We prove multiple generalizations of Fan's combinatorial labeling result for sphere triangulations. This can be seen as a comprehensive extension of the Borsuk–Ulam theorem. In typical applications, the Borsuk–Ulam theorem gives complexity bounds in a suitable sense, whereas our extension additionally provides insight into the structure of objects satisfying the complexity bound. This structure is governed by order types of finite point sets in Euclidean space and more generally by the intersection combinatorics of faces under continuous maps from the simplex. We develop some of those applications for sphere coverings, Kneser-type colorings, Hall-type results for hypergraphs, and hyperplane mass partitions, among other consequences. We provide a new proof of the topological Hall theorem and extend it into a result that simultaneously generalizes hypergraph Hall theorems and topological lower bounds for chromatic numbers.

We prove a colorful generalization of the Borsuk–Ulam theorem and derive colorful consequences from it, such as a colorful generalization of the ham sandwich theorem. Even in the uncolored case this specializes to a strengthening of the ham sandwich theorem, which given an additional condition, contains a result of Bárány, Hubard, and Jerónimo on well-separated measures as a special case. We prove a colorful generalization of Fan’s antipodal sphere covering theorem, we derive a short proof of Gale’s colorful KKM theorem, and we prove a colorful generalization of Brouwer’s fixed point theorem. Our results also provide an alternative between Radon-type intersection results and KKM-type covering results. Finally, we prove colorful Borsuk–Ulam theorems for higher symmetry.

A landmark result of Erdős, Ginzburg, and Ziv (EGZ) states that any sequence of $2n-1$ elements in ${\mathbb {Z}}/n$ contains a zero-sum subsequence of length n. While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result is a topological criterion for determining when any ${\mathbb {Z}}/n$ -coloring of an n-uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson’s generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we give a fractional generalization of the EGZ theorem with applications to balanced set families and provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.

When quantifying the topological properties of a metric dataset using persistent homology, the first step is to produce a simplicial filtration on the data. The Čech and Vietoris-Rips filtrations are two of the workhorses of persistent homology, but over time, various other filtrations which capture different properties of data or have different computational burdens have been introduced. Towards a program of characterizing all the possible simplicial filtrations on a metric dataset, we introduce and develop the framework of valuation-induced stable filtration functors. This framework is based on the concept of curvature sets due to Gromov, and encapsulates the Vietoris-Rips and various other filtrations while simultaneously providing a model for generating families of novel filtration functors that capture diverse features present in datasets. We further extend this foundation by incorporating the notion of basepoint-dependent filtration functors and proving the associated functoriality and stability properties. This rich theoretical framework provides a unifying language for various extant simplicial filtrations, and is also a mechanism for generating arbitrarily large families of novel filtration functors with control over basepoint dependence/independence as well as the locality of the filtration. We exemplify our constructions on both toy datasets and on 3D shapes from a publicly available shape database. Our paper is accompanied by a Matlab software package incorporating an interactive platform for visualizing and testing new filtrations on datasets.

Recent progress on flow polytopes indicates many interesting families with product formulas for their volume. These product formulas are all proved using analytic techniques. Our work breaks from this pattern. We define a family of closely related flow polytopes $F_{(\lambda, {\bf a})}$ for each partition shape $\lambda$ and netflow vector ${\bf a}\in Z^n_{> 0}$. In each such family, we prove that there is a polytope (the limiting one in a sense) which is a product of scaled simplices, explaining their product volumes. We also show that the combinatorial type of all polytopes in a fixed family $F_{(\lambda, {\bf a})}$ is the same. When $\lambda$ is a staircase shape and ${\bf a}$ is the all ones vector the latter results specializes to a theorem of the first author with Morales and Rhoades, which shows that the combinatorial type of the Tesler polytope is a product of simplices.

We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the Knaster-Kuratowski-Mazurkiewicz (KKM) theorem as well as their variants. We prove extensions of Alon's necklace splitting result in certain regimes and relate it to hyperplane mass partitions. We show the existence of fair cake division and rental harmony in the sense of Su even in the absence of full information. Furthermore, we extend Sperner's lemma and the KKM theorem to (colorful) quantitative versions for polytopes and pseudomanifolds. For simplicial polytopes our results turn out to be improvements over the earlier work of De Loera, Peterson, and Su on a polytopal version of Sperner's lemma. Moreover, our results extend the work of Musin on quantitative Sperner-type results for piecewise linear manifolds.

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

The so called Čech and Vietoris-Rips simplicial filtrations are designed to capture information about the topological structure of metric datasets. These filtrations are two of the workhorses in the field of topological data analysis. They enjoy stability with respect to the Gromov-Hausdorff (GH) distance, and this stability property allows us to estimate the GH distance between finite metric space representations of the underlying datasets. Via the concept of Gromov's curvature sets we construct a rich theoretical framework of valuation-induced stable filtration functors. This framework includes the Čech and Vietoris-Rips filtration functors as well as many novel filtration functors that capture diverse features present in datasets. We further explore the concept of basepoint filtrations functors and use it to provide a classification of the filtration functors that we identify.

Submitted for the DAMOP16 Meeting of The American Physical Society Measurement-induced control using a nondestructive quantum gas microscope1 IVAYLO S. MADJAROV, MINWOO JUNG, JACOB RABINOWITZ, ZOE WELLNER, HUIYAO Y. CHEN, HIL F. H. CHEUNG, YOGESH SHARAD PATIL, MUKUND VENGALATTORE, Cornell University —We present progress toward a quantum gas microscope that extends the paradigm of single-site imaging to one of single-site control. The basis for this scheme is our work on nondestructive lattice imaging and measurement-induced localization [1,2], where we show that lattice dynamics can be influenced by continuous measurement. The combination of nondestructive in situ imaging, single-site resolution, and spatially nonuniform measurement landscapes presents exciting prospects for new experimental studies of quantum thermodynamic processes that utilize information, such as the Szilard engine [3]. We present our recent results towards experimental realizations of such systems. [1] Y. S. Patil et al. PRA 90, 033422 (2014) [2] Y. S. Patil et al. PRL 115, 140402 (2015) [3] S. W. Kim et al. PRL 106, 070401 (2011) 1This work is supported by the ARO MURI on non-equilibrium dynamics. Yogesh Sharad Patil Cornell University Date submitted: 28 Jan 2016 Electronic form version 1.4