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Thomas Lam

Publications and source records attributed to Thomas Lam.

At least 19 recordsLinked to original sources
arXiv2026

Dual Lattice Functions of Polytopes

We define the dual lattice function of a rational polytope $P$ via the discrete Laplace transform of the exponential of its support function. This definition is a discrete analogue of the dual volume function of a polytope that the authors studied in previous work. We show that the dual lattice function is valuative, and by multiplying with the torus form, it becomes the canonical form of the exponential polytope $\mathrm{exp}(P)$ as a positive geometry. This result suggests the study of the class of toric polytopes, which are certain semialgebraic subsets of projective toric varieties. Our work is a first step towards discretization of positive geometries in the simplest case of polytopes.

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arXiv2026

Scaffolds for Higher Tropical Grassmannians: Foundations

Scaffolds are the one-dimensional skeleta of high-dimensional flag simplicial complexes of nonpositive curvature. They generalize the phylogenetic trees of Trop G(2,n) to arbitrary $k$, drawing together SL(k)-web bases, affine buildings, the combinatorics of the positive tropical Grassmannian and low-dimensional topology. We prove that scaffolds model points in all tropical Grassmannians via a $k$-point distance function. In this paper, we study in detail CAT(0) planar graphs, which are positive scaffolds for the tropical Grassmannian of three-planes. CAT(0) planar graphs are directed versions of the diskoids of Fontaine-Kamnitzer-Kuperberg, planar dual to SL(3)-webs. Our main result is the construction of a unique representation of any given integer positive tropical Plucker vector by a normal CAT(0) planar graph. We show that any normal CAT(0) planar graph embeds into the tropical linear space as a Lam-Postnikov membrane, and embeds into the Keel-Tevelev membrane within the affine building. We show that Early's planar basis expansion can be computed directly from the strand combinatorics of the dual web, and connect this expansion to Petersen-Pylyavskyy-Speyer's noncrossing tableaux, explored further in our companion paper.

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arXiv2026

Noncrossing Duality and the Geometry of Positive Tropical Linear Spaces

While the positive Grassmannian is deeply understood through the rich combinatorics of plabic graphs and positroid cells, its tropical counterpart, the positive tropical Grassmannian Trop$_{>0}G(k,n)$, has lacked a comparable structural framework for general $k$. Both the global face structure of Trop$_{>0}G(k,n)$ and the internal metric geometry of the tropical linear spaces it parametrizes have remained largely uncharted. This paper develops a systematic algebraic and polyhedral foundation that resolves this gap. The engine of our framework is a fundamental tropical duality, analogous to the duality between cluster variables (or more precisely, their $u$-coordinates) and $\mathbf{g}$-vectors, pairing two families of objects introduced by the first author: the planar basis of tropical Pl\"ucker vectors and the planar cross-ratios on the positive configuration space. We prove that this duality links the fan structure of the positive tropical Grassmannian to the noncrossing fan of Santos, Stump, and Welker, yielding a global bijection between integer points of $Trop_{>0}G(k,n)$ and noncrossing tableaux. We then study how this discrete combinatorial data controls the continuous metric geometry of positive tropical linear spaces. We realize the bounded complex of an integer positive tropical linear space as the subdifferential of a central roof function on the hypersimplex, and use this realization to embed it into a dilate of the fundamental alcoved simplex. The dilation factor, and hence the geometric diameter of the complex, is governed by a single invariant, the planar kinematics ($K) weight, which we show equals the number of columns in the associated noncrossing tableau. The results of this work are applied in our parallel work on scaffolds for higher tropical Grassmannians.

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arXiv2026

Combinatorial invariance for the coefficient of $q$ in Kazhdan-Lusztig polynomials

We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan--Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture, of Lusztig and of Dyer, also for Bruhat intervals of length at most $6$. We also prove the Gabber--Joseph conjecture for the second-highest $Ext$ group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group, and of the numbers of frozen and of mutable variables in the cluster structure on Richardson varieties.

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arXiv2025

The combinatorial geometry of particle physics

Recent breakthroughs in the study of scattering amplitudes have uncovered profound and unexpected connections with combinatorial geometry. These connections range from classical structures -- such as polytopes, matroids, and Grassmannians -- to more modern developments including positroid varieties and the amplituhedron. Together they point toward the unifying framework of positive geometry, in which geometric domains canonically determine analytic functions governing scattering processes. This survey traces the emergence of positive geometry from the physics of amplitudes, building towards recent progress on amplitudes for matroids.

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arXiv2025

A New Two-Sample Test for Covariance Matrices in High Dimensions: U-Statistics Meet Leading Eigenvalues

We propose a two-sample test for covariance matrices in the high-dimensional regime, where the dimension diverges proportionally to the sample size. Our hybrid test combines a Frobenius-norm-based statistic as considered in Li and Chen (2012) with the leading eigenvalue approach proposed in Zhang et al. (2022), making it sensitive to both dense and sparse alternatives. The two statistics are combined via Fisher's method, leveraging our key theoretical result: a joint central limit theorem showing the asymptotic independence of the leading eigenvalues of the sample covariance matrix and an estimator of the Frobenius norm of the difference of the two population covariance matrices, under suitable signal conditions. The level of the test can be controlled asymptotically, and we show consistency against certain types of both sparse and dense alternatives. A comprehensive numerical study confirms the favorable performance of our method compared to existing approaches.

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arXiv2025

Canonical forms of oriented matroids

Positive geometries are semialgebraic sets equipped with a canonical differential form whose residues mirror the boundary structure of the geometry. Every full-dimensional projective polytope is a positive geometry. Motivated by the canonical forms of polytopes, we construct a canonical form for any tope of an oriented matroid, inside the Orlik--Solomon algebra of the underlying matroid. Using these canonical forms, we construct bases for the Orlik--Solomon algebra of a matroid, and for the Aomoto cohomology. These bases of canonical forms are a foundational input in the theory of matroid amplitudes introduced by the second author.

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arXiv2024

Matroids and amplitudes

In the 1990s, Kita--Yoshida and Cho--Matsumoto introduced intersection forms on the twisted (co)homologies of hyperplane arrangement complements. We give a closed combinatorial formula for these intersection pairings. We show that these intersection pairings are obtained from (continuous and discrete) Laplace transforms of subfans of the Bergman fan of the associated matroid. We compute inverses of these intersection pairings, allowing us to identify (variants of) these intersection forms with the contravariant form of Schechtman--Varchenko, and the bilinear form of Varchenko. Building on parallel joint work with C. Eur, we define a notion of scattering amplitudes for matroids. We show that matroid amplitudes satisfy locality and unitarity, and recover biadjoint scalar amplitudes in the case of the complete graphic matroid. We apply our formulae for twisted intersection forms to deduce old and new formulae for scattering amplitudes.

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arXiv2024

Dual Mixed Volume

We define and study the dual mixed volume rational function of a sequence of polytopes, a dual version of the mixed volume polynomial. This concept has direct relations to the adjoint polynomials and the canonical forms of polytopes. We show that dual mixed volume is additive under mixed subdivisions, and is related by a change of variables to the dual volume of the Cayley polytope. We study dual mixed volume of zonotopes, generalized permutohedra, and associahedra. The latter reproduces the planar $\phi^3$-scalar amplitude at tree level.

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arXiv2024

Moduli spaces in positive geometry

These are lecture notes for five lectures given at MPI Leipzig in May 2024. We study the moduli space M_{0,n} of n distinct points on P^1 as a positive geometry and a binary geometry. We develop mathematical formalism to study Cachazo-He-Yuan's scattering equations and the associated scalar and Yang-Mills amplitudes. We discuss open superstring amplitudes and relations to tropical geometry.

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arXiv2024

On the face stratification of the $m=2$ amplituhedron

We define and study the face stratification of the m=2 amplituhedron. We show that the face poset is an upper order ideal in the face poset of the totally nonnegative Grassmannian. Our construction is consistent with earlier work of Lukowski, and we confirm various predictions of Lukowski.

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arXiv2023

Monotone links in DAHA and EHA

We define monotone links on a torus, obtained as projections of curves in the plane whose coordinates are monotone increasing. Using the work of Morton-Samuelson, to each monotone link we associate elements in the double affine Hecke algebra and the elliptic Hall algebra. In the case of torus knots (when the curve is a straight line), we recover symmetric function operators appearing in the rational shuffle conjecture. We show that the class of monotone links viewed as links in $\mathbb R^3$ coincides with the class of Coxeter links, studied by Oblomkov-Rozansky in the setting of the flag Hilbert scheme. When the curve satisfies a convexity condition, we recover positroid links that we previously studied. In the convex case, we conjecture that the associated symmetric functions are Schur positive, extending a recent conjecture of Blasiak-Haiman-Morse-Pun-Seelinger, and we speculate on the relation to Khovanov-Rozansky homology. Our constructions satisfy a skein recurrence where the base case consists of piecewise almost linear curves. We show that convex piecewise almost linear curves give rise to algebraic links.

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arXiv2022

Solution to the Number Rotation Puzzle

The Number Rotation Puzzle (NRP) is a combination puzzle in which the goal is to rearrange a scrambled rectangular grid of numbers back into order via moves that consist of rotating square blocks of numbers of fixed size. Over all possible boards and rotating block sizes, we find all solvable initial configurations and provide algorithms to solve such configurations. For sufficiently large board and rotating block sizes, solvability conditions depend only on parity restrictions, with special additional conditions for smaller sizes. One special case leads to a novel construction of the exotic outer automorphism on $S_6$.

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arXiv2022

Effect of Boundary Conditions on Second-Order Singularly-Perturbed Phase Transition Models on $\mathbb{R}$

The second-order singularly-perturbed problem concerns the integral functional $\int_\Omega \varepsilon_n^{-1}W(u) + \varepsilon_n^3\|\nabla^2u\|^2\,dx$ for a bounded open set $\Omega \subseteq \mathbb{R}^N$, a sequence $\varepsilon_n \to 0^+$ of positive reals, and a function $W:\mathbb{R} \to [0,\infty)$ with exactly two distinct zeroes. This functional is of interest since it models the behavior of phase transitions, and its Gamma limit as $n \to \infty$ was studied by Fonseca and Mantegazza. In this paper, we study an instance of the problem for $N=1$. We find a different form for the Gamma limit, and study the Gamma limit under the addition of boundary data.

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arXiv2022

Braid variety cluster structures, I: 3D plabic graphs

We introduce $3$-dimensional generalizations of Postnikov's plabic graphs and use them to establish cluster structures for type $A$ braid varieties. Our results include known cluster structures on open positroid varieties and double Bruhat cells, and establish new cluster structures for type $A$ open Richardson varieties.

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arXiv2022

Electrical networks and the Grove algebra

We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analogue of the Pl\"ucker ring studied classically in invariant theory. We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gr\"obner basis for the grove ideal.

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