arXiv research

Proves stability in Weyl polytopes using optimal transport.

The paper solves Monge-Ampère equations on reflexive polytopes, linking solvability to SYZ conjecture and tropical geometry.

Deep models generate geometric objects with global properties.

Ehrhart's conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including reflexive polytopes. In particular, they showed that the complex projective space has t…

In this paper, we study the asymptotic behavior of the Hermitian-Yang-Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder-Narasimhan-Seshadri filtration, this answers a question by Bando and Siu.

A simple quantitative example of a reflexive feedback process and the resulting price dynamics after an exogenous price shock to a financial network is presented. Furthermore, an outline of a theory that connects financial reflexivity, which stems from cross-ownership and delayed or incomplete information, and no-arbit…

Characterizes stable sheaves for equality in orbifold BG inequality.

We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geom…

Study shows awareness of reflexivity improves LLMs' financial forecasting accuracy.

In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian-Yang-Mills connections over any reflexive sheaves.

For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption o…

The mathematical model proposed by George Soros for his theory of reflexivity is analyzed under the framework of discrete dynamical systems. We show the importance of the notion of fixed points for explaining the behavior of a reflexive system governed by its cognitive and manipulative functions. The interrelationship …

This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.

In this paper, we study semistable Higgs sheaves over compact Kähler manifolds, we prove that there is an approximate admissible Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolove type inequality holds on a semi-stable reflexive Higgs sheaf.

Study proves existence of precotangent bundles for Grassmannians.

This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-…

It is known that given a stable holomorphic pair $(E ,φ)$, where $E$ is a holomorphic vector bundle on a compact Kähler manifold $X$ and $φ$ is a holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(\E ,φ)$, where $\E$ is a reflexive shea…

The paper studies foliations on smooth projective varieties and their properties.

New examples of Cappell-Shaneson knot pairs with same Alexander polynomial found.

Establishes Hermite-Einstein metrics on complex spaces with singularities.

Bayesian reflex models AI learning like the autonomic nervous system.

We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is at the core of kernel methods in machine learning as it makes…

The paper proves a theorem and characterizes connections over normal varieties.

Bipartite data is common in data engineering and brings unique challenges, particularly when it comes to clustering tasks that impose on strong structural assumptions. This work presents an unsupervised method for assessing similarity in bipartite data. Similar to some co-clustering methods, the method is based on regu…

We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If $\mathscr{E}$ is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a $μ$-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Ya…

The study broadens the concept of cyclic polytopes to Veronese polytopes.

Existence of metrics on non-Kähler varieties, generalizing previous work.

Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.

The paper studies deformation spaces of Coxeter truncation polytopes.

Neural networks approximate unit spheres as polytopes.

The study classifies all compact hyperbolic polytopes with eight facets.

Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…

We complete the TOP classification of 2-knots with torsion-free, solvable knot group by showing that fibred 2-knots with closed fibre the Hantzsche-Wendt flat 3-manifold $HW$ are not reflexive, while every fibred 2-knot with closed fibre a $\mathbb{N}il^3$-manifold with base orbifold $S^2(3,3,3)$ is reflexive, and by g…

This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…

The study classifies all compact 5D polytopes with 9 facets.

Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.

The study classifies 331 specific 4D polytopes with 7 facets.

Contact manifolds' momentum polytopes are convex.

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v…

Smooth approximations bound dihedral angles of convex polytopes.

Polytopic Matrix Factorization models data as latent vectors from a polytope, maximizing determinant for identifiability.

Examines nonrational polytopes and fans in toric geometry.

The article studies factorization structures in geometry and their applications to cones and polytopes.

New methods classify hyperbolic polytopes with up to 40 facets.

The paper explains how to parameterize facets of moment polytopes in real symplectic geometry.

We give a vertex algebra proof of the Berglund-Hübsch duality of nondegenerate invertible potentials. We suggest a way to unify it with the Batyrev-Borisov duality of reflexive Gorenstein cones.

We introduce two operations named biflip and puzzle-move on simple polytopes producing polytopes with diffeomorphic moment-angle manifolds.