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Catalin Zara

Publications and source records attributed to Catalin Zara.

18 recordsLinked to original sources
arXiv2026

Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space

We study colorings of the Euclidean space constrained by \emph{hypersphere forcing conditions}: if an admissible hypersphere, $S_r(p)$, centered at a point $p$ and of radius $r$ contains a monochromatic set of points satisfying a certain property $\mathcal{P}$, then the center of the hypersphere must have that color. These forcing conditions may be restricted in applicability to a specific set of hyperspheres $S_r(p)$. For cardinality-based forcing conditions we prove a general theorem: for countably many colors and any uncountable set of admissible radii $\mathcal{R}$, such a coloring is locally monochromatic on any admissible center set $Ω\subseteq \mathbb{R}^n$ (hence constant, for connected $Ω$). For rigid geometric properties (simplex shape, edge-length, volume constraints) we show that forcing conditions alone are insufficient without regularity assumptions. Our main result shows that for colorings satisfying a certain Baire regularity condition rigid geometric properties enforce local monochromaticity and, in the presence of a certain \emph{``uniform cap" condition}, global monochromaticity. Applications include dichotomies for edge-length and volume constraints in terms of $\inf(\mathcal{L})$ and $\inf(\mathcal{V})$, and a comeagerness criterion in the ``all edges in $\mathcal{L}$'' regime.

math.CO↗
arXiv2016

Polynomial Assignments for Bott-Samelson manifolds

Polynomial assignments for a torus $T$-action on a smooth manifold $M$ were introduced by Ginzburg, Guillemin, and Karshon in 1999; they form a module over $\mathbb{S}(\mathfrak{t}^*)$, the algebra of polynomial functions on $\mathfrak{t}$, the Lie algebra of $T$. In this paper we describe the assignment module $\mathcal{A}_T(M)$ for a natural $T$-action on a Bott-Samelson manifold $M = BS^I$ and present a method for computing generators.

math.AT↗
arXiv2013

The Prouhet-Tarry-Escott Problem and Generalized Thue-Morse Sequences

We present new methods of generating Prouhet-Tarry-Escott partitions of arbitrarily large regularity. One of these methods generalizes the construction of the Thue-Morse sequence to finite alphabets with more than two letters. We show how one can use such partitions to (theoretically) pour the same volume coffee from an urn into a finite number of cups so that each cup gets almost the same amount of caffeine.

math.CO↗
arXiv2013

Cardinality of $\ell_1$-Segments and Genocchi Numbers

We prove that the Genocchi numbers of first and second kind give the cardinality of certain segments in permutation spaces, with respect to the $\ell_1$-distance. Experimental data suggests that those segments have maximal cardinality among all segments in the corresponding spaces.

math.CO↗
arXiv2013

Polynomial Assignments

The concept of assignments was introduced in [GGK99] as a method for extracting geometric information about group actions on manifolds from combinatorial data encoded in the infinitesimal orbit-type stratification. In this paper we will answer in the affirmative a question posed in [GGK99] by showing that the equivariant cohomology ring of $M$ is to a large extent determined by this data.

math.AT↗
arXiv2013

Cardinality of Balls in Permutation Spaces

For a right invariant distance on a permutation space $S_n$ we give a sufficient condition for the cardinality of a ball of radius $R$ to grow polynomially in $n$ for fixed $R$. For the distance $\ell_1$ we show that for an integer $k$ the cardinality of a sphere of radius $2k$ in $S_n$ (for $n \geqslant k$) is a polynomial of degree $k$ in $n$ and determine the high degree terms of this polynomial.

math.CO↗
arXiv2012

Equivariant $K$-theory of GKM bundles

Given a fiber bundle of GKM spaces, $π\colon M\to B$, we analyze the structure of the equivariant $K$-ring of $M$ as a module over the equivariant $K$-ring of $B$ by translating the fiber bundle, $π$, into a fiber bundle of GKM graphs and constructing, by combinatorial techniques, a basis of this module consisting of $K$-classes which are invariant under the natural holonomy action on the $K$-ring of $M$ of the fundamental group of the GKM graph of $B$. We also discuss the implications of this result for fiber bundles $π\colon M\to B$ where $M$ and $B$ are generalized partial flag varieties and show how our GKM description of the equivariant $K$-ring of a homogeneous GKM space is related to the Kostant-Kumar description of this ring.

math.KT↗
arXiv2011

Balanced fiber bundles and GKM theory

Let $T$ be a torus and $B$ a compact $T-$manifold. Goresky, Kottwitz, and MacPherson show in \cite{GKM} that if $B$ is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring $H_T^*(B)$ as a subring of $H_T^*(B^T)$. In this paper we prove an analogue of this result for $T-$equivariant fiber bundles: we show that if $M$ is a $T-$manifold and $π\colon M \to B$ a fiber bundle for which $π$ intertwines the two $T-$actions, there is a simple combinatorial description of $H_T^*(M)$ as a subring of $H_T^*(π^{-1}(B^T))$. Using this result we obtain fiber bundle analogues of results of \cite{GHZ} on GKM theory for homogeneous spaces.

math.AT↗
arXiv2011

Cohomology of GKM Fiber Bundles

The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.

math.CO↗
arXiv2009

Positivity of Equivariant Schubert Classes Through Moment Map Degeneration

For a flag manifold $M=G/B$ with the canonical torus action, the $T-$equivariant cohomology is generated by equivariant Schubert classes, with one class $τ_u$ for every element $u$ of the Weyl group $W$. These classes are determined by their restrictions to the fixed point set $M^T \simeq W$, and the restrictions are polynomials with nonnegative integer coefficients in the simple roots. The main result of this article is a positive formula for computing $τ_u(v)$ in types A, B, and C. To obtain this formula we identify $G/B$ with a generic co-adjoint orbit and use a result of Goldin and Tolman to compute $τ_u(v)$ in terms of the induced moment map. Our formula, given as a sum of contributions of certain maximal ascending chains from $u$ to $v$, follows from a systematic degeneration of the moment map, corresponding to degenerating the co-adjoint orbit. In type A we prove that our formula is manifestly equivalent to the formula announced by Billey in \cite{Bi}, but in type C, the two formulas are not equivalent.

math.SG↗
arXiv2006

Complete Padovan sequences in finite fields

Given a prime $p\ge 5$, and given $1<κ<p-1$, we call a sequence $(a_n)_{n}$ in $\mathbb{F}_p$ a $Φ_κ$-sequence if it is periodic with period $p-1$, and if it satisfies the linear recurrence $a_n+a_{n+1}=a_{n+κ}$ with $a_0=1$. Such a sequence is said to be a complete $Φ_κ$-sequence if in addition $\{a_0,a_1,...,a_{p-2}\}=\{1,...,p-1\}$. For instance, every primitive root $b$ mod $p$ generates a complete $Φ_κ$-sequence $a_n=b^n$ for some (unique) $κ$. A natural question is whether every complete $Φ_κ$-sequence is necessarily defined by a primitive root. For $κ=2$ the answer is known to be positive. In this paper we reexamine that case and investigate the case $κ=3$ together with the associated cases $κ=p-2$ and $κ=p-3$.

math.NT↗
arXiv2001

A GKM description of the equivariant cohomology ring of a homogeneous space

Let $T$ be a torus of dimension $n>1$ and $M$ a compact $T-$manifold. $M$ is a GKM manifold if the set of zero dimensional orbits in the orbit space $M/T$ is zero dimensional and the set of one dimensional orbits in $M/T$ is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about $M$ is encoded in this graph. In this paper we prove that every compact homogeneous space $M$ of non-zero Euler characteristic is of GKM type and show that the graph associated with $M$ encodes \emph{geometric} information about $M$ as well as topological information. For example, from this graph one can detect whether $M$ admits an invariant complex structure or an invariant almost complex structure.

math.SG↗
arXiv2000

Morse theory on graphs

Let $Γ$ be a finite d-valent graph and G an n-dimensional torus. An ``action'' of G on $Γ$ is defined by a map, $α$, which assigns to each oriented edge e of $Γ$ a one-dimensional representation of G (or, alternatively, a weight, $α_e$, in the weight lattice of G). For the assignment, $e \to α_e$, to be a schematic description of a ``G-action'', these weights have to satisfy certain compatibility conditions: the GKM axioms. We attach to $(Γ, α)$ an equivariant cohomology ring, $H_G(Γ)=H(Γ,α)$. By definition this ring contains the equivariant cohomology ring of a point, $\SS(\fg^*) = H_G(pt)$, as a subring, and in this paper we will use graphical versions of standard Morse theoretical techniques to analyze the structure of $H_G(Γ)$ as an $\SS(\fg^*)$-module.

math.CO↗
arXiv2000

G-actions on graphs

Let G be an n-dimensional torus and $τ$ a Hamiltonian action of G on a compact symplectic manifold, M. If M is pre-quantizable one can associate with $τ$ a representation of G on a virtual vector space, Q(M), by $\spin^{\CC}$-quantization. If M is a symplectic GKM manifold we will show that several well-known theorems about this ``quantum action'' of G: for example, the convexity theorem, the Kostant multiplicity theorem and the ``quantization commutes with reduction'' theorem for circle subgroups of G, are basically just theorems about G-actions on graphs.

math.SG↗
arXiv2000

Combinatorial formulas for products of Thom classes

Let G be a torus of dimension n > 1 and M a compact Hamiltonian G-manifold with $M^G$ finite. A circle, $S^1$, in G is generic if $M^G = M^{S^1}$. For such a circle the moment map associated with its action on M is a perfect Morse function. Let $\{ W_p^+ ; p \in M^G\}$ be the Morse-Whitney stratification of M associated with this function, and let $τ_p^+$ be the equivariant Thom class dual to $W_p^+$. These classes form a basis of $H_G^*(M)$ as a module over $\SS(\fg^*)$ and, in particular, $$τ_p^+ τ_q^+ = \sum c_{pq}^r τ_r^+$$ with $c_{pq}^r \in \SS(\fg^*)$. For manifolds of GKM type we obtain a combinatorial description of these $τ_p^+$'s and, from this description, a combinatorial formula for $c_{pq}^r$.

math.SG↗
arXiv2000

One-skeleta, Betti numbers and equivariant cohomology

The one-skeleton of a G-manifold M is the set of points p in M where $\dim G_p \geq \dim G -1$; and M is a GKM manifold if the dimension of this one-skeleton is 2. Goresky, Kottwitz and MacPherson show that for such a manifold this one-skeleton has the structure of a ``labeled" graph, $(Γ, α)$, and that the equivariant cohomology ring of M is isomorphic to the ``cohomology ring'' of this graph. Hence, if M is symplectic, one can show that this ring is a free module over the symmetric algebra $\SS(\fg^*)$, with $b_{2i}(Γ)$ generators in dimension 2i, $b_{2i}(Γ)$ being the ``combinatorial'' 2i-th Betti number of $Γ$. In this article we show that this ``topological'' result is , in fact, a combinatorial result about graphs.

math.DG↗
arXiv1998

Equivariant de Rham Theory and Graphs

Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be theorems about graphs. In this paper we show that for some familiar theorems, this is indeed the case.

math.DG↗