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Brian Hepler

Publications and source records attributed to Brian Hepler.

7 recordsLinked to original sources
arXiv2025

Modular versus Hierarchical: A Structural Signature of Topic Popularity in Mathematical Research

Mathematical researchers, especially those in early-career positions, face critical decisions about topic specialization with limited information about the collaborative environments of different research areas. The aim of this paper is to study how the popularity of a research topic is associated with the structure of that topic's collaboration network, as observed by a suite of measures capturing organizational structure at several scales. We apply these measures to 1,938 algorithmically discovered topics across 121,391 papers sourced from arXiv metadata during the period 2020--2025. Our analysis, which controls for the confounding effects of network size, reveals a structural dichotomy--we find that popular topics organize into modular "schools of thought," while niche topics maintain hierarchical core-periphery structures centered around established experts. This divide is not an artifact of scale, but represents a size-independent structural pattern correlated with popularity. We also document a "constraint reversal": after controlling for size, researchers in popular fields face greater structural constraints on collaboration opportunities, contrary to conventional expectations. Our findings suggest that topic selection is an implicit choice between two fundamentally different collaborative environments, each with distinct implications for a researcher's career. To make these structural patterns transparent to the research community, we developed the Math Research Compass (https://mathresearchcompass.com), an interactive platform providing data on topic popularity and collaboration patterns across mathematical topics.

cs.SI↗
arXiv2022

Moderate Growth and Rapid Decay Nearby Cycles via Enhanced Ind-Sheaves

For any holomorphic function $f\colon X\to \mathbb{C}$ on a complex manifold $X$, we define and study moderate growth 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 irregular Riemann--Hilbert correspondence of D'Agnolo--Kashiwara, these objects recover the classical de Rham complexes with moderate growth and rapid decay associated to a holonomic $\mathcal{D}_X$-module. In order to prove the latter, we resolve a recent conjectural duality of Sabbah between these de Rham complexes of holonomic $\mathcal{D}_X$-modules with growth conditions along a normal crossing divisor by making the connection with a classic duality result of Kashiwara--Schapira between certain topological vector spaces. Via a standard d\'evissage argument, we then prove Sabbah's conjecture for arbitrary divisors. As a corollary, we then recover the well-known perfect pairing between the algebraic de Rham cohomology and rapid decay homology associated to integrable connections on smooth varieties due to Bloch--Esnault and Hien.

math.AG↗
arXiv2018

The Weight Filtration on the Constant Sheaf on a Parameterized Space

On an $n$-dimensional locally reduced complex analytic space $X$ on which the shifted constant sheaf $\Q_X^\bullet[n]$ is perverse, it is well-known that, locally, $\Q_X^\bullet[n]$ underlies a mixed Hodge module of weight $\leq n$ on $X$, with weight $n$ graded piece isomorphic to the intersection cohomology complex $\Idot_X$ with constant $\Q$ coefficients. In this paper, we identify the weight $n-1$ graded piece $\Gr_{n-1}^W \Q_X^\bullet[n]$ in the case where $X$ is a "parameterized space", using the comparison complex, a perverse sheaf naturally defined on any space for which the shifted constant sheaf $\Q_X^\bullet[n]$ is perverse. In the case where $X$ is a parameterized surface, we can completely determine the remaining terms in the weight filtration on $\Q_X^\bullet[2]$, where we also show that the weight filtration is a local topological invariant of $X$. These examples arise naturally as affine toric surfaces in $\C^3$, images of finitely-determined maps from $\C^2$ to $\C^3$, as well as in a well-known conjecture of L\^{e} D\~{u}ng Tr\'{a}ng regarding the equisingularity of parameterized surfaces in $\C^3$.

math.AG↗
arXiv2018

Rational Homology Manifolds and Hypersurface Normalizations

We prove a criterion for determining whether the normalization of a complex analytic space on which the 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 "parameterized space", and has several interesting connections with the Milnor monodromy and mixed Hodge Modules.

math.AG↗
arXiv2017

Deformation Formulas for Parameterized Hypersurfaces

We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. 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 comparison, a perverse sheaf naturally associated to any reduced complex analytic space on which the constant sheaf $\Q_X^\bullet[\dim X]$ is perverse. This generalizes the classical formula for the Milnor number of a plane curve in terms of double points as well as Mond's image Milnor number. We also recover results of Gaffney and Bobadilla using this framework. We obtain similar deformation formulas for maps from $\C^2$ to $\C^3$, and provide an ansatz for obtaining deformation formulas for all dimensions within Mather's nice dimensions.

math.AG↗
arXiv2015

Some Special Cases of Bobadilla's Conjecture

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 ambient topological-type of the hypersurface, and the vanishing of the beta invariant is equivalent to the hypotheses of Bobadilla's conjecture.

math.AG↗