# Brian HeplerUniversity of Wisconsin–Madison | UW · Department of Mathematics

Brian Hepler

Doctor of Philosophy

I am on the job market for a second postdoctoral position or equivalent, as of summer 2022. I'd like to work with you!

## About

10

Publications

291

Reads

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15

Citations

Introduction

Complex analytic singularities, perverse sheaves, vanishing cycles, cohomology of the Milnor fibration for non-isolated singularities, and more recently enhanced perverse ind-sheaves à la the Riemann-Hilbert correspondence for holonomic D-modules.

Additional affiliations

Education

September 2014 - April 2019

September 2012 - April 2014

September 2008 - May 2012

## Publications

Publications (10)

We prove two special cases of a conjecture of J. Fern\'andez de Bobadilla for
hypersurfaces with $1$-dimensional critical loci.
We do this via a new numerical invariant for such hypersurfaces, called the
beta invariant, first defined and explored by the second author in 2014. The
beta invariant is an algebraically calculable invariant of the local...

We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set inuences t...

We investigate one-parameter deformations of functions on affine space which define parametrizable hyper surfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the L\^e numbers of the special fiber in terms of the L\^e numbers of the generic fiber and the characteristic polar multiplicities of the...

For any holomorphic function $f\colon X\to\mathbb{C}$ on a complex manifold $X$, we define and study moderate and rapid decay objects associated to an enhanced ind-sheaf on $X$. These will be sheaves on the real oriented blow-up space of $X$ along $f$. We show that in the context of the Riemann-Hilbert functor due to D'Agnolo-Kashiwara, these objec...

On an $n$-dimensional locally reduced hypersurface $V(f)$ inside a complex manifold, the shifted constant sheaf $\mathbb{Q}_{V(f)}^\bullet[n]$ is perverse, and it is well-known that, locally, $\mathbb{Q}_{V(f)}^\bullet[n]$ underlies a mixed Hodge module of weight $\leq n$ on $V(f)$, with weight $n$ graded piece isomorphic to the intersection cohomo...

We prove a criterion for determining whether the normalization of a complex analytic space on which the shifted constant sheaf is perverse is a rational homology manifold, using a perverse sheaf known as the multiple-point complex. This perverse sheaf is naturally associated to any space on which the shifted constant sheaf is perverse, and has seve...

We prove a criterion for determining whether the normalization of a local complete intersection is a rational homology manifold, using a perverse sheaf known as the multiple-point complex. This perverse sheaf is naturally associated to any "parameterized space", and has several interesting connections with the Milnor monodromy and mixed Hodge Modul...

We investigate one-parameter deformations of functions on affine space whose defining hypersurfaces can be parameterized by a finite morphism that is generically one-to-one. Such hypersurfaces must necessarily have critical loci of codimension one. With the standard assumption of isolated polar activity at the origin, we are able to completely expr...

We investigate one-parameter deformations of functions on affine space whose defining hypersurfaces can be parameterized by a finite morphism that is generically one-to-one. Such hypersurfaces must necessarily have critical loci of codimension one. With the standard assumption of isolated polar activity at the origin, we are able to completely expr...

We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf, called the multiple-point complex of the parameterization, which is in a sense the most natural choice of coefficients for Milnor fiber (...

## Projects

Project (1)