** Real Analytic Milnor Fibrations and the Strong Łojasiewicz Inequality**

Real analytic versions of Milnor's classic fibration for complex analytic singularities are of great current interest. I defined the strong Łojasiewicz inequality and proved that the Milnor fibration inside a ball exists for functions satisfying this inequality. Much more research is needed to possibly weaken the inequality, or to find practical methods for proving that a given real analytic function satisfies the strong Łojasiewicz inequality.

Required background: basic analysis, linear algebra, basic stratification theory, differential topology, basics of real analytic geometry

** Aligned Hypersurface Singularities**

An aligned hypersurface singularity is an affine complex analytic hypersurface V(f) which possesses a Thom a_{f} stratification in which the closure of every connected component is smooth. Hyperplane arrangements are an important example, and the big question is to what extent one can prove results for arbitrary aligned hypersurfaces which generalize the results for hyperplane arrangements.

Required background: basic stratification theory, differential topology, basic commutative algebra, basics of complex analytic geometry, complex Milnor fibrations

** Perverse Sheaves and the Characteristic Polar Complex**

Given a perverse sheaf on a complex analytic subset of an open subset of affine space, one can can effectively calculate the characteristic polar complex, at each point, by selecting a collection of generic linear functions and iterating the nearby and vanishing cycles. The cohomology of this complex yields the stalk cohomology of the original perverse sheaf; in particular, this implies that Morse inequalities are satisfied between the ranks of the terms in the characteristic polar complex and the stalk cohomology. Understanding the differentials in the characteristic polar complex is a difficult, but important, problem. For instance, why are the differentials all zero for the push-forward of the shifted constant sheaf on the complement of a hyperplane arrangement?

Required background: stratification theory, commutative algebra, complex analytic geometry, basic intersection theory, complex Milnor fibrations, the derived category and perverse sheaves, nearby cycles, vanishing cycles, characteristic cycles

** Vanishing Cycles, the Swing, and the Nexus Diagram**

Lê's swing is a fundamental device used in the study of the topology of the Milnor fiber of an affine hypersurface. For any intersection cohomology complex on any complex analytic space, there is a functorial version of the swing, used in the study of the vanishing cycles. This functorial version of the swing is encoded in the nexus diagram. My fairly recent result with Lê says, in the affine, constant sheaf case, that the nexus diagram has an extra property which yields improved bounds on the Betti numbers of Milnor fiber. The question is whether or not this property exists in the nexus diagram for the vanishing cycles of intersection cohomology complexes.

Required background: stratification theory, commutative algebra, complex analytic geometry, basic intersection theory, complex Milnor fibrations, the derived category and perverse sheaves, nearby cycles, vanishing cycles

** Categorical Distinguished Bases for the Vanishing Cycles**

The notion of a distinguished basis for an isolated affine hypersurface singularity is fundamental in analyzing and calculating the Milnor monodromy. One would hope that there is a categorical characterization of a distinguished basis, which generalizes nicely to the vanishing cycles, regardless of the dimension of the support, i.e., regardless of the dimension of the generalized critical locus. A "correct" general notion of a distinguished basis would have many applications.

Required background: stratification theory, commutative algebra, complex analytic geometry, basic intersection theory, complex Milnor fibrations, the derived category and perverse sheaves, nearby cycles, vanishing cycles

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