@article{10071, author = {Adams, Henry and Kourimska, Hana and Heiss, Teresa and Percival, Sarah and Ziegelmeier, Lori}, issn = {1088-9477}, journal = {Notices of the American Mathematical Society}, number = {9}, pages = {1511--1514}, publisher = {American Mathematical Society}, title = {{How to tutorial-a-thon}}, doi = {10.1090/noti2349}, volume = {68}, year = {2021}, } @inproceedings{10367, abstract = {How information is created, shared and consumed has changed rapidly in recent decades, in part thanks to new social platforms and technologies on the web. With ever-larger amounts of unstructured and limited labels, organizing and reconciling information from different sources and modalities is a central challenge in machine learning. This cutting-edge tutorial aims to introduce the multimodal entailment task, which can be useful for detecting semantic alignments when a single modality alone does not suffice for a whole content understanding. Starting with a brief overview of natural language processing, computer vision, structured data and neural graph learning, we lay the foundations for the multimodal sections to follow. We then discuss recent multimodal learning literature covering visual, audio and language streams, and explore case studies focusing on tasks which require fine-grained understanding of visual and linguistic semantics question answering, veracity and hatred classification. Finally, we introduce a new dataset for recognizing multimodal entailment, exploring it in a hands-on collaborative section. Overall, this tutorial gives an overview of multimodal learning, introduces a multimodal entailment dataset, and encourages future research in the topic.}, author = {Ilharco, Cesar and Shirazi, Afsaneh and Gopalan, Arjun and Nagrani, Arsha and Bratanič, Blaž and Bregler, Chris and Liu, Christina and Ferreira, Felipe and Barcik, Gabriek and Ilharco, Gabriel and Osang, Georg F and Bulian, Jannis and Frank, Jared and Smaira, Lucas and Cao, Qin and Marino, Ricardo and Patel, Roma and Leung, Thomas and Imbrasaite, Vaiva}, booktitle = {59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts}, isbn = {9-781-9540-8557-2}, location = {Bangkok, Thailand}, pages = {29--30}, publisher = {Association for Computational Linguistics}, title = {{Recognizing multimodal entailment}}, doi = {10.18653/v1/2021.acl-tutorials.6}, year = {2021}, } @article{10608, abstract = {We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.}, author = {Weighill, Thomas and Yamauchi, Takamitsu and Zava, Nicolò}, issn = {2199-6768}, journal = {European Journal of Mathematics}, publisher = {Springer Nature}, title = {{Coarse infinite-dimensionality of hyperspaces of finite subsets}}, doi = {10.1007/s40879-021-00515-3}, year = {2021}, } @inproceedings{9296, abstract = { matching is compatible to two or more labeled point sets of size n with labels {1,…,n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.}, author = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit}, booktitle = {15th International Conference on Algorithms and Computation}, isbn = {9783030682101}, issn = {16113349}, location = {Yangon, Myanmar}, pages = {221--233}, publisher = {Springer Nature}, title = {{On compatible matchings}}, doi = {10.1007/978-3-030-68211-8_18}, volume = {12635}, year = {2021}, } @article{9465, abstract = {Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.}, author = {Edelsbrunner, Herbert and Nikitenko, Anton and Osang, Georg F}, issn = {14208997}, journal = {Journal of Geometry}, number = {1}, publisher = {Springer Nature}, title = {{A step in the Delaunay mosaic of order k}}, doi = {10.1007/s00022-021-00577-4}, volume = {112}, year = {2021}, } @inproceedings{9345, abstract = {Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction.}, author = {Edelsbrunner, Herbert and Heiss, Teresa and Kurlin , Vitaliy and Smith, Philip and Wintraecken, Mathijs}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, issn = {1868-8969}, location = {Virtual}, pages = {32:1--32:16}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The density fingerprint of a periodic point set}}, doi = {10.4230/LIPIcs.SoCG.2021.32}, volume = {189}, year = {2021}, } @inproceedings{9604, abstract = {Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.}, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {Leibniz International Proceedings in Informatics}, isbn = {9783959771849}, issn = {18688969}, location = {Online}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Counting cells of order-k voronoi tessellations in ℝ3 with morse theory}}, doi = {10.4230/LIPIcs.SoCG.2021.16}, volume = {189}, year = {2021}, } @inproceedings{9824, abstract = {We define a new compact coordinate system in which each integer triplet addresses a voxel in the BCC grid, and we investigate some of its properties. We propose a characterization of 3D discrete analytical planes with their topological features (in the Cartesian and in the new coordinate system) such as the interrelation between the thickness of the plane and the separability constraint we aim to obtain.}, author = {Čomić, Lidija and Zrour, Rita and Largeteau-Skapin, Gaëlle and Biswas, Ranita and Andres, Eric}, booktitle = {Discrete Geometry and Mathematical Morphology}, isbn = {9783030766566}, issn = {16113349}, location = {Uppsala, Sweden}, pages = {152--163}, publisher = {Springer Nature}, title = {{Body centered cubic grid - coordinate system and discrete analytical plane definition}}, doi = {10.1007/978-3-030-76657-3_10}, volume = {12708}, year = {2021}, } @article{8317, abstract = {When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.}, author = {Aichholzer, Oswin and Akitaya, Hugo A. and Cheung, Kenneth C. and Demaine, Erik D. and Demaine, Martin L. and Fekete, Sándor P. and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane}, issn = {09257721}, journal = {Computational Geometry: Theory and Applications}, publisher = {Elsevier}, title = {{Folding polyominoes with holes into a cube}}, doi = {10.1016/j.comgeo.2020.101700}, volume = {93}, year = {2021}, } @article{8773, abstract = {Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.}, author = {Brown, Adam and Romanov, Anna}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, keywords = {Applied Mathematics, General Mathematics}, number = {1}, pages = {37--52}, publisher = {American Mathematical Society}, title = {{Contravariant forms on Whittaker modules}}, doi = {10.1090/proc/15205}, volume = {149}, year = {2021}, }