arXiv subjects

Tudor Popescu

Publications and source records attributed to Tudor Popescu.

6 recordsLinked to original sources

arXiv2026

Distinguished standard modules for $\mathrm{GL}_{2m}(\mathbb{C})/\mathrm{GL}_m(\mathbb{H})$

We characterize the standard modules of $\GL_{2m}(\C)$ that are distinguished by $\GL_m(\HH)$. Let $\delta_1,\dots, \delta_{2m}$ be characters of $\mathbb{C}^\times$. Assume that $S = \delta_1 \times \cdots \times \delta_{2m}$ is a standard module of $\GL_{2m}(\mathbb{C})$. For each $i$, define ${\delta_i^*} (z) = \delta_i(\overline{z})^{-1}$ for $ z \in \mathbb{C}^\times$. In particular, we conclude that a standard module for $\GL_{2m}(\mathbb{C})$ is distinguished by $\GL_{m}(\mathbb{H})$ if and only if there exists an involution $p\in S_{2m}$ without fixed points such that $\delta_{p(i)}=\delta_i^*$ for every $i$. We first verify the hypotheses of the multiplicity estimate theorem of Suzuki and Tamori in \cite{ST}. The orbit calculation of Matringe, Offen, and Yang in \cite{MOYglobal} then gives the necessary condition and a dimension bound. Then local intertwining periods prove sufficiency.

math.RT↗

arXiv2025

Matchings with Prescribed Color Counts

In this note, we prove an interesting result about perfect matchings in a complete bipartite graph with 2n vertices on each side, whose edges are colored in red and blue such that each vertex is part of n red edges and n blue edges.

math.CO↗

arXiv2024

Walking to Infinity Along Some Number Theory sequences

An interesting open conjecture asks whether it is possible to walk to infinity along primes, where each term in the sequence has one digit more than the previous. We present different greedy models for prime walks to predict the long-time behavior of the trajectories of orbits, one of which has similar behavior to the actual backtracking one. Furthermore, we study the same conjecture for square-free numbers, which is motivated by the fact that they have a strictly positive density, as opposed to primes. We introduce stochastic models and analyze the walks' expected length and frequency of digits added. Lastly, we prove that it is impossible to walk to infinity in other important number-theoretical sequences or on primes in different bases.

math.NT↗

arXiv2022

Walking to Infinity on the Fibonacci Sequence

An interesting open problem in number theory asks whether it is possible to walk to infinity on primes, where each term in the sequence has one more digit than the previous. In this paper, we study its variation where we walk on the Fibonacci sequence. We prove that all walks starting with a Fibonacci number and the following terms are Fibonacci numbers obtained by appending exactly one digit at a time to the right have a length of at most two. In the more general case where we append at most a bounded number of digits each time, we give a formula for the length of the longest walk.

math.NT↗

arXiv2022

Modeling Random Walks to Infinity on Primes in $\mathbb{Z}[\sqrt{2}]$

An interesting question, known as the Gaussian moat problem, asks whether it is possible to walk to infinity on Gaussian primes with steps of bounded length. Our work examines a similar situation in the real quadratic integer ring $\mathbb{Z}[\sqrt{2}]$ whose primes cluster near the asymptotes $y = \pm x/\sqrt{2}$ as compared to Gaussian primes, which cluster near the origin. We construct a probabilistic model of primes in $\mathbb{Z}[\sqrt{2}]$ by applying the prime number theorem and a combinatorial theorem for counting the number of lattice points whose absolute values of their norms are at most $r^2$. We then prove that it is impossible to walk to infinity if the walk remains within some bounded distance from the asymptotes. Lastly, we perform a few moat calculations to show that the longest walk is likely to stay close to the asymptotes; hence, we conjecture that there is no walk to infinity on $\mathbb{Z}[\sqrt{2}]$ primes with steps of bounded length.

math.NT↗

arXiv2021

Filtering cohomology of ordinary and Lagrangian Grassmannians

This paper studies, for a positive integer $m$, the subalgebra of the cohomology ring of the complex Grassmannians generated by the elements of degree at most $m$. We build in two ways upon a conjecture for the Hilbert series of this subalgebra due to Reiner and Tudose. The first reinterprets it in terms of the operation of $k$-conjugation, suggesting two conjectural bases for the subalgebras that would imply their conjecture. The second introduces an analogous conjecture for the cohomology of Lagrangian Grassmannians.

math.CO↗