Stephen Melczer

Postdoctoral Fellow, University of Pennsylvania
4C3 David Rittenhouse Lab
209 South 33rd Street
Philadelphia, PA 19104, USA

About Me

I'm a Postdoctoral Fellow at the University of Pennsylvania in Philadelphia, interested in answering computability and complexity questions in enumerative combinatorics using tools from symbolic computation, complex analysis, topology, and other areas of algebra and geometry.

In 2017 I completed dual (cotutelle) doctorates at the University of Waterloo and the École normale supérieure de Lyon, supervised by Bruno Salvy and George Labahn.

Click here to view my PhD Thesis

My thesis focuses on effective methods for determining asymptotics of sequences which arise in several areas of mathematics, computer science, and related domains like physics. The thesis also gives a pedagogical introduction to the (relatively new) topic of Analytic Combinatorics in Several Variables, with a focus on explicit computations. Enumerating lattice paths restricted to certain regions gives a large collection of problems to which the methods of ACSV can be applied, and this relationship is explored in depth.

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Recent Projects

  • Partition Asymptotics and Unimodality of q-Objects

    We give probabilistic methods for determining asymptotics of partitions in a rectangle and q-binomial coefficients, leading to an asymptotic refinement of Sylvester's unimodality theorem. This gives asymptotics of Kronecker coefficients with increasing parts. With Robin Pemantle and Greta Panova.

  • Morse-Theoretic Methods for Singularity Analysis

    Morse Theory provides powerful tools for contour deformations in the singularity analysis of multivariate generating functions. Existence of non-proper height functions necessitates the use of additional algebro-geometric tools. With Yuliy Baryshnikov and Robin Pemantle.

  • Algorithms for Multivariate Generating Functions

    The methods of analytic combinatorics in several variables are now sufficiently well developed to be automated in computer algebra systems. We give the first fully rigorous algorithms and complexity analyses of these methods in arbitrary numbers of variables. With Bruno Salvy.

  • Non-Negativity of Multivariate Series

    Shockingly, it is unknown whether there is an algorithm to determine when the Taylor series of a rational function has a zero coefficient. Easy cases can be decided through asymptotics, and new advances in multivariate asymptotics give an approach to resolving certain multivariate conjectures. With Yuliy Baryshnikov, Robin Pemantle, and Armin Straub.

  • Lattice Walks in High-Dimensional Cones

    Combinatorial arguments often count walks restricted to cones using sub-series expansions of multivariate rational functions. We give asymptotics for walks restricted to orthants whose step sets have different symmetry properties, connecting combinatorial and analytic behaviour, and proving several previous conjectures in two dimensions. With Mark Wilson.

  • Fast Calculation of Recurrent Sequences

    The Taylor coefficients of a rational function G(z)/H(z) satisfy a linear recurrence with constant coefficients. We improve the complexity of calculating the Nth term of such a sequence by showing only the degree of the square-free part of H(z) enters the complexity. This has application to computing terms in multivariate series. With K. Hyun, É. Schost, and C. St-Pierre.

Coauthors

Yuliy Baryshnikov, Alin Bostan, Mireille Bousquet-Mélou, Sophie Burrill, Erin Compaan, Julien Courtiel, Éric Fusy, Kevin Hyun, Veronika Irvine, Manuel Kauers, Marni Mishna, Greta Panova, Robin Pemantle, Kilian Raschel, Frank Ruskey, Bruno Salvy, Éric Schost, Catherine St-Pierre, Armin Straub, Mark Wilson