** Notes on Perverse Sheaves and Vanishing Cycles (updated June 2014)**

My continually updated notes on perverse sheaves and vanishing cycles.
This is sort of a working mathematician's guide to these results. There are very few proofs given.

** Notes on Analytic Intersection Theory**

An appendix, from a book that I am working on, which contains the basics of proper intersections of analytic cycles in a smooth manifold.

** Critical Points of Functions on Singular Spaces**

We investigate different notions of the critical locus of a complex analytic function whose domain is an arbitrarily singular complex analytic space. We generalize the Milnor number to the case where arbitrary perverse sheaves are used as coefficients. We give topological and numerical conditions which imply that Thom's a_f condition holds for a complex analytic function on an arbitary complex analytic space.

** The Sebastiani-Thom Isomorphism in the Derived Category**

The title says it all.

** Numerical Control over Complex Analytic Singularities**

A book describing my generalization of the L\^e cycles and numbers to the case where the underlying space is arbitrarily singular. To appear in
Memoirs of the American Mathematical Society.

** A Little Microlocal Morse Theory**

If a complex analytic function, f, has a stratified isolated critical point, then it is known that the cohomology of the Milnor fibre of
f has a direct sum decomposition in terms of the normal Morse data to the strata. We use microlocal Morse theory to obtain the same
result under the weakened hypothesis that the vanishing cycles along f have isolated support. We also investigate an index-theoretic
proof of this fact.

** Perverse Cohomology and the Vanishing Index Theorem**

The characteristic cycle of a complex of sheaves on a complex analytic space provides weak information about the complex; essentially, it
yields the Euler characteristics of the hypercohomology of normal data to strata. We show how perverse cohomology actually allows one to
extract the individual Betti numbers of the hypercohomology of normal data to strata, not merely the Euler characteristics. We apply this
to the ``calculation'' of the vanishing cycles of a complex, and relate this to the work of Parusi\'nski and Brian\c con, Maisonobe, and
Merle on Thom's $a_f$ condition.

** Invariant Subspaces of the Monodromy**

We show that there are obstructions to the existence of certain types of invariant subspaces of the Milnor monodromy; this
places restrictions on the cohomology of Milnor fibres of non-isolated hypersurface singularities.

** Singularities and Enriched Cycles (updated 5/30/03) **

We introduce graded, enriched characteristic cycles as a method for encoding Morse modules of strata with respect to a
constructible complex of sheaves. Using this new device, we obtain results for arbitrary complex analytic
functions on arbitrarily singular complex analytic spaces.

** The Nexus Diagram and Integral Restrictions on the Monodromy **

Given a complex analytic function with a one-dimensional critical locus at the origin,
we use a new device -- the nexus diagram -- to examine the monodromy action on the integral cohomology of the Milnor fiber.
The nexus diagram relates this monodromy to that of a generic hyperplane slice through the origin, and to that of a generic
hyperplane slice near the origin. We thereby obtain number-theoretic restrictions on the monodromy and on the cohomology of
the original Milnor fiber.

** Intersection Cohomology, Monodromy, and the Milnor Fiber (updated January 14, 2005) **

We say that a complex analytic space, $X$, is an intersection cohomology manifold if and only if
the shifted constant sheaf on $X$ is isomorphic to intersection cohomology; this is quickly seen
to be equivalent to $X$ being a homology manifold. Given an analytic function $f$ on an intersection
cohomology manifold, we describe a simple relation between $V(f)$ being an intersection cohomology manifold
and the vanishing cycle Milnor monodromy of $f$. We then describe how the Sebastiani-Thom isomorphism
allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Finally,
as an easy application, we obtain restrictions on the cohomology of the Milnor fiber of a
hypersurface with a special type of one-dimensional critical locus.

** Semi-simple Carrousels and the Monodromy (updated October 12, 2004) **

Let $\Cal U$ be an open neighborhood of the origin in $\Bbb C^{n+1}$ and let $f:(\Cal U, \bold 0)\rightarrow(\Bbb C, 0)$ be complex analytic. Let $z_0$ be a generic linear form on $\Bbb C^{n+1}$. If the relative polar curve $\Gamma^1_{f, z_0}$ at the origin is irreducible and the intersection number $\big(\Gamma^1_{f, z_0}\cdot V(f))_\bold 0$ is prime, then there are severe restrictions on the possible degree $n$ cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when $\big(\Gamma^1_{f, z_0}\cdot V(f))_\bold 0$ is not prime.

** L\^e Modules and Traces (updated February 13, 2005) **

We show how some of our recent results clarify the relationship between the L\^e numbers and the cohomology of the Milnor fiber of a non-isolated hypersurface singularity. The L\^e numbers are actually the ranks of the free Abelian groups -- the L\^e modules -- appearing in a complex whose cohomology is that of the Milnor fiber. Moreover, the Milnor monodromy acts on the L\^e module complex, and we describe the traces of these monodromy actions in terms of the topology of the critical locus.

** Hypersurface Singularities and the Swing (updated February 8, 2005) **

Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of $f$. This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.

** Notes on Real and Complex Analytic and Semianalytic Singularities (updated May 10, 2006) **

Notes of lectures given with Lê Dũng Tráng at an ICTP summer school in Trieste, Italy in 2005.

** Stratified Morse Theory: Past and Present (updated May 10, 2006) **

A survey article, written in honor of Bob MacPherson's 60th birthday.

** Vanishing Cycles and Thom's $a_f$ Condition (updated March 20, 2007) **

We give a complete description of the relationship between the vanishing cycles of a complex of sheaves along a function $f$ and Thom's $a_f$ condition.

** Enriched Relative Polar Curves and Discriminants (updated May 31, 2007) **

Let $(f, g)$ be a pair of complex analytic functions on a singular analytic space $X$. We give ``the correct'' definition of the relative polar curve of $(f, g)$, and we give a very formal generalization of L\^e's attaching result, which relates the relative polar curve to the relative cohomology of the Milnor fiber modulo a hyperplane slice. We also give the technical arguments which allow one to work with a derived category version of the discriminant and Cerf diagram of a pair of functions. From this, we derive a number of generalizations of results which are classically proved using the discriminant.

** Natural Commuting of Vanishing Cycles and the Verdier Dual (February 2010) **

We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a {\bf natural} isomorphism, even when the coefficients are not in a field.

** A Strong \L ojasiewicz Inequality and Real Analytic Milnor Fibrations (updated January 30, 2009) **

We give a a strong version of a classic inequality of \L ojasiewicz; one which collapses to the usual inequality in the complex analytic case. We show that this inequality for real analytic functions allows us to construct a real Milnor fibration.

** Natural Commuting of Vanishing Cycles and the Verdier Dual (updated August 19, 2009) **

We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a {\bf natural} isomorphism, even when the coefficients are not in a field.

** Calculations with Characteristic Cycles (updated January 22, 2011) **

We discuss and prove a number of results for calculating characteristic cycles, or graded, enriched characteristic cycles. We concentrate particularly on results related to hypersurfaces.

** The Perverse Study of the Milnor Fiber (updated July 29, 2012) **

In this note, we provide a quick introduction to the study of the Milnor fibration via the derived category and perverse sheaves. This is primarily a dictionary for translating from the standard topological setting to the derived category and/or the Abelian category of perverse sheaves.

** Iterated Vanishing Cycles (updated August 24, 2012) **

If $\Adot$ is a bounded, constructible complex of sheaves on a complex analytic space $X$, and $f:X\rightarrow\C$ and $g:X\rightarrow\C$ are complex analytic functions, then the iterated vanishing cycles $\phi_g[-1](\phi_f[-1]\Adot)$ are important for a number of reasons. We give a formula for the stalk cohomology $H^*(\phi_g[-1]\phi_f[-1]\Adot)_x$ in terms of relative polar curves, algebra, and the normal Morse data and micro-support of $\Adot$.

** Milnor Fibers and Links of Local Complete Intersections (updated October 8, 2014) **

There are essentially no previously-known results which show how Milnor fibers, real links, and complex links ``detect'' the dimension of the singular locus of a local complete intersection. In this paper, we show how a good understanding of the derived category and the perverse $t$-structure quickly yields such results for local complete intersections with singularities of arbitrary dimension.

** Non-isolated Hypersurface Singularities and L\^e Cycles (updated October 8, 2014) **

In this series of lectures, I will discuss results for complex hypersurfaces with non-isolated singularities.
In Lecture 1, I will review basic definitions and results on complex hypersurfaces, and then present classical material on the Milnor fiber and fibration. In Lecture 2, I will present basic results from Morse theory, and use them to prove some results about complex hypersurfaces, including a proof of L\^e's attaching result for Milnor fibers of non-isolated hypersurface singularities. This will include defining the relative polar curve. Lecture 3 will begin with a discussion of intersection cycles for proper intersections inside a complex manifold, and then move on to definitions and basic results on L\^e cycles and L\^e numbers of non-isolated hypersurface singularities. Lecture 4 will explain the topological importance of L\^e cycles and numbers, and then I will explain, informally, the relationship between the L\^e cycles and the complex of sheaves of vanishing cycles.

** A New Conjecture, a New Invariant, and a New Non-splitting Result (updated May 2015) **

We prove a new non-splitting result for the cohomology of the Milnor fiber, reminiscent of the classical result proved independently by Lazzeri, Gabrielov, and L\^e in 1973-74.
We do this while exploring a conjecture of Bobadilla about a stronger version of our non-splitting result. To explore this conjecture, we define a new numerical invariant for hypersurfaces with 1-dimensional critical loci: the beta invariant. The beta invariant is an invariant of the ambient topological-type of the hypersurface, is non-negative, and is algebraically calculable. Results about the beta invariant remove the topology from Bobadilla's conjecture and turn it into a purely algebraic question.

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