propose a conjecture.