The arrangement of point configurations in metric spaces, whether deterministic or random, is a truly interdisciplinary topic of great interest in mathematics, physics and computer science. Mathematical aspects involve optimization, discretization of manifolds, best packing and cubature, among others. For physics, such configurations arise in the study of crystallization, point processes connected with random matrices, self-assembling materials, jammed states, hyperuniformity and phase transitions. For computer science, extremal point configurations play a fundamental role in coding and information theory, and lattice-based protocols in cryptography and related computational complexity issues are of growing importance. Furthermore, there has been recent and substantial progress on related age-old problems (such as the Kepler conjecture).
The investigation of the above topics often evolves from the development of efficient computational methods that enable extensive numerical experiments. In turn, these experiments suggest conjectures that become the focus of rigorous theorem proving, which may also require some computer assistance. The program topics include random point configurations, computation and optimization of energy, packing and covering, multi-pole methods, sparsity, and frames, and the theory of lattices with applications to coding and cryptography.
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Publications
Eva Bayer-Fluckiger, Cyclotomic Fields Are Characterized by Their Trace Lattices, Mathematical Notes 107 (2020), 844-846.
Matt Sourisseau, Yu Guang Wang, Hau-Tieng Wu, Wei-Hsuan Yu, Optimization-based quasi-uniform spherical t-design and generalized multitaper for complex physiological time series, arXiv preprint arXiv:1907.13493 (2019).
P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff, M. M. Stoyanova, Bounds for spherical codes: The Levenshtein framework lifted, Mathematics of Computation 90 (2021) no. 329, 1323–1356.
Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, Maya Stoyanova, On polarization of spherical codes and designs, boyvalenkov2022polarization:2207.08807, 2022.
Djalil Chafaï, Edward B. Saff, Robert S. Womersley, On the solution of a Riesz equilibrium problem and integral identities for special functions, Journal of Mathematical Analysis and Applications 515 (2022) no. 1, 126367.
Peter D. Dragnev, Alan R. Legg, Edward B. Saff, Point Source Equilibrium Problems with Connections to Weighted Quadrature Domains, dragnev2022point:2203.09421, 2022.
Peter Dragnev, Ramon Orive, Edward B. Saff, Franck Wielonsky, Riesz energy problems with external fields and related theory, dragnev2022riesz:2104.03733, 2022.
Djalil Chafaï, Edward B. Saff, Robert S. Womersley, Threshold condensation to singular support for a Riesz equilibrium problem, Analysis and Mathematical Physics 13 (2023) no. 1.
Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, Maya Stoyanova, Universal minima of discrete potentials for sharp spherical codes, boyvalenkov2023universal:2211.00092, 2023.
Matthew de Courcy-Ireland, Suresh Eswarathasan, Effective bounds for monochromatic connectivity measures in two dimensions, Nonlinearity 34 (2021) no. 10, 7144.
Rafael D{\'\i}az Fuentes, Marco Donatelli, Caterina Fenu, Giorgio Mantica, Estimating the trace of matrix functions with application to complex networks, Numerical Algorithms 92 (2023) no. 1, 503-522.
Sarth Chavan, Christophe Vignat, Evaluation of multivariate integrals based on a duality identity for the Stieltjes transform, arXiv preprint arXiv:2105.11771 (2021).
Maria Dostert, David De Laat, Philippe Moustrou, Exact semidefinite programming bounds for packing problems, SIAM Journal on Optimization 31 (2021) no. 2, 1433-1458.
Christian Hess, Raffaello Seri, Generic consistency for approximate stochastic programming and statistical problems, SIAM Journal on Optimization 29 (2019) no. 1, 290-317.
Eva Bayer-Fluckiger, Isometries of lattices and automorphisms of K3 surfaces, arXiv preprint arXiv:2107.07583 (2021).
Peter D Dragnev, Oleg R Musin, Log-optimal (d+ 2)-configurations in d-dimensions, arXiv preprint arXiv:1909.09909 (2019).
Raffaello Seri, Mario Martinoli, Davide Secchi, Samuele Centorrino, Model calibration and validation via confidence sets, Econometrics and Statistics 20 (2021), 62-86.
Maria Dostert, Frank Vallentin, New dense superball packings in three dimensions, Advances in Geometry 20 (2020) no. 4, 473-482.
Matthew de Courcy-Ireland, Maria Dostert, Maryna Viazovska, Six-dimensional sphere packing and linear programming, arXiv preprint arXiv:2211.09044 (2022).
Uju{\'e} Etayo, Spherical cap discrepancy of the diamond ensemble, Discrete \& Computational Geometry 66 (2021) no. 4, 1218-1238.
Tanay Wakhare, Christophe Vignat, Structural properties of multiple zeta values, International Journal of Number Theory 17 (2021) no. 08, 1873-1897.
Carlos Beltr{\'a}n, Uju{\'e} Etayo, The Diamond ensemble: a constructive set of points with small logarithmic energy, arXiv preprint arXiv:1809.09416 (2018).
Fabr{\'\i}cio Caluza Machado, Sinai Robins, The null set of a polytope, and the Pompeiu property for polytopes, arXiv preprint arXiv:2104.01957 (2021).
Ali Abdolali, Tyler J Hesser, Mary Anderson Bryant, Aron Roland, Arslaan Khalid, Jane Smith, Celso Ferreira, Avichal Mehra, Mathieu Dutour Sikiric, Wave Attenuation by Vegetation: Model Implementation and Validation Study, Frontiers in Built Environment (2022), 88.
Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, Maya Stoyanova, Next levels universal bounds for spherical codes: the Levenshtein framework lifted, arXiv preprint arXiv:1906.03062 (2019).
PG Boyvalenkov, PD Dragnev, DP Hardin, EB Saff, MM Stoyanova, On polarization of spherical codes and designs, Journal of Mathematical Analysis and Applications 524 (2023) no. 1, 127065.
Djalil Chafa{\"\i}, Edward B Saff, Robert S Womersley, On the solution of a Riesz equilibrium problem and integral identities for special functions, Journal of Mathematical Analysis and Applications 515 (2022) no. 1, 126367.
Peter D Dragnev, Alan R Legg, Edward B Saff, Point source equilibrium problems with connections to weighted quadrature domains, Indagationes Mathematicae 34 (2023) no. 2, 200-218.
PD Dragnev, Ramon Orive, Edward B Saff, Franck Wielonsky, Riesz energy problems with external fields and related theory, Constructive Approximation 57 (2023) no. 1, 1-43.
Djalil Chafa{\"\i}, Edward B Saff, Robert S Womersley, Threshold condensation to singular support for a Riesz equilibrium problem, Analysis and Mathematical Physics 13 (2023) no. 1, 19.
Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, Maya Stoyanova, Universal minima of discrete potentials for sharp spherical codes, arXiv preprint arXiv:2211.00092 (2022).
Sinai Robins, A friendly introduction to Fourier analysis on polytopes, arXiv preprint arXiv:2104.06407 (2021).
Carlos Beltr{\'a}n, Uju{\'e} Etayo, Jordi Marzo, Joaquim Ortega-Cerd{\`a}, A sequence of polynomials with optimal condition number, Journal of the American Mathematical Society 34 (2021) no. 1, 219-244.
Mathieu Dutour Sikiri{\'c}, Achill Sch{\"u}rmann, Frank Vallentin, A simplex algorithm for rational cp-factorization, Mathematical Programming 187 (2021) no. 1-2, 25-45.
Sergiy Borodachov, Absolute Minima of Potentials of a Certain Class of Spherical Designs, arXiv preprint arXiv:2212.04594 (2022).
Sergiy Borodachov, Absolute minima of potentials of certain regular spherical configurations, arXiv preprint arXiv:2210.04295 (2022).
Eva Bayer-Fluckiger, Alexander polynomials and signatures of some high-dimensional knots, arXiv preprint arXiv:2202.03027 (2022).
Zachary P Bradshaw, Christophe Vignat, An Operational Calculus Generalization of Ramanujan's Master Theorem, Journal of Mathematical Analysis and Applications (2023), 127029.
Giorgio Mantica, Asymptotic behavior of Jacobi matrices of IFS: a long-standing conjecture, Analysis, Probability and Mathematical Physics on Fractals, World Scientific, 2020, pp. 243-271.
Raffaello Seri, Asymptotic Distributions of Covering and Separation Measures on the Hypersphere, Discrete \& Computational Geometry 69 (2023) no. 2, 285-299.
Raffaello Seri, Mario Martinoli, Asymptotic properties of the plug-in estimator of the discrete entropy under dependence, IEEE Transactions on Information Theory 67 (2021) no. 12, 7659-7683.
Eva Bayer-Fluckiger, Automorphisms of K3 surfaces, signatures, and isometries of lattices, arXiv preprint arXiv:2209.06698 (2022).
Alexander Barg, Peter Boyvalenkov, Maya Stoyanova, Bounds for the sum of distances of spherical sets of small size, Discrete Mathematics 346 (2023) no. 5, 113346.
Raffaello Seri, Computing the asymptotic distribution of second-order U-and V-statistics, Computational Statistics \& Data Analysis 174 (2022), 107437.
Giorgio Mantica, CT--scans of fractal and non fractal measures in the plane coded by affine homogeneous iterated function systems, Communications in Nonlinear Science and Numerical Simulation 91 (2020), 105410.
Parth Chavan, Sarth Chavan, Christophe Vignat, Tanay Wakhare, Dirichlet series under standard convolutions: variations on Ramanujan’s identity for odd zeta values, The Ramanujan Journal 59 (2022) no. 4, 1245-1285.
Sergiy Borodachov, Douglas Hardin, Edward Saff, Discrete Energy on Rectifiable Sets, 2019.
Yu Guang Wang, Robert S. Womersley, Hau-Tieng Wu, Wei-Hsuan Yu, Numerical computation of triangular complex spherical designs with small mesh ratio, arXiv e-prints (2019), arXiv:1907.13493.
Johann S. Brauchart, Peter J. Grabner, Wöden Kusner, Jonas Ziefle, Hyperuniform point sets on the sphere: probabilistic aspects, Monatshefte für Mathematik 192 (2020) no. 4, 763–781.
Karl Dilcher, Armin Straub, Christophe Vignat, Identities for Bernoulli polynomials related to multiple Tornheim zeta functions, Journal of Mathematical Analysis and Applications (2019).
Tanay Wakhare, Christophe Vignat, Multiple zeta values for classical special functions, The Ramanujan Journal 51 (2020).
Maria Dostert, Frank Vallentin, New dense superball packings in three dimensions, Advances in Geometry 20 (2020) no. 4, 473–482.
Eiichi Bannai, Mikio Nakahara, Da Zhao, Yan Zhu, On the explicit constructions of certain unitary t-designs, Journal of Physics A: Mathematical and Theoretical 52 (2019) no. 49, 495301.
Mathieu Dutour Sikirić, Alexey Garber, Alexander Magazinov, On the Voronoi Conjecture for Combinatorially Voronoi Parallelohedra in Dimension 5, SIAM Journal on Discrete Mathematics 34 (2020) no. 4, 2481–2501.
Mathieu Dutour Sikirić, Alexey Garber, Periodic Triangulations of $\mathbbZ^n$, The Electronic Journal of Combinatorics 27 (2020) no. 2.
Giorgio Mantica, Vilmos Totik, Regularity of Minkowski's question mark measure, its inverse and a class of IFS invariant measures: REGULARITY OF MINKOWSKI'S QUESTION MARK MEASURE, Journal of the London Mathematical Society 99 (2018).
Alexey Glazyrin, Josiah Park, Repeated Minimizers of $p$-Frame Energies, SIAM Journal on Discrete Mathematics 34 (2020) no. 4, 2411–2423.
Zhongyang Li, Schur Function at General Points and Limit Shape of Perfect Matchings on Contracting Square Hexagon Lattices with Piecewise Boundary Conditions, International Mathematics Research Notices (2021).
Renaud Coulangeon, Gabriele Nebe, Slopes of Euclidean lattices, tensor product and group actions, Israel Journal of Mathematics 235 (2019), 39-61.
Yu Guang Wang, Xiaosheng Zhuang, Tight framelets on graphs for multiscale data analysis, 2019, pp. 11.
Peter J. Grabner, Tetiana A. Stepanyuk, Upper and lower estimates for numerical integration errors on spheres of arbitrary dimension, Journal of Complexity 53 (2019), 113–132.
Peter Boyvalenkov, Peter Dragnev, D. Hardin, Edward Saff, Maya Stoyanova, Linear Programming Bounds for Cardinality and Energy of Codes of Given Min and Max Distances, 2019.
P. Boyvalenkov, Peter D. Dragnev, Douglas P. Hardin, Edward B. Saff, Maya Stoyanova, Bounds for spherical codes: The Levenshtein framework lifted, Math. Comput. 90 (2021), 1323-1356.
P. Boyvalenkov, Peter D. Dragnev, Douglas P. Hardin, Edward B. Saff, Maya Stoyanova, On spherical codes with inner products in a prescribed interval, Designs, Codes and Cryptography 87 (2019), 299-315.
Peter Boyvalenkov, Peter D. Dragnev, Douglas P. Hardin, Edward B. Saff, Maya Stoyanova, Universal Bounds for Size and Energy of Codes of Given Minimum and Maximum Distances, IEEE Transactions on Information Theory 67 (2021), 3569-3584.
Peter Boyvalenkov, Peter D. Dragnev, Douglas P. Hardin, Edward B. Saff, Maya Stoyanova, Upper bounds for energies of spherical codes of given cardinality and separation, Designs, Codes and Cryptography (2020), 1-16.
Eva Bayer-Fluckiger, Isometries of lattices and Hasse principles, arXiv 2001.07094 (2020).
Quoc Thong Le Gia, Ian H. Sloan, Robert S. Womersley, Yu Guang Wang, Sparse Isotropic Regularization for Spherical Harmonic Representations of Random Fields on the Sphere, arXiv 1801.03212 (2018).
Alexey A. Glazyrin, Moments of isotropic measures and optimal projective codes, arXiv 1904.11159 (2019).
Jordan Stoyanov, Christophe Vignat, Nonconventional limits of random sequences related to partitions of integers, arXiv 1901.04029 (2019), 1.
James Gary Propp, One-Dimensional Packing: Maximality Implies Rationality, arXiv 1704.08785 (2017).
Dmitriy Bilyk, Alexey A. Glazyrin, Ryan Matzke, Josiah Park, O. V. Vlasiuk, Optimal measures for p-frame energies on spheres, arXiv 1908.00885 (2019).
Zheng Ma, Ming Li, Yuguang Wang, PAN: Path Integral Based Convolution for Deep Graph Neural Networks, arXiv 1904.10996 (2019).
Matthew de Courcy-Ireland, Shrinking Scale Equidistribution for Monochromatic Random Waves on Compact Manifolds, arXiv 1902.05271 (2019).
Alexander Kolpakov, Sinai Robins, Spherical Pythagorean triples, and volume rationality of spherical tetrahedra, arXiv 1811.06598 (2018).
Tanay V. Wakhare, Christophe Vignat, Structural properties of multiple zeta values, arXiv 1809.07492 (2018).
Tanay V. Wakhare, Christophe Vignat, Taylor coefficients of the Jacobi \$\theta\_\3\\left( q \right)\$ function, arXiv 1909.01508 (2019).
Douglas P. Hardin, Mircea Petrache, Edward B. Saff, Unconstrained Polarization (Chebyshev) Problems: Basic Properties and Riesz Kernel Asymptotics, arXiv 1902.08497 (2019).
Eiichi Bannai, Gabriel Navarro, Noelia Rizo, Pham Huu Tiep, Unitary \$t\$-groups, arXiv 1810.02507 (2018).
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo V. Radchenko, Maryna S. Viazovska, Universal optimality of the \$E\_8\$ and Leech lattices and interpolation formulas, arXiv 1902.05438 (2019).
Mathieu Dutour Sikiri'c, Philippe Elbaz-Vincent, Alexander Kupers, Jacques Martinet, VORONOI COMPLEXES IN HIGHER DIMENSIONS, COHOMOLOGY OF \$GL\_N (Z)\$ FOR \$N\ge 8\$ AND THE TRIVIALITY OF \$K\_8 (Z)\$, arXiv 1910.11598 (2019).
Tanay Wakhare, Christophe Vignat, Quang-Nhat Le, Sinai Robins, A Continuous Analogue of Lattice Path Enumeration, The Electronic Journal of Combinatorics 26 (2017).
Mathieu Dutour Sikirić, Achill Schürmann, Frank Vallentin, A simplex algorithm for rational cp-factorization, Mathematical Programming 187 (2020) no. 1-2, 25–45.
Lin Jiu, Tanay V. Wakhare, Christophe Vignat, Analytic continuation for multiple zeta values using symbolic representations, International Journal of Number Theory (2019).
Mathieu Dutour Sikirić, David A. Madore, Philippe Moustrou, Frank Vallentin, Coloring the Voronoi tessellation of lattices, Journal of the London Mathematical Society 104 (2021) no. 3, 1135–1171.
Peter J. Grabner, Tetiana A. Stepanyuk, Comparison of probabilistic and deterministic point sets on the sphere, Journal of Approximation Theory 239 (2019), 128–143.
A. Alahmadi, Michel Deza, Mathieu Sikiric, Patrick Solé, Covering aspects of the Niemeier lattices, European Journal of Combinatorics 80 (2018).
Alexey Glazyrin, Covering by homothets and illuminating convex bodies, Proceedings of the American Mathematical Society (2021), 1.
Geoffrey R. Grimmett, Zhongyang Li, Cubic graphs and the golden mean, Discret. Math. 343 (2020), 111638.
Shao-Bo Lin, Yu Guang Wang, Ding-Xuan Zhou, Distributed filtered hyperinterpolation for noisy data on the sphere, SIAM J. Numer. Anal. 59 (2021), 634-659.
AHRAM S. FEIGENBAUM, PETER J. GRABNER, DOUGLAS P. HARDIN, Eigenfunctions of the Fourier transform with specified zeros, Mathematical Proceedings of the Cambridge Philosophical Society 171 (2021) no. 2, 329–367.
Dmitriy Bilyk, Alexey A. Glazyrin, Ryan Matzke, Josiah Park, O. V. Vlasiuk, Energy on spheres and discreteness of minimizing measures, Journal of Functional Analysis (2021).
Yen-chi Roger Lin, Wei-Hsuan Yu, Equiangular lines and the Lemmens-Seidel conjecture, Discret. Math. 343 (2020), 111667.
M. Dostert, David de Laat, Philippe Moustrou, Exact Semidefinite Programming Bounds for Packing Problems, SIAM J. Optim. 31 (2021), 1433-1458.
Theophile Caby, Giorgio Mantica, Extreme value theory of evolving phenomena in complex dynamical systems: Firing cascades in a model of a neural network, Chaos: An Interdisciplinary Journal of Nonlinear Science 30 (2020) no. 4, 043118.
Ming Li, Zheng Ma, Yu Guang Wang, Xiaosheng Zhuang, Fast Haar Transforms for Graph Neural Networks, Neural networks : the official journal of the International Neural Network Society 128 (2020), 188-198.
Dmitriy Bilyk, Feng Dai, Stefan Steinerberger, General and refined Montgomery Lemmata, Mathematische Annalen 373 (2018), 1283-1297.
Théophile Caby, Davide Faranda, Giorgio Mantica, Sandro Vaienti, Pascal Yiou, Generalized dimensions, large deviations and the distribution of rare events, Physica D: Nonlinear Phenomena 400 (2019), 132143.
Johann Brauchart, Peter Grabner, Woden Kusner, Hyperuniform Point Sets on the Sphere: Deterministic Aspects, Constructive Approximation 50 (2019).
D. P. Hardin, T. J. Michaels, E. B. Saff, ASYMPTOTIC LINEAR PROGRAMMING LOWER BOUNDS FOR THE ENERGY OF MINIMIZING RIESZ AND GAUSS CONFIGURATIONS, Mathematika 65 (2018) no. 1, 157–180.
P. Boyvalenkov, Peter D. Dragnev, Douglas P. Hardin, Edward B. Saff, Maya Stoyanova, Energy bounds for codes in polynomial metric spaces, Analysis and Mathematical Physics (2019), 1-28.
Jan Hamann, Quoc Thong Le Gia, Ian H. Sloan, Y. G. Wang, Robert S. Womersley, A New Probe of Gaussianity and Isotropy for CMB Maps, arXiv 1911.11442 (2019).
Tanay V. Wakhare, Christophe Vignat, A symbolic approach to the poly-Bernoulli numbers., arXiv 1903.05270 (2019).
Lin Jiu, Christophe Vignat, Connection Coefficients for Higher-order Bernoulli and Euler Polynomials: A Random Walk Approach, arXiv 1809.04636 (2018).
Henry Cohn, N. Triantafillou, Dual linear programming bounds for sphere packing via modular forms, arXiv 1909.04772 (2019).
Ming Li, Philip Broadbridge, Andriy Olenko, Yu Guang Wang, Fast Tensor Needlet Transforms for Tangent Vector Fields on the Sphere, arXiv 1907.13339 (2019).
Quoc Thong Le Gia, Ming Li, Yu Guang Wang, FaVeST: Fast Vector Spherical Harmonic Transforms, arXiv 1908.00041 (2019).
Zhongyang Li, Fluctuations of dimer heights on contracting square-hexagon lattices, arXiv 1809.08727 (2018).
Yu Guang Wang, Ming Li, Zheng Ma, Guido Mont{\'u}far, Xiaosheng Zhuang, Yanan Fan, HaarPooling: Graph Pooling with Compressive Haar Basis, arXiv 1909.11580 (2019).
