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Ernesto Lupercio

Publications and source records attributed to Ernesto Lupercio.

At least 19 recordsLinked to original sources
arXiv2026

The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge1$, \[ ν_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ ν_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\mathbb F_2$, giving the exact valuation.

math.CO↗
arXiv2026

Regular Bundles on Orbifolds: A Short Proof of Presentability

Let $\X$ be a Hausdorff second-countable smooth orbifold without boundary, possibly noncompact and ineffective. Suppose $\dim\X\leq n$ and $|G_x|\leq B$ for integers $n\geq0$ and $B\geq1$, where $G_x$ is the full stabilizer at $x$. We construct a smooth Hermitian bundle of explicit rank $R(n,B)$ whose fibre at $x$ is a positive multiple of $\C[G_x]$. Adams operations cancel the Bott obstructions on the boundary spheres of a locally finite good triangulation, and the connectivity of Stiefel manifolds gives actual bundles representing the resulting virtual classes. Smoothing preserves all stabilizer representations. Unitary frames give $\X\simeq[M/U(R(n,B))]$ for a smooth manifold $M$ with a proper locally free action; $M$ is compact exactly when $|\X|$ is compact. Thus every compact smooth orbifold is presentable.

math.AT↗
arXiv2026

Knots, black holes, databases, and birthdays: Collision entropy of knot invariants

A knot invariant is a fingerprint shared by equivalent knots: unequal values certify inequivalence, while equal values may conceal different knots. Under two rooted random-diagram models, we prove that the incomplete normalized Alexander polynomial separates random pairs but collides in a growing database. If $D_n$ and $D_n'$ are independent $n$-crossing diagrams, then $\Pr\{Δ_{K(D_n)}=Δ_{K(D_n')}\}=O(n^{-1/2})$, and every fixed normalized Alexander polynomial is exponentially rare. With exponentially high probability, the diagram shadow contains linearly many disjoint opened-trefoil slots. Conditional on the shadow and exterior crossing signs, their indicators are independent Bernoulli$(1/4)$ variables whose sum is a binomial coordinate in the $3$-adic determinant valuation. Consequently, $\sup_{a\geq 1}\Pr\{\det K(D_n)=a\}=O(n^{-1/2})$. For a discrete invariant $I$, let $α_n(I)=\Pr\{I(D_n)=I(D_n')\}$. Its collision entropy is $H_2(I(D_n))=-\logα_n(I)$. An independent sample of size $M$ has $\binom{M}{2}α_n(I)$ colliding pairs on average and birthday scale $α_n(I)^{-1/2}$. If $M_n^2α_n(I)\to\infty$, a repeated value occurs with probability tending to one even when $α_n(I)\to0$. For the determinant, $M=o(n^{1/4})$ suffices for collision freedom with high probability; a matching lower bound is open. We also give exact finite-population formulas for fixed censuses, calculate the expected cost of invariant cascades, and relate collision probability to the frequency of calls to a complete equivalence procedure. On a balanced pair-classification benchmark, the normalized-Alexander rule has balanced accuracy $1-O(n^{-1/2})$ but cannot distinguish inequivalent pairs in one fiber.

math.GT↗
arXiv2026

Many-point tropical relaxation and the Monge--Ampère equation

We prove a quantitative tropical approximation to the planar Aleksandrov Monge--Ampère equation. Let $Ω\subset\mathbb R^2$ be a bounded open convex domain, fix $K\SubsetΩ$, and let $F_N=G_{P_N}(0_Ω)$ be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corner locus containing a universally generic $N$-point set $P_N\subset K$. Set $u_N=N^{-1/2}F_N$ and $μ_N=N^{-1}\sum_{p\in P_N}δ_p$. For every compact $L\SubsetΩ$ we prove $\left|\intφ\,d(\mathrm{MA}(u_N)-μ_N)\right|\le C(Ω,K,L)N^{-1/2}(\|φ\|_\infty+\|\nablaφ\|_\infty)$ for $φ\in C_c^1(Ω)$ supported in $L$. If $μ_N\rightharpoonupμ$, where $μ$ is a probability measure supported in $K$, then $u_N$ converges uniformly on $\overlineΩ$ to the unique continuous concave zero-boundary Aleksandrov solution of $\mathrm{MA}(F)=μ$, and $\mathrm{MA}(u_N)\rightharpoonupμ$ vaguely in $Ω$. No regularity or strict convexity of $\partialΩ$ is assumed. For bounded rational convex polygons, strong genericity suffices. If $P\subset K$ is strongly generic with $|P|=N$ and $F_P=G_P(0_Ω)$, its tropical curve has exactly $N$ bounded cells; the duals of the uncut marked carriers form a spanning tree; every compact internal edge has weight one; and $\mathrm{MA}(F_P)(Ω^\circ)=N-1+\tfrac12D_{\mathrm{term}}(F_P)$, with $D_{\mathrm{term}}(F_P)=O_{Ω,K}(\sqrt N)$. For strongly generic sequences satisfying the same empirical-measure hypothesis, the normalized curvature measures converge weakly on the closed polygon. We also obtain almost-sure limits for i.i.d. samples from absolutely continuous laws supported in $K$, affine covariance of the continuum solution, and, for source sequences covered by the polygonal theorem, a configuration-dependent Abelian-sandpile diagonal.

math.AP↗
arXiv2026

Symmetry Emergence in Self-Organized Criticality

We describe a mechanism of affine symmetry emergence in the maximal density regime of the prototypical model of self-organized criticality when the inverse square of the mesh of the underlying lattice is much larger than the number of random perturbation points distributed according to a prescribed probability measure supported in the interior of the ambient convex domain. Moreover, an appropriate scaling limit of the toppling function (aka odometer), which counts the number of operations per site, is a solution to a non-linear partial differential equation well known in the context of optimal transport and differential geometry, making it possible to accurately estimate the deviation of the density from its maximal value in any macroscopic window. The mechanism for the affine symmetry emergence is due to the novel empirical fact, supported in addition by inductive arguments that have recently being upgraded to a rigorous proof, that the scaling limit of the toppling function is the unique concave solution of the Monge-Ampère equation with Dirichlet boundary condition on the convex domain with the potential given by the probability measure used above as the infinite-perturbation profile.

math-ph↗
arXiv2026

Residues of a tropical zeta function for convex domains

We define an $\operatorname{SL}_n(\mathbb{Z})$-invariant tropical zeta function of a convex domain. In dimension 2 it admits boundary Dirichlet-series representation with summands indexed by Farey pairs. For $C^3$ strictly convex domains, it extends meromorphically to $\Re(s)>3/5$, holomorphic there except for a simple pole at $s=2/3$, with residue universally proportional to equiaffine perimeter. A Tauberian argument yields the $t^{1/3}$ wave-front lattice-perimeter asymptotic for $t\rightarrow 0$. In addition, for a special domain $L$, which is a limit shape of lattice polygons in a square, with its tropical zeta function being expressed in terms of Witten SU(3) zeta function, we compute the exact coefficient in the asymptotic expansion of the integer-averaged lattice point counting for the leading term $N^{1/2}$.

math.NT↗
arXiv2024

Revisiting the Classical McKay Correspondence, Derived Equivalences and the Spectrum of Kleinian Surface Singularities: A Look Through the Mirror

In this article, we revisit the classical McKay correspondence via homological mirror symmetry. Specifically, we demonstrate how this correspondence can be articulated as a derived equivalence between the category of vanishing cycles associated with a Kleinian surface singularity and the category of perfect complexes on the corresponding quotient orbifold. We further illustrate how this equivalence allows for the interpretation of the spectrum of a Kleinian surface singularity solely in terms of the representation-theoretic data of the associated binary polyhedral group.

math.AG↗
arXiv2020

Non-commutative Geometry Indomitable

This paper is a very brief and gentle introduction to non-commutative geometry geared primarily towards physicists and geometers. It starts with a brief historical description of the motivation for non-commutative geometry and then goes on to motivate the subject from the point of view of the the understanding of local symmetries affordee by the theory of groupoids. The paper ends with a very rapid survey of recent developments and applications such as non-commutative toric geometry, the standard model for particle physics and the study of the Riemann Hypothesis.

hep-th↗
arXiv2020

Quantum (Non-commutative) Toric Geometry: Foundations

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we define their category and show that it is equivalent to a category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.

math.SG↗
arXiv2019

Nearly Frobenius Algebras

In this introductory paper we study nearly Frobenius algebras which are generalizations of the concept of a Frobenius algebra which appear naturally in topology: nearly Frobenius algebras have no traces (co-units). We survey the most basic foundational results and some of the applications they encounter in geometry, topology and representation theory.

math.RA↗
arXiv2018

Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry

Tropical Geometry, an established field in pure mathematics, is a place where String Theory, Mirror Symmetry, Computational Algebra, Auction Theory, etc, meet and influence each other. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena, and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy-model (cf. Turing reaction-diffusion model), requiring further investigation.

nlin.AO↗
arXiv2017

The exponential map of the complexification of {\em Ham} in the real-analytic case

Let $(M, ω, J)$ be a Kähler manifold and K its group of hamiltonian symplectomorphisms. The complexification of K introduced by Donadson is not a group, only a "formal Lie group". However it still makes sense to talk about the exponential map in the complexification. In this note we show how to construct geometrically the exponential map (for small time), in case the initial data are real-analytic. The construction is motivated by, but does not use, semiclassical analysis.

math-ph↗
arXiv2014

T-duality and exceptional generalized geometry through symmetries of dg-manifolds

We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework we construct the T-duality map as a degree -1 map between the cohomologies of the T-dual dg-manifolds and we show an explicit isomorphism between the differential graded algebra of the symmetries of the T-dual dg-manifolds. We furthermore show how the algebraic structure underlying B_n generalized geometry could be recovered as derived dg-Leibniz algebra of the fixed points of the T-dual automorphism acting on the symmetries of a self T-dual dg-manifold, and we show how other types of algebraic structures underlying exceptional generalized geometry could be obtained as derived symmetries of certain dg-manifolds.

math.DG↗
arXiv2013

Non-commutative Toric Varieties

In this note we introduce a new family of non-commutative spaces that we call non-commutative toric varieties and we describe some of their main properties. The main technical tool in this investigation is a natural extension of LVM-theory for the irrational case. In order to introduce the moduli space of (non-commutative) toric varieties we use variations on the notion of diffeology as models for non-commutative spaces.

math.SG↗
arXiv2008

Orbifold String Topology

In this paper we study the string topology (á la Chas-Sullivan) of an orbifold. We define the string homology ring product at the level of the free loop space of the classifying space of an orbifold. We study its properties (introducing an operad to do so) and do some explicit calculations.

math.AT↗
arXiv2006

The loop orbifold of the symmetric product

By using the loop orbifold of the symmetric product, we give a formula for the Poincaré polynomial of the free loop space of the Borel construction of the symmetric product. We also show that the Chas-Sullivan product structure in the homology of the free loop space of the Borel construction of the symmetric product induces a ring structure in the homology of the inertia orbifold of the symmetric product. This ring structure is compared to the one in cohomology defined through the usual field theory formalism as in the theory of Chen and Ruan.

math.AT↗
arXiv2006

Topological Quantum Field Theories, Strings, and Orbifolds

In this expository paper written for physicists and geometers we introduce the notions of TQFT and of orbifold. Then we survey the construction of TQFT's originating from orbifolds such as Chen-Ruan theory and Orbifold String Topology.

hep-th↗