# Hannah Alpert's research while affiliated with Auburn University and other places

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## Publications (23)

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 conver...

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,\varep...

The routing number is a graph invariant introduced by Alon, Chung, and Graham in 1994, and it has been studied for trees and other classes of graphs such as hypercubes. It gives the minimum number of routing steps needed to sort a set of distinct tokens, placed one on each vertex, where each routing step swaps a set of disjoint pairs of adjacent to...

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 o...

We study the topology of the configuration spaces \(\mathcal {C}(n,w)\) of n hard disks of unit diameter in an infinite strip of width w. We describe ranges of parameter or “regimes”, where homology \(H_j [\mathcal {C}(n,w)]\) behaves in qualitatively different ways. We show that if \(w \ge j+2\), then the homology \(H_j[\mathcal {C}(n, w)]\) is is...

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...

The routing number is a graph invariant introduced by Alon, Chung, and Graham in 1994, and it has been studied for trees and other classes of graphs such as hypercubes. It gives the minimum number of routing steps needed to sort a set of distinct tokens, placed one on each vertex, where each routing step swaps a set of disjoint pairs of adjacent to...

We show that closed arithmetic hyperbolic [Formula: see text]-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose [Formula: see text]-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic [Formula: see...

How many chess rooks or queens does it take to guard all 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 $$\lfloor {\frac{n}{2}} \rfloor $$ ⌊ n 2 ⌋ rooks or $$\lfloor {\frac{n}{3}}...

We study the configuration spaces C(n;p,q) of n labeled hard squares in a p by 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 wh...

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 configurat...

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...

We study the topology of the configuration space $C(n,w)$ 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 inclusion $i$ into the configuration space of $n$ points in the pla...

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. A related question: can 1--skeletons of triangulations of a sphere form an expander family?

We introduce rook visibility, which is a new model of visibility for the art gallery problem when the gallery is a polyomino, the union of square tiles from a square grid. We show that floor(n/2) chess rooks are sufficient and sometimes necessary to guard a polyomino with n tiles. We also prove that finding the minimum number of rooks needed to gua...

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...

In this paper we prove the following. Let $\Sigma$ be an $n$--dimensional closed hyperbolic manifold and let $g$ be a Riemannian metric on $\Sigma \times \mathbb{S}^1$. Given an upper bound on the volumes of unit balls in the Riemannian universal cover $(\widetilde{\Sigma\times \mathbb{S}^1},\widetilde{g})$, we get a lower bound on the area of the...

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) o...

Given a closed Riemannian manifold of dimension $n$ and a Morse-Smale
function, there are finitely many $n$-part broken trajectories of the negative
gradient flow. We show that if the manifold admits a hyperbolic metric, then
the number of $n$-part broken trajectories is always at least the hyperbolic
volume. The proof combines known theorems in Mo...

For a non-vanishing gradient-like vector field on a compact manifold
$X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the
boundary with reduced multiplicity $n$, which is the maximum possible. (Among
them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We
prove a lower bound on the number of such traje...

The waist inequality states that for a continuous map from S^n to R^q, not
all fibers can have small (n-q)-dimensional volume. We construct maps for which
most fibers have small (n-q)-dimensional volume and all fibers have bounded
(n-q)-dimensional volume.

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and in this note we show that this problem is NP-hard. We use a reduction from Partial Orientation, a problem recently shown by Pálvölgyi to be...

In 2003, it was claimed that the following problem was solvable in polynomial
time: do there exist k edge-disjoint paths of length exactly 3 between vertices
s and t in a given graph? The proof was flawed, and we show that this problem
is NP-hard even if we disallow multiple edges. We use a reduction from Partial
Orientation, a problem recently sho...

## Citations

... Of course, for "small" this set is connected but very little else is known. Motivated in part by these suggestions, in [AKM21] we studied hard disks in an infinite strip. In that article, we obtained estimates for the rate of growth of the Betti numbers, and also introduced definitions of homological solid, liquid, and gas regimes. ...

... The proof is inspired by Guth's proof of the isosystolic inequality, see [18]. So we assume in the sequel n = 3. Furthermore suppose that we have ordered our cohomological classes such that L(α 1 ...

... A close reading of Guth's proof of the volume theorem yields a proof of Theorem 1.1 for closed aspherical manifolds whose smallest non-contractible loop (systole) is of length at least 1 (see the discussion [1,Section 4]). If the fundamental group is residually finite, then we can pass to a finite cover whose systole is of length at least 1. ...

Reference: Volume and macroscopic scalar curvature

... In this paper, we analyze the weak and strong parity properties, and maximally connectivity of boards with different shapes with hexagonal tiles-see Figure 4 for some examples of these boards. Of the families of hexagonal boards that we study here, only one family, parallelogram shaped boards with hexagonal tiles, have previously been define and studied by H. Alpert [2]. In her paper, Alpert analyzes, with respect of the size of the boards, asymptotically how fast these parallelogram shaped hexagonal sliding puzzles can be solved. ...

Reference: Parity Property of Hexagonal Sliding Puzzles

... (4) Estimates for the Betti numbers of configuration spaces of hard disks in a box (rather than hard squares) would be interesting. These configuration spaces were studied earlier, for example in [BBK14] and [Alp17], but so far little seems to be known. For instance, for n unit disks in a p by q rectangle, can we find values of x = n pq and y = j pq for which we can prove the conclusion of Theorem 1.1? ...

... Application to manifolds. Theorem 1. 2 gives new examples of manifolds that satisfy integral approximation for simplicial volume [14; 16; 29; 36, Section 6]: Corollary 1.4. Let M be an oriented closed connected smooth aspherical manifold with cat Am M ≤ dim M and residually finite fundamental group. ...

... This result extends the classical vanishing theorem by Ivanov [Iva85,Iva20] to the relative setting. By now there are several alternative proofs of Ivanov's result [FM18,LS20,Rap21]; moreover, in the case of aspherical manifolds, the vanishing of simplicial volume can also be obtained directly through the amenable reduction lemma [AK16,FM18,LMS21]. We will follow the approach via classifying spaces by Löh and Sauer [LS20]. ...

... If the function f in Theorem 1.1 is assumed to be merely continuous, then Gromov showed that there exists some t ∈ C k such that the set Z = f −1 (t) satisfies (1). From Theorem 1.1 we thus gather that when f : C n → C k is not only continuous but also holomorphic, it is always possible to select t = f (0) in the Gaussian waist inequality. ...

... The problem is NP-hard even if G + D (the graph obtained by taking the disjoint union of the edge sets) is Eulerian and D consists of at most three set of parallel edges, as shown by Vygen [33]. If no restrictions are made on G, then the problem is NP-hard for one set of parallel edges which should be mapped to edge-disjoint paths of length exactly 3, see [3]. ...