To go to my new site please visit
http://melczer.ca
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.
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
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,
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 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.
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.
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.
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.
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.
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