About

I am an Assistant Professor in the Department of Combinatorics and Optimization at the University of Waterloo. My work sits between algebraic combinatorics, optimization, and theoretical computer science, especially around log-concavity, polynomial methods, entropy, counting, and sampling.

Before coming to Waterloo, I was a Dirichlet Postdoctoral Fellow at TU Berlin, where I worked in Peter Bürgisser's group. Earlier postdoctoral and fellowship stops included Institut Mittag-Leffler, KTH Stockholm, and the Simons Institute; in Stockholm I worked with Petter Brändén.

I received my PhD in mathematics from UC Berkeley, advised by Olga Holtz. Before that, I earned an MS in mathematics and BS degrees in computer engineering and applied mathematics from Texas A&M University, and worked as a software developer for the Teacher Retirement System of Texas.

Teaching

I teach undergraduate courses in combinatorics, optimization, and coding theory, as well as graduate topics courses on Lorentzian polynomials, polynomial capacity, and related methods. Public course pages are linked where available.

Graduate and topics courses

Undergraduate courses

Research

I develop methods in the polynomial paradigm/geometry of polynomials which extract analytic inequalities from combinatorial structure, and I then use those inequalities for counting, optimization, approximation, and sampling.

Publications by research area

Use the cards below to jump to a research area; each paper has a short blurb and compact link badges.

Lorentzian polynomials, log-concavity, and high-dimensional expanders

Papers where the main object is Lorentzian/C-Lorentzian structure, log-concavity, or expansion phenomena.

  1. Lorentzian Polynomials on Cones

    Petter Brändén and Jonathan Leake

    Forum of Math, Sigma (2026) arXiv (2023) arXiv short note (2021) talk + slides (2023)

    Develops a cone-based theory of Lorentzian polynomials and uses it to prove Hodge-type inequalities, including Alexandrov–Fenchel inequalities and the Heron–Rota–Welsh log-concavity theorem for matroids. The 2021 arXiv note is a shorter, direct entry point focused on the Heron–Rota–Welsh theorem; the 2023 arXiv/journal version contains the fuller cone-based theory.

    Lorentzian polynomials Matroids Hodge-type inequalities Alexandrov–Fenchel

    For a friendly undergraduate/beginning-graduate entry point to the Heron–Rota–Welsh application, Eric Katz’s “Volume Polynomials and Log-concavity of the Characteristic Polynomials of Matroids” explains the volume-polynomial perspective and how it leads to log-concavity for characteristic polynomials of matroids. It builds the needed background in chromatic polynomials, matroids, and linear algebra, then walks through the Lorentzian-polynomial proof in a more tutorial style.

  2. Optimal Trickle-Down Theorems for Path Complexes via C-Lorentzian Polynomials with Applications to Sampling and Log-Concave Sequences

    Jonathan Leake, Kasper Lindberg, and Shayan Oveis Gharan

    FOCS (2025) arXiv (2025) talk (2025)

    Builds C-Lorentzian machinery for path complexes, turning codimension-two local expansion into fast mixing and log-concavity results, with applications to linear extensions, modular lattices, sparse complexes, and matroid-type log-concavity.

    High-dimensional expanders Sampling Log-concavity

  3. Trickle-down Theorems via C-Lorentzian Polynomials II: Pairwise Spectral Influence and Improved Dobrushin's Condition

    Jonathan Leake and Shayan Oveis Gharan

    arXiv (2025)

    Introduces pairwise spectral influence as a refined local-expansion criterion and uses it to prove rapid mixing for Glauber dynamics, including near-threshold results for multi-state hardcore models.

    Glauber dynamics Spectral influence Dobrushin condition

  4. Lower Bounds for Contingency Tables via Lorentzian Polynomials

    Petter Brändén, Jonathan Leake, and Igor Pak

    Israel Journal of Mathematics (2023) arXiv (2020) talk (2021) slides (2021)

    Uses Lorentzian polynomial techniques to improve and extend lower bounds for numbers of contingency tables, with applications to volumes of transportation and flow polytopes.

    Contingency tables Transportation polytopes Lorentzian methods

Capacity, counting, and combinatorial optimization

Capacity inequalities and related methods for enumeration, polytopes, matchings, and TSP-style bounds.

  1. Capacity Bounds on Integral Flows and the Kostant Partition Function

    Jonathan Leake and Alejandro H. Morales

    Advances in Applied Mathematics (2026) arXiv (2024)

    Studies asymptotics of the type A Kostant partition function, improving lower bounds and settling conjectures of O’Neill and Yip via Lorentzian/capacity methods and flow-polytope formulas.

    Kostant partition function Integer flows Flow polytopes

  2. From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP

    Leonid Gurvits, Nathan Klein, and Jonathan Leake

    ICALP (2024) arXiv (2023) talk (2024)

    Strengthens productization-style capacity bounds for strongly Rayleigh distributions, identifies a forest structure in extremal cases, and slightly improves and simplifies the analysis of metric TSP approximation.

    Strongly Rayleigh Productization Metric TSP

  3. Capacity Lower Bounds via Productization

    Leonid Gurvits and Jonathan Leake

    STOC (2021) arXiv (2020) talk (2020) talk (2021)

    Introduces the productization technique for real stable polynomials and proves sharp capacity lower bounds from gradient data, with consequences for scaling, permanent approximation, and TSP-related inequalities.

    Polynomial capacity Real stability Productization

  4. Counting Matchings via Capacity Preserving Operators

    Leonid Gurvits and Jonathan Leake

    Combinatorics, Probability and Computing (2021) arXiv (2018) talk (2019)

    Unifies the theory of capacity-preserving operators for real stable polynomials and applies it to matching-count lower bounds, including a new route to Csikvári’s result on Friedland’s lower matching conjecture.

    Matchings Capacity preservers Real stability

  5. Deterministic Approximation Algorithms for Volumes of Spectrahedra

    Mahmut Levent Doğan, Jonathan Leake, and Mohan Ravichandran

    arXiv (2022)

    Adapts maximum-entropy volume methods from polytopes to spectrahedra, yielding deterministic approximation algorithms and asymptotic formulas for PSD-cone slices, including quantum-state examples.

    Spectrahedra Volume approximation Maximum entropy

Maximum entropy, sampling, and continuous symmetry

Algorithmic work around entropy-maximizing distributions, Lie-group orbits, HCIZ densities, and spectrahedra.

  1. Sampling Matrices from Harish-Chandra–Itzykson–Zuber Densities with Applications to Quantum Inference and Differential Privacy

    Jonathan Leake, Colin S. McSwiggen, and Nisheeth K. Vishnoi

    STOC (2021) Random Structures & Algorithms (2025) arXiv (2020) talk (2021) code

    Gives efficient algorithms for approximate sampling from HCIZ densities, with applications to matrix Langevin distributions, quantum maximum-entropy ensembles, and differentially private low-rank approximation.

    HCIZ distributions Quantum inference Differential privacy

  2. On the Computability of Continuous Maximum Entropy Distributions with Applications

    Jonathan Leake and Nisheeth K. Vishnoi

    STOC (2020) SIAM Journal on Computing (2022) arXiv (2020) talk (2020)

    Develops algorithms for continuous maximum-entropy distributions with prescribed marginals, with applications to Goemans–Williamson rounding, entropic barriers, and barycentric quantum entropy.

    Maximum entropy Convex optimization Quantum entropy

  3. On the Computability of Continuous Maximum Entropy Distributions: Adjoint Orbits of Lie Groups

    Jonathan Leake and Nisheeth K. Vishnoi

    arXiv (2020)

    Extends the maximum-entropy algorithmic framework to adjoint orbits of compact Lie groups, using tools such as the Harish-Chandra integral formula and Kostant convexity theorem.

    Lie groups Adjoint orbits Maximum entropy

  4. Optimization and Sampling Under Continuous Symmetry: Examples and Lie Theory

    Jonathan Leake and Nisheeth K. Vishnoi

    arXiv (2021) talk (2021)

    Explains how continuous symmetries and Lie theory reduce certain nonconvex optimization and sampling problems to convex polytopes and Weyl-group formulas.

    Lie theory Symmetry Optimization

Real-rootedness, interlacing, and finite free convolution

Polynomial-root techniques, interlacing, matching and independence polynomials, and finite free probability.

  1. Compatibility of Real-Rooted Polynomials with Mixed Signs

    Jonathan Leake and Nick Ryder

    arXiv (2024)

    Characterizes compatible families of real-rooted polynomials when leading coefficients may have mixed signs, generalizing the Chudnovsky–Seymour same-sign criterion.

    Real-rootedness Compatibility Interlacing

  2. Connecting the q-Multiplicative Convolution and the Finite Difference Convolution

    Jonathan Leake and Nick Ryder

    Advances in Mathematics (2020) arXiv (2017)

    Proves a root-mesh preservation conjecture for finite-difference convolution by connecting additive/Walsh and multiplicative/Grace–Szegő convolution methods.

    Polynomial convolution Root mesh Finite differences

  3. Mixed Determinants and the Kadison–Singer Problem

    Jonathan Leake and Mohan Ravichandran

    Mathematische Annalen (2020) arXiv (2016)

    Adapts Marcus–Spielman–Srivastava interlacing-polynomial ideas to Anderson paving and mixed determinants, producing improved paving estimates related to Kadison–Singer.

    Kadison–Singer Mixed determinants Interlacing polynomials

  4. Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

    Jonathan Leake and Nick Ryder

    Algebraic Combinatorics (2019) arXiv (2016)

    Extends real-rootedness and Heilmann–Lieb-type bounds from matching polynomials to multivariate independence polynomials, characterizing claw-freeness via a stability-like property.

    Matching polynomials Independence polynomials Claw-free graphs

  5. On the Further Structure of the Finite Free Convolutions

    Jonathan Leake and Nick Ryder

    arXiv (2018) talk (2019)

    Pushes the finite-free-convolution/barrier-method framework beyond its original uses by generalizing root bounds to differential operators and identifying limits of natural multivariate extensions.

    Finite free convolution Barrier method Root bounds

Representation theory, algebraic groups, and stability preservers

Algebraic applications involving Verma modules, unipotent orbits, and representation-theoretic viewpoints on stability preservers.

  1. New Bounds for Integer Flows and Verma Modules, via Denormalized Lorentzian Laurent Series

    Jonathan Leake and Maryam Mohammadi Yekta

    arXiv (2026)

    Introduces denormalized Lorentzian Laurent series, a power-series analog of denormalized Lorentzian polynomials, and uses them to bound integral flows on directed acyclic graphs and weight spaces of parabolic Verma modules.

    DL Laurent series Integer flows Verma modules

  2. Inequalities Characterizing Distinguished Unipotent Orbits

    Alexander Bertoloni Meli, Teruhisa Koshikawa, and Jonathan Leake

    arXiv (2024) resources

    Proves a new inequality characterization of distinguished unipotent orbits, using combinatorial checks for classical groups and computer verification for exceptional groups.

    Unipotent orbits Reductive groups Representation theory

  3. A Representation Theoretic Explanation of the Borcea–Brändén Characterization

    Jonathan Leake

    Mathematische Zeitschrift (2022) arXiv (2017) talk + slides (2018)

    Reinterprets the Borcea–Brändén characterization of stability-preserving linear operators through representation theory, giving a conceptual unification of related polynomial-stability results.

    Stable polynomials Representation theory Grace theorem