Harry Bateman Postdoctoral Scholar
Division of Physics, Mathematics and Astronomy, Caltech
Faculty mentor: Matilde Marcolli
Research interests: Algebraic and Non-Archimedean Geometry, Commutative Algebra, Combinatorics, Mathematical Physics.
1200 E California Blvd, Pasadena CA 91125, USA • maazouz [at] caltech [dot] edu
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Symplectic Grassmannian description of the Coulomb branch three and four point amplitudes
(with V. Calvo Cortes, S. Hegde and A. Suthar). Journal of High Energy Physics.
[arxiv] [journal] [abstract] We present a formulation of the three- and four-point amplitudes on the Coulomb branch of $N=4$ SYM as integrals over the symplectic Grassmannian. We demonstrate that their kinematic spaces are equivalent to symplectic Grassmannians $\mathrm{SpGr}(n,2n)$. For the three-point case, we express the amplitude as an integral over the symplectic Grassmannian in a specific little group frame. In the four-point case, we show that the integral yields the amplitude up to a known kinematic factor. Building on the four-dimensional analysis, we also express the six-dimensional $N = (1,1)$ SYM amplitude in terms of four-dimensional variables in a form that makes its symplectic Grassmannian structure manifest.
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Spinor-Helicity Varieties
(with A. Pfister and B. Sturmfels). SIAM J. Applied Algebra and Geometry.
[arxiv] [journal] [abstract] The spinor-helicity formalism in particle physics gives rise to natural subvarieties in the product of two Grassmannians. These include two-step flag varieties for subspaces of complementary dimension. Taking Hadamard products leads to Mandelstam varieties. We study these varieties through the lens of combinatorics and commutative algebra, and we explore their tropicalization, positive geometry, and scattering correspondence.
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The positive orthogonal Grassmannian
(with Y. Mandelshtam). Le Matematiche, Vol. 80 No. 1 (2025).
[arxiv] [journal] [abstract] The Plücker positive region $\mathrm{OGr}_{+}(k,2k)$ of the orthogonal Grassmannian emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian $\mathrm{OGr}_{+}(k,n)$ for general values of $k,n$. We determine the boundary structure of the quadric $\mathrm{OGr}_{+}(1,n)$ in $\mathbb{P}^{n−1}_{+}$ and show that it is a positive geometry. We show that $\mathrm{OGr}_{+}(k,2k+1)$ is isomorphic to $\mathrm{OGr}_{+}(k+1,2k+2)$ and connect its combinatorial structure to matchings on $[2k+2]$. Finally, we show that in the case $n>2k+1$, the \emph{positroid cells} of $\mathrm{Gr}_{+}(k,n)$ do not induce a CW cell decomposition of $\mathrm{OGr}_{+}(k,n)$.
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Sampling from $p$-adic algebraic manifolds
(with E. Kaya). SIAM J. Applied Algebra and Geometry, Vol 9, Issue 2 (2025).
[arxiv] [journal] [abstract] We present a method for sampling points from an algebraic manifold, either affine or projective, defined over a local field, with a prescribed probability distribution. Inspired by the work of Breiding and Marigliano on sampling real algebraic manifolds, our approach leverages slicing the given variety with random linear spaces of complementary dimension. We also provide an implementation of this sampling technique and demonstrate its applicability to various contexts, including sampling from linear $p$-adic algebraic groups, abelian varieties, and modular curves.
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$\mathrm{GL}(n,\mathbb{Z}_p)$-invariant Gaussian measures on the space of $p$-adic polynomials
(with A. Lerario). Electronic J. Probability, Vol. 30 (2025).
[arxiv] [journal] [abstract] We prove that if $p > d$ there is a unique gaussian distribution (in the sense of Evans) on the space $\mathbb{Q}_p[x_1,…,x_n]_{(d)}$ which is invariant under the action of $\mathrm{GL}(n,\mathbb{Z}_p)$ by change of variables. This gives the nonarchimedean counterpart of Kostlan's Theorem on the classification of orthogonally (respectively unitarily) invariant gaussian measures on the space $\mathbb{R}[x_1,…,x_n]_{(d)}$ (respectively $\mathbb{C}[x_1,…,x_n]_{(d)}$). More generally, if $V$ is an $n$-dimensional vector space over a nonarchimedean local field $K$ with ring of integers $R$, and if $\lambda$ is a partition of an integer $d$, we study the problem of determining the invariant lattices in the Schur module $S_{\lambda}(V)$ under the action of the group $\mathrm{GL}(n,R)$.
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On the topology of the moduli of tropical unramified $p$-covers
(with P.A. Helminck, F. Röhrle, P. Souza, and C. Yun). Selecta Mathematica, Vol. 31, N. 14 (2025).
[arxiv] [journal] [abstract] We study the topology of the moduli space of unramified $\mathbb{Z}/p$-covers of tropical curves of genus $g \geq 2$, where $p$ is a prime number. We use recent techniques by Chan–Galatius–Payne to identify contractible subcomplexes of the moduli space. We then use this contractibility result to show that this moduli space is simply connected. In the case of genus $2$, we determine the homotopy type of this moduli space for all primes $p$. This work is motivated by prospective applications to the top-weight cohomology of the space of prime cyclic étale covers of smooth algebraic curves.
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The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution
(with J. Pitman). Combinatorics, Probability and Computing, Vol. 33, Issue 2, March 2024, pp. 210–237.
[arxiv] [journal] [abstract] The factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n >0$ is the anti-derivative of $b_{n-1}(x)$ subject to $\int_0^1 b_n(x) dx = 0$.
We offer a related characterization: $b_1(x) = x - 1/2$ and $(-1)^{n-1} b_n(x)$ for $n >0$ is the $n$-fold circular convolution of $b_1(x)$ with itself.
Equivalently, $1 - 2^n b_n(x)$ is the probability density at $x \in (0,1)$ of the fractional part of a sum of $n$ independent random variables, each with the beta$(1,2)$ probability density $2(1-x)$ at $x \in (0,1)$.
This result has a novel combinatorial analog, the {\em Bernoulli clock}: mark the hours of a $2 n$ hour clock by a uniform random permutation of the multiset $\{1,1, 2,2, \ldots, n,n\}$, meaning pick two different hours uniformly at random from the $2 n$ hours and mark them $1$, then pick two different hours uniformly at random from the remaining $2 n - 2$ hours and mark them $2$, and so on.
Starting from hour $0 = 2n$, move clockwise to the first hour marked $1$, continue clockwise to the first hour marked $2$, and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked $n$ is encountered, at a random hour $I_n$ between $1$ and $2n$.
We show that for each positive integer $n$, the event $( I_n = 1)$ has probability $(1 - 2^n b_n(0))/(2n)$, where $n! b_n(0) = B_n(0)$ is the $n$th Bernoulli number.
For $ 1 \le k \le 2 n$, the difference $\delta_n(k):= 1/(2n) - \mathbb{P}( I_n = k)$ is a polynomial function of $k$ with the surprising symmetry $\delta_n( 2 n + 1 - k) = (-1)^n \delta_n(k)$, which is a combinatorial analog of the well known symmetry of Bernoulli polynomials $b_n(1-x) = (-1)^n b_n(x)$.
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Tropical invariants for binary quintics and reduction types of Picard curves
(with P.A. Helminck and E. Kaya). Glasgow Math. J., Vol. 66, Issue 1, Jan 2024, pp. 65–87.
[arxiv] [journal] [abstract] We express the reduction types of Picard curves in terms of tropical invariants associated to binary quintics. We also give a general framework for tropical invariants associated to group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne-Mumford compactification $\overline{M}_{0,n}$.
- Lines on $p$-adic and real cubic surfaces
(with R. Ait El Manssour, K. Rose and E. Kaya). Abh. Math. Sem. Univ. Hamburg, Vol. 93, Sep 2023, pp. 149–162.
[arxiv] [journal] [abstract] We study lines on smooth cubic surfaces over the field of $p$-adic numbers, from a theoretical and computational point of view. Segre showed that the possible counts of such lines are $0,1,2,3,5,7,9,15$ or $27$. We show that each of these counts is achieved. Probabilistic aspects are also investigated by sampling both $p$-adic and real cubic surfaces from different distributions and estimating the probability of each count. We link this to recent results on probabilistic enumerative geometry. Some experimental results on the Galois groups attached to $p$-adic cubic surfaces are also discussed.
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Bolytrope orders
(with G. Nebe and M. Stanojkovski). Int. J. Number Theory, Vol. 19, Issue 05, Jun 2023.
[arxiv] [journal] [abstract] Bolytropes are bounded subsets of an affine building that consist of all points that have distance at most r from some polytrope. We prove that the points of a bolytrope describe the set of all invariant lattices of a bolytrope order, generalizing the correspondence between polytropes and graduated orders.
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Orders and Polytropes: Matrix Algebras from Valuations
(with M.A. Hahn, G. Nebe, M. Stanojkovski and B. Sturmfels). Beitr. Algebra Geom., 2021, pp. 515–531.
[arxiv] [journal] [abstract]We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings.
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Valued rank-metric codes
(with M.A. Hahn, A. Neri and M. Stanojkovski). J. Algebra Appl., Vol 24 (2025), no. 4.
[arxiv] [journal] [abstract]In this paper, we study linear spaces of matrices defined over discretely valued fields and discuss their dimension and minimal rank drops over the associated residue fields. To this end, we take first steps into the theory of rank-metric codes over discrete valuation rings by means of skew algebras derived from Galois extensions of rings. Additionally, we model projectivizations of rank-metric codes via Mustafin varieties, which we then employ to give sufficient conditions for a decrease in the dimension.
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The Gaussian entropy map in valued fields
Algebraic Statistics, Vol. 13, Issue 1, Dec 2022, pp. 1–18.
[arxiv] [journal] [abstract]We exhibit the analog of the entropy map for multivariate Gaussian distributions on local fields. As in the real case, the image of this map lies in the supermodular cone and it determines the distribution of the valuation vector. In general, this map can be defined for non-archimedian valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension 3.