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Brian Freidin

Publications and source records attributed to Brian Freidin.

11 recordsLinked to original sources
arXiv2024

Transverse-freeness in finite geometries

We study projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth hypersurfaces in $n$-dimensional projective space that are tangent to every $k$-dimensional subspace, for some value of $n$ and $k$. We then study projective surfaces that serve as models of finite inversive and hyperbolic planes, finite analogs of spherical and hyperbolic geometries. In these surfaces we construct curves tangent to each of the lowest degree curves defined over the base field.

math.AG↗
arXiv2023

Fast winning strategies in a generalized van der Waerden game

Consider the following Maker-Breaker game. Fix a finite subset $S\subset\mathbb{N}$ of the naturals. The players Maker and Breaker take turns choosing previously unclaimed natural numbers. Maker wins by eventually building a homothetic copy $aS+b$ of $S$, where $a\in\mathbb{N}\setminus\{0\}$ and $b\in\mathbb{Z}$. This is a generalization of the van der Waerden game analyzed by Beck. By the Hales-Jewett theorem, there exists a constant $c$ depending only on $|S|$ such that Maker can win in $c$ or less moves. We show that Maker can win in $|S|$ moves if $|S|\leq 3$. When $|S|=4$, we show that Maker can always win in $5$ or less moves and describe all $S$ such that Maker can win in $4$ moves. If $|S|\geq 5$, Maker has no winning strategy in $|S|$ moves.

math.CO↗
arXiv2022

Balanced homogeneous harmonic maps between cones

We study the degrees of homogeneous harmonic maps between simplicial cones. Such maps have been used to model the local behavior of harmonic maps between singular spaces, where the degrees of homogeneous approximations describe the regularity of harmonic maps. In particular the degrees of homogeneous harmonic maps are related to eigenvalues of discrete normalized graph Laplacians.

math.DG↗
arXiv2021

Harmonic maps between 2-dimensional simplicial complexes 2

We study metrics on two-dimensional simplicial complexes that are conformal either to flat Euclidean metrics or to the ideal hyperbolic metrics described by Charitos and Papadopoulos. Extending the results of our previous paper, we prove existence, uniqueness, and regularity results for harmonic maps between two such metrics on a complex.

math.DG↗
arXiv2020

On the proportion of transverse-free plane curves

We study the asymptotic proportion of smooth plane curves over a finite field $\mathbb{F}_q$ which are tangent to every line defined over $\mathbb{F}_q$. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over $\mathbb{F}_q$ admits a transverse $\mathbb{F}_q$-line with very high probability.

math.AG↗
arXiv2019

A Liouville-type theorem and Bochner formula for harmonic maps into metric spaces

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT($κ$) spaces for $κ\in\{0,1\}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT($κ$) spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an $L^p$ Liouville-type theorem for harmonic maps from admissible polyhedra into convex CAT($κ$) spaces. Another consequence we derive from the target variation formula is the Eells-Sampson Bochner formula for CAT(1) targets.

math.DG↗
arXiv2018

Harmonic maps between ideal 2-dimensional simplicial complexes

We prove existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes. The spaces in question were introduced by Charitos-Papadopoulos, who describe their Teichmüller spaces and some compactifications. This work is a first step in introducing harmonic map theory into the Teichmüller theory of these spaces.

math.DG↗
arXiv2018

Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

We prove that the area of a free boundary minimal surface $Σ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $Σ$ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows work of Brendle and Fraser-Schoen in the euclidean setting.

math.DG↗
arXiv2017

A Bochner Formula for Harmonic Maps into Non-Positively Curved Metric Spaces

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson Bochner formula in this more general setting. In particular, we show that harmonic maps from spaces of non-negative Ricci curvature into non-positively curved spaces have subharmonic energy density. When the domain is compact the energy density is constant, and if the domain has a point of positive Ricci curvature every harmonic map into an NPC space must be constant.

math.DG↗
arXiv2016

Free Boundary Minimal Surfaces in the Unit Ball With Low Cohomogeneity

We study free boundary minimal surfaces in the unit ball of low cohomogeneity. For each pair of positive integers $(m,n)$ such that $m, n >1$ and $m+n\geq 8$, we construct a free boundary minimal surface $Σ_{m, n} \subset B^{m+n}$(1) invariant under $O(m)\times O(n)$. When $m+n<8$, an instability of the resulting equation allows us to find an infinite family $\{Σ_{m,n, k}\}_{k\in \mathbb{N}}$ of such surfaces. In particular, $\{Σ_{2, 2, k}\}_{k\in \mathbb{N}}$ is a family of solid tori which converges to the cone over the Clifford Torus as $k$ goes to infinity. These examples indicate that a smooth compactness theorem for Free Boundary Minimal Surfaces due to Fraser and Li does not generally extend to higher dimensions. For each $n\geq 3$, we prove there is a unique nonplanar $SO(n)$-invariant free boundary minimal surface (a "catenoid") $Σ_n \subset B^n(1)$. These surfaces generalize the "critical catenoid" in $B^3(1)$ studied by Fraser and Schoen.

math.DG↗