Dynamics, topology and bifurcations of McMullen maps

We give a survey of many diff erent topological structure arise in the dynamical and parameter planes of McMullen maps.     

Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps

For metric spaces (X, d _x) and (Y, d _y) we consider the Hausdorff metric topology on the set (CL(X × Y), ρ) of closed subsets of the product metrized by the product (box) metric ρ and consider the proximal topology defined on CL(X × Y). These topologies are inherited by the set G(X, Y) of closed-graph multifunctions from X to Y, if we identify each multifunction with its graph. Finally, we consider the topology of uniform convergence τ _uc on the set F(X, 2^Y) of all closed-valued multifunctions, i.e. functions from X to the set (CL(Y),) of closed subsets of Y metrized by the Hausdorff metric . We show the relationship between these topologies on the space G(X, Y) and also on the subspaces of minimal USCO maps and locally bounded densely continuous forms. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank.ubica Holá for suggestions and comments.     

Quasicontinuous functions,minimal usco maps and topology of pointwise convergence

In [HOLá, Ľ.—HOLY, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces, Rocky Mountain J. Math.] a complete answer is given, for a Baire space X, to the question of when the pointwise limit of a sequence of real-valued quasicontinuous functions defined on X is quasicontinuous. In [HOLá, Ľ.—HOLY, D.: Minimal USCO maps, densely continuous forms and upper semicontinuous functions, Rocky Mountain J. Math. 39 (2009), 545–562], a characterization of minimal USCO maps by quasicontinuous and subcontinuous selections is proved. Continuing these results, we study closed and compact subsets of the space of quasicontinuous functions and minimal USCO maps equipped with the topology of pointwise convergence. We also study conditions under which the closure of the graph of a set-valued mapping which is the pointwise limit of a net of set-valued mappings, is a minimal USCO map.     

PBW Degeneration: Algebra,Geometry, and Combinatorics

We review the recent progress in the theory of Poincaré–Birkhoff–Witt degenerations of irreducible representations of simple Lie algebras. We describe algebraic, geometric, and combinatorial aspects of the theory.     

Extended topology: Filters and convergence I

Conclusion The above-mentioned results show some of the advantages of using extended filters. It has already been shown that the family of all proper extended filters forms a complete lattice. Many of the results obtained by using Cartan's filters can be obtained by using proper extended filters; it is not necessary to impose the condition that they should be closed with respect to finite intersections. And for most purposes it is sufficient if extended filters have the pair-wise intersection property; for instance limits are unique under this condition.In the next paper I propose to take up the question of convergence. Convergence will be dealt with in a setting more general than a topology of which convergence is extended topologies and topologies will be special cases. Wherever convenient relation theory will be used in dealing with extended topology which generalizes extended topology defined in terms of expansive functions.This research was supported by the National Science Foundation Research Participation Program in Mathematics at the University of Oklahoma, Norman, Oklahoma.     

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

The Hard Lefschetz Theorem and the topology of semismall maps

A line bundle on a complex projective manifold is said to be lef if one of its powers is globally generated and defines a semismall map in the sense of Goresky-MacPherson. As in the case of ample bundles the first Chern class of lef line bundles satisfies the Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations. As a consequence, we prove a generalization of the Grauert contractibility criterion: the Hodge Index Theorem for semismall maps, Theorem 2.4.1. For these maps the Decomposition Theorem of Beilinson, Bernstein and Deligne is equivalent to the non-degeneracy of certain intersection forms associated with a stratification. This observation, joint with the Hodge Index Theorem for semismall maps gives a new proof of the Decomposition Theorem for the direct image of the constant sheaf. A new feature uncovered by our proof is that the intersection forms involved are definite.     

Combinatorics,graph theory,and computing

Journal of Algebraic Combinatorics -     

Compactness of the product topology and the solvability of infinite systems of infinite linear equations

If every finite subsystem of an infinite system of linear equations (say, over the field of real numbers) each with finitely many unknowns has a solution then the entire system has a solution. The situation is not so if the equations contain infinitely many unknowns. In this case, as shown below, the solvability of every finite subsystem implies the solva. bility of the entire system provided finite subsystems have solution with common upper and lower bounds and the coefficients of ever equation satisfy some boundedness or convergence conditions. The passage from the solvability of finite subsystem to the solvability of the entire system is achieved based on Tychnoff’s theorem stating that any product of compact topological spaces is compact in their product topology.     

4-Manifold topology I: Subexponential groups

The technical lemma underlying the 5-dimensional topologicals-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating ₁-null immersions of disks. These conjectures are theorems precisely for those fundamental groups (good groups) where the ₁-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations: (1) extension and (2) direct limit. The finitely generated groups in this class are amenable and no amenable group is known to lie outside this class.Oblatum 20-II-1995 & 26-V-1995The first author is supported by the IHES, the Guggenheim foundation and the NSF.The second author is supported by the IHES and the Humboldt foundation.