Stephen Melczer

CRM-ISM Postdoctoral Fellow
PK-4130, Laboratoire de Combinatoire et d’informatique Mathématique (LaCIM)
Université du Québec à Montréal

About Me

I'm a CRM-ISM postdoc in Montreal, interested in developing effective tools for mathematics and combinatorics using techniques from computational algebraic geometry, complex analysis, and topology. Previously (2017-2019) I was a postdoctoral fellow at the University of Pennsylvania. In 2017 I completed doctorates at the University of Waterloo and the École normale supérieure de Lyon. A short overview of some of my research interests can be found in this article, to appear in Notices of the AMS.

2020 AMS Math Research Community

In 2020 I am co-organizing an MRC, Combinatorial Applications of Computational Geometry and Algebraic Topology, aimed at introducing young researchers (between -2 and +5 years post PhD) in computational algebra, combinatorics, analysis, or algebraic topology to new research directions in enumerative combinatorics. The main activity of this program is a one week (funded) workshop / summer school from May 31 - June 6 in Rhode Island. Details and application procedures can be found here: https://www.ams.org/programs/research-communities/2020MRC-CompGeom

ACSV Project and Upcoming Textbooks

I am a long-term organizer of the analytic combinatorics in several variables project and help maintain its website, available here . Currently I am under contract to publish two textbooks on this topic,

  • An Invitation to Analytic Combinatorics in Several Variables by S. Melczer, expected in print late 2020 / early 2021. (Introductory treatment for graduate students, based off my PhD thesis)
  • Analytic Combinatorics in Several Variables, 2nd Edition by R. Pemantle, M. C. Wilson and S. Melczer, Cambridge Studies in Advanced Mathematics, expected late 2021. (More advanced treatment using Morse theory, multivariate residues, and similar constructions)
I am on the job market in Fall 2019, if you know of a position that might be of interest feel free to let me know.

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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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

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