arXiv subjects

Manisha Garg

Publications and source records attributed to Manisha Garg.

4 recordsLinked to original sources

arXiv2026

On universal elements for doubling geodesic trees

For $n\ge 3$ and $c\in(0,1)$, let $\mathcal{GT}(n,c)$ denote the class of geodesic metric trees of valence at most $n$ whose branch points are uniformly relatively separated with constant $c$. We prove that $\mathcal{GT}(n,c)$ has no bi-Lipschitz universal element. More precisely, we construct a family $(T_a)_{a\in[1/4,1/3]}\subset\mathcal{GT}(n,c)$ such that, for every $n_M\geq 3, c_M\in(0,1)$ and every $M\in\mathcal{GT}(n_M, c_M)$, there are at most countably many parameters $a$ for which $T_a$ admits a bi-Lipschitz embedding into $M$, whereas each $T_a$ admits a bi-Lipschitz embedding into $\mathbb R^2$. Thus the obstruction is neither dimensional nor caused by a failure of planar embeddability. This gives a negative answer to a question of Chrontsios-Garitsis, Ioannidis, and Vellis~\cite[Question~1.11]{CGIV2024}. Furthermore, we show a complementary positive result for ultrametric spaces: every bounded ultrametric space $X$ admits a bi-Lipschitz embedding into every complete metric space $Y$ satisfying $\dim_A X<\dim_{LA} Y$ where $\dim_A X$ and $\dim_{LA} X$ are Assouad and lower Assouad dimensions, respectively.

math.MG↗

arXiv2026

Analytic and quasiregular distortion of Nagata dimension

We study how analytic functions, and more generally quasiregular mappings, distort Nagata dimension. Quasiconformal mappings of domains preserve the Nagata dimension of compact subsets, in view of a result of Lang and Schlichenmaier. We establish the same conclusion for analytic functions defined on general planar domains. On the other hand, polynomials (and more generally, rational maps) preserve the Nagata dimension of arbitrary subsets of their domain. In the absence of the compactness assumption, we provide examples to show that an entire function can increase or decrease the Nagata dimension of subsets of the domain. Some of these results generalize to meromorphic functions, and separately to planar quasiregular maps in view of Stoilow factorization. We also show that conformal mappings can change the porosity behavior of noncompact subsets of their domain; this yields examples of planar conformal maps which take sets of Nagata dimension strictly less than two onto set of Nagata dimension two. We conclude with open questions and potential future work related to the distortion of Nagata dimension by higher-dimensional quasiregular maps.

math.CV↗

arXiv2025

On zero-divisors and units in group rings of torsion-free CAT$(0)$ groups

This paper addresses two of Kaplansky's conjectures concerning group rings $K[G]$, where $K$ is a field and $G$ is a torsion-free group: the zero-divisor conjecture, which asserts that $K[G]$ has no non-trivial zero-divisors, and the unit conjecture, which asserts that $K[G]$ has no non-trivial units. While the zero-divisor conjecture still remains open, the unit conjecture was disproven by Gardam in 2021. The search for more counterexamples remains an open problem. Let $m$ and $n$ be the cardinality of support of two non-trivial elements $\alpha, \beta \in \mathbb{F}_2[G]$, respectively. We address these conjectures by introducing a process called \text{left alignment} and recursively constructing the taikos of size $(m,n)$ which would yield counterexamples to both conjectures over the field $\mathbb{F}_2$ if they satisfy conditions $\mathsf{T}_1-\mathsf{T}_4$ given in \cite{Mineyev2024}. We also present a computer-search method that can be utilized to search for counterexamples of a certain geometry by significantly pruning the search space. We prove that a class of CAT(0) groups with certain geometry cannot be counterexamples to these conjectures. Moreover, we prove that for $ 1\le m \le 5$ and $n$ any positive integer, there are no counterexamples to the conjectures such that the associated oriented product structures are of type $(m,n)$. With the aid of computer, we prove that, in fact, there are no such counterexamples of the length combination $(m,n)$ where $1\le m \le 13$ and $1\le n \le 13$.

math.GR↗

arXiv2023

SF-SFD: Stochastic Optimization of Fourier Coefficients to Generate Space-Filling Designs

Due to the curse of dimensionality, it is often prohibitively expensive to generate deterministic space-filling designs. On the other hand, when using na{\"i}ve uniform random sampling to generate designs cheaply, design points tend to concentrate in a small region of the design space. Although, it is preferable in these cases to utilize quasi-random techniques such as Sobol sequences and Latin hypercube designs over uniform random sampling in many settings, these methods have their own caveats especially in high-dimensional spaces. In this paper, we propose a technique that addresses the fundamental issue of measure concentration by updating high-dimensional distribution functions to produce better space-filling designs. Then, we show that our technique can outperform Latin hypercube sampling and Sobol sequences by the discrepancy metric while generating moderately-sized space-filling samples for high-dimensional problems.

stat.ME↗