This thematic semester aims at exploring those areas of topology where the research challenges stem from scientific and engineering problem and computer experiments rather than the intrinsic development of the topology proper. In this context, topology is a toolbox of mathematical results and constructions which impacts and inspires developments in other areas. Born as a supporting discipline, aimed at creating a foundation of intuitive notions immensely useful in differential equations and complex analysis, algebraic topology remains indispensable in many disciplines.
Our goal here is to concentrate on relatively recent areas of research enabled, in particular, by the computational revolution in mathematical discovery.
The past decade has seen a surge of interest in applied topology, a vaguely defined area that became a focus of attraction for several communities neighboring mathematics proper - biology, engineering, computer science, data analysis, to name a few. The central lure of this emerging area is the feeling that the flexibility afforded by the passage from geometric constructions to analysis situs, analysis of (mutual) positions, can be a powerful force in applications. The meta-goal of this project is to contribute to the development of the original view of topology as analysis situs: the extraction of discrete, combinatorial invariants from continuous shapes.
Here is a sample of the topics to be covered: moduli spaces of linkages, hinges and origami; configuration spaces of hard disks and coverings; topological robotics; topology of random simplicial complexes and random varieties; topology in condensed matter; high-dimensional expanders; equipartitions and Tverberg type theorems.
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Publications
Deniz Kutluay, Winding homology of knotoids., arXiv 2002.07871 (2020).
Dusko Jojic, W. Marzantowicz, S. Vrecica, R. Zivaljevi{\'c}, Unavoidable complexes, via an elementary equivariant index theory, Journal of Fixed Point Theory and Applications 22 (2020), 1-12.
Fedor Manin, {\'E}rika Rold{\'a}n, Benjamin Schweinhart, Topology and local geometry of the Eden model, arXiv 2005.12349 (2020).
Greg Malen, 'Erika Rold'an, Topology and Geometry of Crystallized Polyominoes., arXiv 1910.10342 (2019).
Aleksandar Kartelj, Vladimir Filipovi{\'c}, Sinisa T. Vrecica, Rade T. Zivaljevi{\'c}, Topologically sensitive metaheuristics, arXiv 2002.11164 (2020).
Martin Raussen, Pair component categories for directed spaces, Journal of Applied and Computational Topology 4 (2020), 101-139.
Ciprian S. Borcea, Ileana Streinu, New Principles for Auxetic Periodic Design, SIAM Journal on Applied Algebra and Geometry 1 1 (2017), 442-458.
Michael A. Burr, Martin Schmoll, Christian Wolf, On the computability of rotation sets and their entropies, Ergodic Theory and Dynamical Systems 40 (2020), 367 - 401.
Wojciech Chach{\'o}lski, Ran Levi, Roy Meshulam, On the topology of complexes of injective words, Journal of Applied and Computational Topology 4 (2020), 29-44.
Chaim Even-Zohar, Patterns in Random Permutations, Combinatorica 40 (2020) no. 6, 775–804.
Ciprian S. Borcea, Ileana Streinu, Periodic Auxetics: Structure and Design, The Quarterly Journal of Mechanics and Applied Mathematics 71 (2018), 125 - 138.
Ciprian Borcea, Ileana Streinu, Periodic tilings and auxetic deployments, Mathematics and Mechanics of Solids 26 (2020), 108128652094811.
Benjamin Schweinhart, Persistent Homology and the Upper Box Dimension, arXiv 1802.00533 (2021).
Filip D. Jevti{\'c}, Marinko Timotijevi{\'c}, Rade T. Zivaljevi{\'c}, Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles, Discrete \& Computational Geometry 65 (2021), 1275-1286.
Greg Malen, {\'E}rika {Rold{\'a}n}, Polyiamonds Attaining Extremal Topological Properties, Part I, Geombinatorics 30 (2020) no. 1, 14-25.
Magdalena Metlicka, Mojtaba Nouri, Ileana Streinu, Repairing gaps in Kinari-2 for large scale protein and flexibility analysis applications, 2017 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), Kansas City, MO, 2017, pp. 577-580.
Davide Lofano, Andrew Newman, The worst way to collapse a simplex, Israel Journal of Mathematics (2021).
Federico Ardila, Federico Castillo, Jos{\'e} Alejandro Samper, The Topology of the External Activity Complex of a Matroid, Electronic Journal of Combinatorics 23 (2016), P3.8.
Martin Raussen, Strictifying and taming directed paths in Higher Dimensional Automata, Mathematical Structures in Computer Science 31 (2021), 193 - 213.
OB Ericok, JK Mason, Quotient Maps and Configuration Spaces of Hard Disks, arXiv 2101.00780 (2021).
Sergey A. Dyachenko, Vera Mikyoung Hur, Stokes waves with constant vorticity: folds, gaps and fluid bubbles, Journal of Fluid Mechanics 878 (2019), 502–521.
Sergey A. Dyachenko, Vera Mikyoung Hur, Stokes waves with constant vorticity: I. Numerical computation, Studies in Applied Mathematics (2019).
Sergey A. Dyachenko, Vera Mikyoung Hur, Stokes Waves in a Constant Vorticity Flow, Nonlinear Water Waves (2019).
Andrew Newman, Small Simplicial Complexes with Prescribed Torsion in Homology, Discrete \& Computational Geometry (2019), 1-28.
Hannah Alpert, Regina Barnes, Sue Bell, Antonio Mauro, Nir Nevo, N. Tucker, H. Yang, Routing by matching on convex pieces of grid graphs, arXiv 2106.10751 (2021).
Greg Malen, Erika Roldan-Roa, Extremal Topological and Geometric Problems for Polyominoes, The Electronic Journal of Combinatorics 27 (2020).
Federico Castillo, Jos{\'e} Alejandro Samper, Finiteness theorems for matroid complexes with prescribed topology, European Journal of Combinatorics 92 (2021), 103239.
Duvsko Joji'c, Gaiane Panina, Sinivsa T. Vre'cica, Rade T. vZivaljevi'c, Generalized chessboard complexes and discrete Morse theory, arXiv 2003.04018 (2020).
H. Alpert, Generalized representation stability for disks in a strip and no-k-equal spaces, arXiv 2006.01240 (2020).
Filip D. Jevti'c, R. Zivaljevi{\'c}, Generalized Tonnetz and discrete Abel-Jacobi map, arXiv 2002.09184 (2020).
James McInerney, Bryan Gin-ge Chen, Louis Theran, Christian D. Santangelo, D. Zeb Rocklin, Hidden symmetries generate rigid folding mechanisms in periodic origami, Proceedings of the National Academy of Sciences 117 (2020) no. 48, 30252–30259.
Federico Castillo, Jeremy L. Martin, Jos{\'e} Alejandro Samper, Hopf monoids of ordered simplicial complexes, arXiv 2004.07308 (2020).
Ciprian S Borcea, Ileana Streinu, Infinitesimal Periodic Deformations and Quadrics, Symmetry 13 (2021) no. 9, 1719.
A. Costa, Michael Farber, Large Random Simplicial Complexes, III; The Critical Dimension, arXiv 1512.08714 (2015).
Mark Goresky, Lecture notes on sheaves and perverse sheaves, arXiv 2105.12045 (2021).
Margaret M. Bayer, Bennet Goeckner, Marija Milutinovi{\'c}, MANIFOLD MATCHING COMPLEXES, arXiv 1906.03328 (2019).
Marija Jeli'c Milutinovi'c, Helen Jenne, Alex McDonough, Julianne Vega, Matching complexes of trees and applications of the matching tree algorithm., arXiv 1905.10560 (2019).
Henry Adams, Johnathan Bush, Florian Frick, METRIC THICKENINGS, BORSUK–ULAM THEOREMS, AND ORBITOPES, Mathematika 66 (2019) no. 1, 79–102.
Michael J. Catanzaro, Justin M. Curry, Brittany Terese Fasy, Jānis Lazovskis, Greg Malen, Hans Riess, Bei Wang, Matthew Zabka, Moduli spaces of morse functions for persistence, Journal of Applied and Computational Topology 4 (2020) no. 3, 353–385.
Mark Goresky, Morse theory, stratifications and sheaves, Handbook of geometry and topology of singularities. {I}, Springer, Cham, 2020, pp. 275-319.
Irit Dinur, Roy Meshulam, Near Coverings and Cosystolic Expansion - an example of topological property testing, Electronic Colloquium on Computational Complexity 26 (2019), 126.
Pavel M. Lushnikov, Sergey A. Dyachenko, Denis A. Silantyev, New conformal mapping for adaptive resolving of the complex singularities of Stokes wave, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473 (2017) no. 2202, 20170198.
Mojtaba Nouri Bygi, Ileana Streinu, Efficient pebble game algorithms engineered for protein rigidity applications, 2017 IEEE 7th International Conference on Computational Advances in Bio and Medical Sciences (ICCABS), IEEE, 2017, pp. 1-6.
Ciprian S. Borcea, Ileana Streinu, Deforming Diamond, arXiv 1501.03551 (2015).
Ciprian S. Borcea, Ileana Streinu, Auxetic deformations and elliptic curves, Computer Aided Geometric Design 61 (2018), 9-19.
Ciprian S.
and Streinu Borcea, Auxetic Regions in Large Deformations of Periodic Frameworks, Interdisciplinary Applications of Kinematics, Springer International Publishing, Cham, 2019, pp. 197-204.
Ciprian S. Borcea, Ileana Streinu, Auxetics Abounding, 2018 International Conference on Reconfigurable Mechanisms and Robots (ReMAR), IEEE, 2018, pp. 1-7.
Filip D Jevti{\'c}, Rade T {\v{Z}}ivaljevi{\'c}, Bier spheres of extremal volume and generalized permutohedra, 2108.00618 (2021).
Sakura Kawano, Jeremy K. Mason, Classification of atomic environments via the Gromov–Wasserstein distance, Computational Materials Science 188 (2021), 110144.
Matthew Kahle, Frank H. Lutz, Andrew Newman, Kyle Parsons, Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes, Experimental Mathematics 29 (2020), 347 - 359.
Greg Malen, Collapsibility of Random Clique Complexes, arXiv 1903.05055 (2019).
Duvsko Joji'c, Gaiane Panina, Rade T. vZivaljevi'c, Colored Tverberg problem, extensions and new results., arXiv 2002.09186 (2020).
Hannah Alpert, Matthew Kahle, R. D. Macpherson, Configuration spaces of disks in an infinite strip, J. Appl. Comput. Topol. 5 (2021), 357-390.
O. B. Ericok, K. Ganesan, J. K. Mason, Configuration spaces of hard spheres, Physical Review E 104 (2021) no. 5.
Hannah Alpert, Ulrich Bauer, Matthew Kahle, R. D. Macpherson, Kelly Spendlove, Configuration spaces of hard squares in a rectangle., arXiv 2010.14480 (2020).
Filip D. Jevti{\'c}, Marija Jeli{\'c}, Rade T. Zivaljevi{\'c}, Cyclohedron and Kantorovich–Rubinstein Polytopes, Arnold Mathematical Journal 4 (2017), 87-112.
Imre B{\'a}r{\'a}ny, Roy Meshulam, Eran Nevo, Martin Tancer, Pach’s Selection Theorem Does Not Admit a Topological Extension, Discrete \& Computational Geometry 60 (2018), 420-429.
Martin Loebl, Jaroslav Ne\v{s}et\v{r}il, Robin Thomas, A journey through discrete mathematics, Springer, Cham, 2017.
Chaim Even-Zohar, Models of random knots, Journal of Applied and Computational Topology 1 (2017) no. 2, 263–296.
Chelsea E. Knittel, Michael Tanis, Amy L. Stoltzfus, Toen Castle, Randall D. Kamien, Genevi{\`e}ve Dion, Modelling textile structures using bicontinuous surfaces, Journal of Mathematics and the Arts 14 (2018), 331 - 344.
Yoav Kallus, Miranda Holmes-Cerfon, Free energy of singular sticky-sphere clusters, Physical Review E 95 (2017) no. 2.
Benjamin Zimmerman, Ivan Abraham, Sara Schmidt, Yuliy Baryshnikov, Fatima Husain, Dissociating Tinnitus Patients from Healthy Controls Using Resting 2 State Cyclicity Analysis and Clustering, Network Neuroscience 3 (2018), 1-36.
Hannah Alpert, Discrete configuration spaces of squares and hexagons, Journal of Applied and Computational Topology 4 (2020), 263-280.
J{\'e}r{\'e}my Cochoy, Steve Oudot, Decomposition of Exact pfd Persistence Bimodules, Discrete \& Computational Geometry 63 (2020), 255-293.
Filip D. Jevti{\'c}, Marija Jeli{\'c}, Rade T. Živaljevi{\'c}, Cyclohedron and Kantorovich–Rubinstein Polytopes, Arnold Mathematical Journal 4 (2017), 87-112.
John D. Wiltshire-Gordon, Models for configuration space in a simplicial complex, Colloquium Mathematicum 155 (2019) no. 1, 127–139.
Marija Jeli{\'c} Milutinovi{\'c}, Dusko Jojic, Marinko Timotijevi{\'c}, Sinisa T. Vrecica, Rade T. Živaljevi{\'c}, Combinatorics of unavoidable complexes, Eur. J. Comb. 83 (2020).
Matthew Kahle, Frank H. Lutz, Andrew Newman, Kyle Parsons, Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes, Experimental Mathematics 29 (2020), 347 - 359.
Steve Oudot, Elchanan Solomon, Barcode embeddings for metric graphs, Algebraic & Geometric Topology 21 (2021) no. 3, 1209–1266.
Thomas Machon, Hillel Aharoni, Yichen Hu, Randall D. Kamien, Aspects of Defect Topology in Smectic Liquid Crystals, Communications in Mathematical Physics 372 (2019) no. 2, 525–542.
Hannah Alpert, {\'E}rika Rold{\'a}n, Art Gallery Problem with Rook and Queen Vision, Graphs Comb. 37 (2021), 621-642.
Lisa Tran, Maxim O. Lavrentovich, Guillaume Durey, Alexandre Darmon, Martin F. Haase, Ningwei Li, Daeyeon Lee, Kathleen J. Stebe, Randall D. Kamien, Teresa Lopez-Leon, Change in Stripes for Cholesteric Shells via Anchoring in Moderation, Physical Review X 7 (2017) no. 4.
Patricia Hersh, Karola M{\'e}sz{\'a}ros, SB-labelings and posets with each interval homotopy equivalent to a sphere or a ball, J. Comb. Theory, Ser. A 152 (2017), 104-120.
Lisa Tran, Hye-Na Kim, Ningwei Li, Shu Yang, Kathleen J. Stebe, Randall D. Kamien, Martin F. Haase, Shaping nanoparticle fingerprints at the interface of cholesteric droplets, Science Advances 4 (2018) no. 10, eaat8597.
Patricia Hersh, Richard W. Kenyon, Shellability of face posets of electrical networks and the CW poset property, Adv. Appl. Math. 127 (2021), 102178.
Elisabetta A Matsumoto, Randall D. Kamien, Gareth P Alexander, Straight round the twist: frustration and chirality in smectics-A, Interface Focus 7 (2017).
Dusko Jojic, Sinisa T. Vrecica, Rade T. Živaljevi{\'c}, Symmetric multiple chessboard complexes and a new theorem of Tverberg type, Journal of Algebraic Combinatorics 46 (2015), 15-31.
Chaim Even-Zohar, Joel Hass, Nathan Linial, Tahl Nowik, The distribution of knots in the Petaluma model, Algebraic & Geometric Topology 18 (2018) no. 6, 3647–3667.
Yuliy Baryshnikov, Robert Ghrist, Stokes's theorem, data, and the polar ice caps, Amer. Math. Monthly 125 (2018) no. 9, 830-834.
Maxim Arnold, Yuliy M. Baryshnikov, Yuriy Mileyko, Typical representatives of free homotopy classes in multi-punctured plane, Journal of Topology and Analysis (2019).
Hillel Aharoni, Yu Xia, Xinyue Zhang, Randall D. Kamien, Shu Yang, Universal inverse design of surfaces with thin nematic elastomer sheets, Proceedings of the National Academy of Sciences 115 (2018) no. 28, 7206–7211.
John D. Wiltshire-Gordon, Configuration space in a product, arXiv 1808.08894 (2018).
Roy Meshulam, Martin Raussen, Homology of Spaces of Directed Paths in Euclidean Pattern Spaces, arXiv 1512.05978 (2017), 593-614.
Yuliy M. Baryshnikov, Robert Ghrist, Minimal Unimodal Decompositions on Trees, arXiv 1806.09673 (2018).
Yuliy M. Baryshnikov, Caroline J. Klivans, Nicholas Kosar, On the topology of no \$k\$-equal spaces, arXiv 1708.00032 (2017).
Matthew Kahle, 'Erika Rold'an, Polyominoes with maximally many holes, arXiv 1807.10231 (2018).
Patricia Hersh, Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology, arXiv 1802.04342 (2018).
Hannah Alpert, Fedor Manin, Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories, arXiv e-prints (2021), arXiv:2107.04574.
