The theory, algorithms, and software of linear algebra are familiar tools across mathematics, the applied sciences, and engineering. This ubiquity of linear algebra masks a fairly recent growth of nonlinear algebra in mathematics and its applications to other disciplines. The proliferation of nonlinear algebra has been fueled by recent theoretical advances, efficient implementations of core algorithms, and an increased awareness of these tools.
The benefits of this nonlinear theory and its tools are manifold. Pushing computational boundaries has led to the development of new mathematical theories, such as homotopy methods for numerical algebraic geometry, tropical geometry and toric deformations, and sums of squares methods for polynomial optimization. This uncovered many concrete nonlinear mathematical objects and questions, many of which are ripe for computer experimentation. In turn, resulting mathematical breakthroughs often lead to more powerful and efficient algorithms for computation.
This semester will work towards a time when ideas of nonlinear algebra, its theory, methods, and software are as ubiquitous as those of linear algebra.
Riley Murray, Helen Naumann, Thorsten Theobald, The X-circuits Behind Conditional SAGE Certificates, 2020.
Hannes Bartz, Lukas Holzbaur, Hedongliang Liu, Sven Puchinger, Julian Renner, Antonia Wachter-Zeh, Rank-Metric Codes and Their Applications, bartz2022rankmetriccodesapplications:2203.12384, 2022.
Petr Hruby, Timothy Duff, Anton Leykin, Tomas Pajdla, Learning to Solve Hard Minimal Problems, hruby2021learning:2112.03424, 2021.
Ricardo Fabbri, Timothy Duff, Hongyi Fan, Margaret Regan, David da Costa de Pinho, Elias Tsigaridas, Charles Wampler, Jonathan Hauenstein, Benjamin Kimia, Anton Leykin, Tomas Pajdla, Trifocal Relative Pose from Lines at Points and its Efficient Solution, fabbri2022trifocal:1903.09755, 2022.
Petr Hrub{\'y}, Timothy Duff, Anton Leykin, Tom{\'a}s Pajdla, Learning to Solve Hard Minimal Problems, 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2021), 5522-5532.
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Angelos Mantzaflaris, Bernard Mourrain, Agnes Szanto, Punctual Hilbert scheme and certified approximate singularities, Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation (2020).
Kathl{\'e}n Kohn, Kristian Ranestad, Projective Geometry of Wachspress Coordinates, Found. Comput. Math. 20 (2020), 1135-1173.
Jonathan D. Hauenstein, Margaret H. Regan, Real monodromy action, Appl. Math. Comput. 373 (2020), 124983.
Taylor Brysiewicz, Jose Israel Rodriguez, Frank Sottile, Thomas Yahl, Solving Decomposable Sparse Systems, Numer. Algorithms 88 (2021), 453-474.
Elisa Gorla, Daniela Mueller, Christophe Petit, Stronger bounds on the cost of computing Gröbner bases for HFE systems, Journal of Symbolic Computation 109 (2022), 386–398.
Diego Cifuentes, Thomas Kahle, Pablo Parrilo, Sums of squares in Macaulay2, Journal of Software for Algebra and Geometry 10 (2020) no. 1, 17–24.
Papri Dey, Daniel Plaumann, Testing Hyperbolicity of Real Polynomials, Mathematics in Computer Science 14 (2020), 111-121.
Irénée Briquel, Peter Bürgisser, The real tau‐conjecture is true on average, Random Structures & Algorithms 57 (2020) no. 2, 279–303.
D. Bernstein, Robert Krone, The Tropical Cayley-Menger Variety, SIAM J. Discret. Math. 33 (2019), 1725-1742.
Peter Bürgisser, Cole Franks, Ankit Garg, Rafael Oliveira, Michael Walter, Avi Wigderson, Towards a Theory of Non-Commutative Optimization: Geodesic 1st and 2nd Order Methods for Moment Maps and Polytopes, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), 2019, pp. 845-861.
Daniel Irving Bernstein, Grigoriy Blekherman, Kisun Lee, Typical ranks in symmetric matrix completion, Journal of Pure and Applied Algebra (2021).
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Winfried Bruns, Pedro A. Garc{\'i}a-S{\'a}nchez, Christopher O'Neill, Dane Wilburne, Wilf's conjecture in fixed multiplicity, Int. J. Algebra Comput. 30 (2020), 861-882.
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Riley Murray, Helen Naumann, Thorsten Theobald, Sublinear Circuits and the Constrained Signomial Nonnegativity Problem, arXiv e-prints (2020), arXiv:2006.06811.
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M. Levent Dogan, Alperen Ali Erg{\"u}r, Jake D. Mundo, Elias P. Tsigaridas, The Multivariate Schwartz-Zippel Lemma, arXiv 1910.01095 (2019).
Taylor Brysiewicz, Fulvio Gesmundo, The Degree of Stiefel Manifolds., arXiv 1909.10085 (2019).
Giorgio Ottaviani, Zahra Shahidi, Tensors with eigenvectors in a given subspace, arXiv 2010.03843 (2020).
Giorgio Ottaviani, Philipp Reichenbach, Tensor Rank and Complexity, arXiv 2004.01492 (2020).
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D. Bernstein, Cameron Farnsworth, J. Rodriguez, The algebraic matroid of the funtf variety, arXiv 1812.10353 (2018).
Mario Kummer, Rainer Sinn, Hyperbolic Secant Varieties of M-Curves, arXiv 2002.00486 (2020).
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Taylor Brysiewicz, Newton polytopes and numerical algebraic geometry, arXiv 2004.12177 (2020).
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Michael A. Burr, Frank Sottile, Elise Walker, Numerical homotopies from Khovanskii bases, arXiv 2008.13055 (2020).
Jonathan D. Hauenstein, Ali Mohammad-Nezhad, Tingting Tang, Tam{\'a}s Terlaky, On computing the nonlinearity interval in parametric semidefinite optimization., arXiv 1908.10499 (2019).
Kisun Lee, Nan Li, Lihong Zhi, On isolation of singular zeros of multivariate analytic systems, arXiv 1904.07937 (2019).
Timothy Duff, Kathl{\'e}n Kohn, Anton Leykin, Tom{\'a}s Pajdla, PLMP - Point-Line Minimal Problems in Complete Multi-View Visibility, 2019 IEEE/CVF International Conference on Computer Vision (ICCV) (2019), 1675-1684.
Samantha N. Sherman, Jonathan D. Hauenstein, Charles W. Wampler, Curve Cognate Constructions Made Easy, 10.1115/DETC2020-22409:https://asmedigitalcollection.asme.org/IDETC-CIE/proceedings-pdf/IDETC-CIE2020/83990/V010T10A024/6586833/v010t10a024-detc2020-22409.pdf, 2020.
Lily Silverstein, Dane Wilburne, J. Yang, Asymptotic Degree of Random Monomial Ideals, arXiv 2009.05174 (2020).
Peter Bürgisser, Felipe Cucker, Pierre Lairez, Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time, Journal of the ACM 66 (2019) no. 1, 1–30.
Michael F. Adamer, Martin Helmer, Families of toric chemical reaction networks, Journal of Mathematical Chemistry 58 (2020), 2061 - 2093.
Heather A. Harrington, Dhagash Mehta, Helen M. Byrne, Jonathan D. Hauenstein, Decomposing the Parameter Space of Biological Networks via a Numerical Discriminant Approach, MC, 2019.
Nidhi Kaihnsa, Yue Ren, Mohab Safey El Din, Johannes W. R. Martini, Cooperativity, absolute interaction, and algebraic optimization, Journal of Mathematical Biology 81 (2020), 1169 - 1191.
Tobias Boege, Thomas Kahle, Construction methods for gaussoids, Kybernetika 56 (2020), 1045-1062.
Papri Dey, Stephan Gardoll, Thorsten Theobald, Conic stability of polynomials and positive maps, Journal of Pure and Applied Algebra (2021).
Josu{\'e} Tonelli-Cueto, Elias P. Tsigaridas, Condition numbers for the cube. I: Univariate polynomials and hypersurfaces, Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation (2020).
Peter Bürgisser, Felipe Cucker, Josué Tonelli-Cueto, Computing the Homology of Semialgebraic Sets. II: General Formulas, Foundations of Computational Mathematics 21 (2021) no. 5, 1279–1316.
Peter Bürgisser, Felipe Cucker, Josué Tonelli-Cueto, Computing the Homology of Semialgebraic Sets. I: Lax Formulas, Foundations of Computational Mathematics 20 (2019) no. 1, 71–118.
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Justin Chen, Papri Dey, Computing symmetric determinantal representations, Journal of Software for Algebra and Geometry 10 (2020) no. 1, 9–15.
Mohab Din, Zhi-Hong Yang, Lihong Zhi, Computing real radicals and S-radicals of polynomial systems, Journal of Symbolic Computation 102 (2019).
Oliver Gäfvert, Computational complexity of learning algebraic varieties, Advances in Applied Mathematics 121 (2020), 102100.
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Samantha Sherman, Jonathan Hauenstein, Charles Wampler, A General Method for Constructing Planar Cognate Mechanisms, Journal of Mechanisms and Robotics 13 (2021), 1-18.
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Masoud Gharahi, Stefano Mancini, Giorgio Ottaviani, Fine-structure classification of multiqubit entanglement by algebraic geometry, arXiv 1910.09665 (2020).
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Taylor Brysiewicz, Jose Israel Rodriguez, Frank Sottile, Thomas Yahl, Decomposable sparse polynomial systems, arXiv 2006.03154 (2021).
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Dmitrii V. Pasechnik, An efficient sum of squares nonnegativity certificate for quaternary quartic, arXiv 1511.03473 (2015).
Samantha N. Sherman, A Complete Theory of Planar Cognate Linkages, 2021.
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Yuhan Jiang, Bernd Sturmfels, Bad Projections of the PSD Cone, arXiv 2006.09956 (2020).
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and Kohn Duff, PL\$\$\_1\$\$P - Point-Line Minimal Problems Under Partial Visibility in Three Views, Computer Vision -- ECCV 2020, Springer International Publishing, Cham, 2020, pp. 175-192.
Michael Burr, Kisun Lee, Anton Leykin, Effective Certification of Approximate Solutions to Systems of Equations Involving Analytic Functions, Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (2019).
Jonathan D. Hauenstein, Anton Leykin, Jose Israel Rodriguez, Frank Sottile, A numerical toolkit for multiprojective varieties, Math. Comput. 90 (2021), 413-440.
Peter Burgisser, Alperen A. Ergur, Josu{\'e} Tonelli-Cueto, On the Number of Real Zeros of Random Fewnomials, SIAM J. Appl. Algebra Geom. 3 (2019), 721-732.
Timothy Duff, Michael Ruddy, Numerical Equality Tests for Rational Maps and Signatures of Curves, Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, ISSAC '20, Association for Computing Machinery, New York, NY, USA, 2020, pp. 154–161.
Daniele Agostini, Carlos Am'endola, Kristian Ranestad, Moment Identifiability of Homoscedastic Gaussian Mixtures, Foundations of Computational Mathematics (2021), 1-30.
Daniel Plaumann, Rainer Sinn, Stephan Weis, Kippenhahn's Theorem for Joint Numerical Ranges and Quantum States, SIAM J. Appl. Algebra Geom. 5 (2021), 86-113.
