arXiv subjects

Hannah Alpert

Publications and source records attributed to Hannah Alpert.

At least 19 recordsLinked to original sources

arXiv2025

1-Uryson width and covers

We investigate the following question: Do there exist Riemannian polyhedra $X$ such that the 1-Uryson width of their universal covers $\mathrm{UW}_1(\widetilde{X})$ is bounded but $\mathrm{UW}_1(X)$ is arbitrarily large? We rule out two specific cases: when $\pi_1(X)$ is virtually cyclic and when $X$ is a Riemannian surface. More specifically, we show that if $X$ is a compact polyhedron with a virtually cyclic fundamental group, then its 1-Uryson width is bounded by the 1-Uryson width of its universal cover $\widetilde{X}$. Precisely: $$\mathrm{UW}_1(X) \leq 6 \cdot \mathrm{UW}_1(\widetilde{X}).$$ We show that if $X$ is a Riemannian surface with boundary then $$\mathrm{UW}_1(X) \leq \mathrm{UW}_1(\widetilde{X}).$$ Furthermore, we show that if there exist spaces $X$ for which $\mathrm{UW}_1(\widetilde{X})$ is bounded while $\mathrm{UW}_1(X)$ is arbitrarily large, then such examples must already appear in low dimensions. In particular, such $X$ can be found among Riemannian $2$-complexes.

math.MG↗

arXiv2024

Growth in the universal cover under large simplicial volume

Consider a closed manifold $M$ with two Riemannian metrics: one hyperbolic metric, and one other metric $g$. What hypotheses on $g$ guarantee that for a given radius $r$, there are balls of radius $r$ in the universal cover of $(M, g)$ with greather-than-hyperbolic volumes? We show that this conclusion holds for all $r \geq 1$ if $(\mathrm{Vol} (M, g))^2$ is less than a small constant times the hyperbolic volume of $M$. This strengthens a theorem of Sabourau and is partial progress toward a conjecture of Guth.

math.DG↗

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

Simplicial volume and 0-strata of separating filtrations

We use Papasoglu's method of area-minimizing separating sets to give an alternative proof, and explicit constants, for the following theorem of Guth and Braun--Sauer: If $M$ is a closed, oriented, $n$-dimensional manifold, with a Riemannian metric such that every ball of radius $1$ in the universal cover of $M$ has volume at most $V_1$, then the simplicial volume of $M$ is at most the volume of $M$ times a constant depending on $n$ and $V_1$.

math.DG↗

arXiv2022

Asymptotic Betti numbers for hard squares in the homological liquid regime

We study configuration spaces $C(n; p, q)$ of $n$ ordered unit squares in a $p$ by $q$ rectangle. Our goal is to estimate the Betti numbers for large $n$, $j$, $p$, and $q$. We consider sequences of area-normalized coordinates, where $(\frac{n}{pq}, \frac{j}{pq})$ converges as $n$, $j$, $p$, and $q$ approach infinity. For every sequence that converges to a point in the "feasible region" in the $(x,y)$-plane, we show that the factorial growth rate of the Betti numbers is the same as the factorial growth rate of $n!$. This implies that (1) the Betti numbers are vastly larger than for the configuration space of $n$ ordered points in the plane, which have the factorial growth rate of $j!$, and (2) every point in the feasible region is eventually in the homological liquid regime.

math.AT↗

arXiv2022

Systolic almost-rigidity modulo 2

No power law systolic freedom is possible for the product of mod $2$ systoles of dimension $1$ and codimension $1$. This means that any closed $n$-dimensional Riemannian manifold $M$ of bounded local geometry obeys the following systolic inequality: the product of its mod $2$ systoles of dimensions $1$ and $n-1$ is bounded from above by $c(n,\varepsilon) \mbox{Vol}(M)^{1+\varepsilon}$, if finite (if $H_1(M; \mathbb{Z}/2)$ is non-trivial).

math.DG↗

arXiv2021

Macroscopic scalar curvature and codimension 2 width

We show that a complete $3$-dimensional Riemannian manifold $M$ with finitely generated first homology has macroscopic dimension $1$ if it satisfies the following "macroscopic curvature" assumptions: every ball of radius $10$ in $M$ has volume at most $4$, and every loop in every ball of radius $1$ in $M$ is null-homologous in the concentric ball of radius $2$.

math.DG↗

arXiv2021

Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories

We give $\mathbb{Z}$-bases for the homology and cohomology of the configuration space $\operatorname{config}(n,w)$ of $n$ unit disks in an infinite strip of width $w$, first studied by Alpert, Kahle and MacPherson. We also study the way these spaces evolve both as $n$ increases (using the framework of representation stability) and as $w$ increases (using the framework of persistent homology). Finally, we include some results about the cup product in the cohomology and about the configuration space of unordered disks.

math.AT↗

arXiv2021

Thickness of skeletons of arithmetic hyperbolic orbifolds

We show that closed arithmetic hyperbolic n-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic n-orbifolds and find nearly optimal geodesic triangulations of arithmetic hyperbolic n-orbifolds.

math.DG↗

arXiv2020

Homology of configuration spaces of hard squares in a rectangle

We study ordered configuration spaces $C(n;p,q)$ of $n$ hard squares in a $p \times q$ rectangle, a generalization of the well-known "15 Puzzle". Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which $n$, $j$, $p$, and $q$ the homology group $H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for "large-scale" nontrivial homology.

math.AT↗

arXiv2020

Generalized representation stability for disks in a strip and no-k-equal spaces

For fixed j and w, we study the j-th homology of the configuration space of n labeled disks of width 1 in an infinite strip of width w. As n grows, the homology groups grow exponentially in rank, suggesting a generalized representation stability as defined by Church--Ellenberg--Farb and Ramos. We prove this generalized representation stability for the strip of width 2, leaving open the case of w > 2. We also prove it for the configuration space of n labeled points in the line, of which no k are equal.

math.AT↗

arXiv2020

Configuration spaces of disks in an infinite strip

We study the topology of the configuration spaces $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j [C(n,w)]$ behaves in qualitatively different ways. We show that if $w \ge j+2$, then the homology $H_j[C(n, w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[C(n, \mathbb{R}^2)]$. The Betti numbers of $C(n, \mathbb{R}^2) $ were computed by Arnold, and so as a corollary of the isomorphism, $β_j[C(n,w)]$ is a polynomial in $n$ of degree $2j$. On the other hand, we show that if $2 \le w \le j+1$, then $β_j [ C(n,w) ]$ grows exponentially with $n$. Most of our work is in carefully estimating $β_j [ C(n,w) ]$ in this regime. We also illustrate, for every $n$, the homological "phase portrait" in the $(w,j)$-plane--- the parameter values where homology $H_j [C(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j [C(n, \mathbb{R}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes.

math.AT↗

arXiv2019

Clocking the formation of today's largest galaxies: Wide field integral spectroscopy of Brightest Cluster Galaxies and their surroundings

The formation and evolution of local brightest cluster galaxies (BCGs) is investigated by determining the stellar populations and dynamics from the galaxy core, though the outskirts and into the intracluster light (ICL). Integral spectroscopy of 23 BCGs observed out to 4 r_e is collected and high signal-to-noise regions are identified. Stellar population synthesis codes are used to determine the age, metallicity, velocity, and velocity dispersion of stars within each region. The intracluster light (ICL) spectra are best modeled with populations that are younger and less metal-rich than those of the BCG cores. The average BCG core age of the sample is 13.3$\pm$ 2.8 Gyr and the average metallicity is [Fe/H] = 0.30 $\pm$ 0.09, whereas for the ICL the average age is 9.2$\pm$3.5 Gyr and the average metallicity is [Fe/H] = 0.18$\pm$0.16. The velocity dispersion profile is seen to be rising or flat in most of the sample (17/23), and those with rising values reach the value of the host cluster's velocity dispersion in several cases. The most extended BCGs are closest to the peak of the cluster's X-ray luminosity. The results are consistent with the idea that the BCG cores and inner regions formed quickly and long ago, with the outer regions and ICL forming more recently, and continuing to assemble through minor merging. Any recent star formation in the BCGs is a minor component, and is associated with the cluster cool core status.

astro-ph.GA↗

arXiv2018

Art gallery problem with rook and queen vision

How many chess rooks or queens does it take to guard all the squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can "see" whichever squares the rook or queen attacks. We show that floor(n/2) rooks or floor(n/3) queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We also prove that finding the minimum number of rooks or the minimum number of queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. We also use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

cs.CC↗

arXiv2017

Macroscopic stability and simplicial norms of hypersurfaces

We introduce a $\mathbb{Z}$--coefficient version of Guth's macroscopic stability inequality for almost-minimizing hypersurfaces. In manifolds with a lower bound on macroscopic scalar curvature, we use the inequality to prove a lower bound on areas of hypersurfaces in terms of the Gromov simplicial norm of their homology classes. We give examples to show that a very positive lower bound on macroscopic scalar curvature does not necessarily imply an upper bound on the areas of minimizing hypersurfaces.

math.DG↗

arXiv2017

Macroscopic scalar curvature and areas of cycles

In this paper we prove the following. Let $Σ$ be an $n$--dimensional closed hyperbolic manifold and let $g$ be a Riemannian metric on $Σ\times \mathbb{S}^1$. Given an upper bound on the volumes of unit balls in the Riemannian universal cover $(\widetilde{Σ\times \mathbb{S}^1},\widetilde{g})$, we get a lower bound on the area of the $\mathbb{Z}_2$--homology class $[Σ\times \ast]$ on $Σ\times \mathbb{S}^1$, proportional to the hyperbolic area of $Σ$. The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar curvature.

math.DG↗

arXiv2017

Discrete configuration spaces of squares and hexagons

We consider generalizations of the familiar fifteen-piece sliding puzzle on the 4 by 4 square grid. On larger grids with more pieces and more holes, asymptotically how fast can we move the puzzle into the solved state? We also give a variation with sliding hexagons. The square puzzles and the hexagon puzzles are both discrete versions of configuration spaces of disks, which are of interest in statistical mechanics and topological robotics. The combinatorial theorems and proofs in this paper suggest followup questions in both combinatorics and topology, and may turn out to be useful for proving topological statements about configuration spaces.

math.MG↗

arXiv2016

Restricting cohomology classes to disk and segment configuration spaces

The configuration space of n labeled disks of radius r inside the unit disk is denoted Conf_{n, r}(D^2). We study how the cohomology of this space depends on r. In particular, given a cohomology class of Conf_{n, 0}(D^2), for which r does its restriction to Conf_{n, r}(D^2) vanish? A related question: given the configuration space Seg_{n, r}(D^2) of n labeled, oriented segments of length r, it has a map to (S^1)^n that records the direction of each segment. For which r does this angle map have a continuous section? The paper consists of a collection of partial results, and it contains many questions and conjectures.

math.MG↗