Dmitriy Bilyk (He/him)
Lecture 1: Lower bounds for the spherical cap discrepancy, sums of distances, positive definite functions, and Welch bounds: We shall revisit a classical result of J. Beck [A1] which guarantees that the discrepancy of any N-point set in the sphere S^d with respect to all spherical caps is at least of the order N^{-1/2-1/(2d)}. We shall give a completely elementary and self-contained proof of this result [A3], along the way discussing several very interesting topics:
• Stolarsky principle: an identity relating the L^2 spherical cap discrepancy and the sum of Euclidean distances.
• positive definite functions and energy minimization.
• Welch bounds (and their variants): estimates for sums of even powers of inner products between unit vectors.
The argument that we present completely removes harmonic analysis from the proof, while the Fourier approach will be discussed in more detail in the lectures of Bianca Gari- boldi.
Lecture 2: Energy minimization, spherical harmonics, positive definiteness, and the linear programming method: We shall briefly review spherical harmonics and Gegenbauer polynomials and will look at their relation to positive definite kernels and energy minimization. This will lead us to the so-called linear programming method for finding bounds on various geometric objects (spherical codes, spherical designs, equiangular lines etc.). In particular, we shall present the classical proof (due to Delsarte, Goethals, Seidel [A5]) of the lower bound on the cardinality of spherical designs, which will complement the results presented by Ujué Etayo.
Project A.1. Unit norm tight frames, frame energy, equiangular tight frames. This project will concentrate on unit norm tight frames (UNTF’s), an im- portant object in signal processing and functional analysis, and the fact that they can be characterized as energy minimizers (global or local) of the so-called “frame energy” [A2, A6]. This can be viewed as a starting point for the Welch bounds from Lecture 1. Students will also learn about equiangular lines, upper bounds on their cardinality, and equiangular tight frames which achieve these bounds, as well as a peculiar difference be- tween the real setting (which is more restrictive and mysterious) and the complex case (in which such maximal equiangular tight frames, known as SIC-POVM’s, are believed to exist in every dimension, according to famous Zauner’s conjecture).
Project A.2. Universal optimality on the sphere. In this project, students will study a seminal paper of Cohn and Kumar [A4] on universal optimality. This is a very interesting phenomenon: certain highly symmetric point configurations minimize energies simultaneously for a very broad class of potentials (absolutely monotone with respect to the inner product). This result is a tour de force of the linear programming method, combining polynomial interpolation, positive definiteness, orthogonal (Gegen- bauer) polynomials, as well as ideas from discrete geometry. Furthermore, we shall look at some known examples of such minimizers (universally optimal sets), in particular, a very important subclass – tight designs: these are exactly those spherical designs that achieve the Delsarte–Goethals–Seidel bound [A5] from Lecture 2. They enjoy numerous fascinating properties, but very few such configurations are known.
[A1] J. Beck, Sums of distances between points on a sphere—an application of the theory of irregularities of distribution to discrete geometry, Mathematika 31 (1984), 33–41.
[A2] J. J. Benedetto and M. C. Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003), 357–385.
[A3] D. Bilyk and J. Brauchart, On the lower bounds for the spherical cap discrepancy, preprint.
[A4] H. L. Cohn and A. Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), 99–148.
[A5] P. Delsarte, J.-M. Goethals, J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), 363–388.
[A6] D. G. Mixon et al., Three proofs of the Benedetto–Fickus theorem, in Sampling, approximation, and signal analysis—harmonic analysis in the spirit of J. Rowland Higgins, 2023.
Ujué Etayo (She/her)
This course is devoted to the existence and construction of optimal point distributions on the sphere, with particular emphasis on spherical t-designs. The main focus is the breakthrough result of Bondarenko, Radchenko, and Viazovska (Annals of Mathematics, 2013), which proves the existence of spherical t-designs on S^d with N \sim C_d\, t^d points, matching the optimal order predicted by dimensional considerations.
A detailed, step-by-step proof of the BRV theorem will be presented, highlighting three main ingredients:
Regular partitions of the sphere with good geometric control
Marcinkiewicz–Zygmund type inequalities for spherical polynomials
A topological fixed point argument ensuring the existence of exact quadrature formulas
Beyond the classical setting, the course discusses how these ideas can be adapted and generalised, leading to recent results on optimal sampling and quadrature on more general manifolds and function spaces. This includes joint work with Marzo and Ortega-Cerdà, and with Ehler, Gigante, Gariboldi, and Peter.
Project B.1. Fixed Point Theorems in the BRV Framework: Study of Marcinkiewicz–Zygmund inequalities for spaces of spherical polynomials. Students will derive discrete norm equivalences associated with regular partitions of the sphere and analyse the dependence of constants on polynomial degree and geometry. Extensions include comparison with sampling results on compact manifolds and connections to work with Marzo and Ortega-Cerdà.
Project B.2. Regular Area Partitions of the Sphere: Investigation of regular equal-area partitions of the sphere used in the BRV argument. Topics include existence, diameter and separation estimates, and the impact of geometric regularity on analytic bounds. Possible extensions involve analogous constructions on compact manifolds or partitions adapted to non-uniform measures.
[B1] A. Bondarenko, D. Radchenko, and M. Viazovska, Optimal asymptotic bounds for spherical designs, Annals of Mathematics 178 (2013), no. 2, 443–452.
[B2] P. Delsarte, J.-M. Goethals, and J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6 (1977), 363–388.
[B3] M. Ehler, U. Etayo, B. Gariboldi, G. Gigante, and T. Peter, Asymptotically optimal cubature formulas on manifolds for prefixed weights, Journal of Approximation Theory 271 (2021), 105632.
[B4] U. Etayo, J. Marzo, and J. Ortega-Cerdà, Asymptotically optimal designs on compact algebraic manifolds, Monatshefte für Mathematik 186 (2018), 235–248.
[B5] J. Marzo, Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics, Journal of Functional Analysis 250 (2007), no. 2, 559–587.
[B6] P. D. Seymour and T. Zaslavsky, Averaging sets: a generalization of mean values and spherical designs, Advances in Mathematics 52 (1984), no. 3, 213–240.
Bianca Gariboldi (she/her)
This course provides an introduction to the theory of irregularities of distribution, starting with a brief overview of classical results in discrepancy theory.
Following Chapter 6 of Montgomery’s book Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, the course first studies squares and disks on the torus, with the goal of deriving lower bounds for discrepancy. A central tool in this analysis is the Cassels lemma.
A detailed proof of the lemma is presented in the setting of the torus, followed by an extension to general manifolds, where it is known as the Cassels–Montgomery lemma. This framework highlights deep connections between discrepancy theory, harmonic analysis, and geometry.
Project C.1. Irregularities of distribution for spherical caps on the sphere (following Montgomery). This project follows Montgomery’s approach to derive lower bounds for the discrepancy of spherical caps on the sphere. A key ingredient is the systematic use of the Cassels–Montgomery lemma. A useful reference is Irregularities of distribution on two-point homogeneous spaces by Brandolini, Gariboldi, and Gigante (2025), where analogous results are proved in a more general setting.
Project C.2. Single Radius Spherical Cap Discrepancy: This project focuses on spherical cap discrepancy on the sphere when the cap radius is fixed. The goal is to understand how restricting to a single radius affects uniformity measurements and how geometric properties interact with discrepancy theory. Relevant references include work by Bilyk, Mastrianni, and Steinerberger (2024), and extensions to compact two-point homogeneous spaces by Brandolini, Gariboldi, Gigante, and Monguzzi (2024).
[C1] J. Beck, W. W. L. Chen, Irregularities of Distribution, Cambridge Tracts in Mathematics, Cambridge University Press (1987).
[C2] D. Bilyk, M. Mastrianni, S. Steinerberger, Single radius spherical cap discrepancy via Gegenbauer-approximable numbers, Adv. Math. 452 (2024).
[C3] L. Brandolini, B. Gariboldi, G. Gigante, On a sharp lemma of Cassels and Montgomery on manifolds, Math. Ann. 379(3–4), 1807–1834 (2021).
[C4] L. Brandolini, B. Gariboldi, G. Gigante, Irregularities of distribution on two-point homogeneous spaces, in The Mathematical Heritage of Guido Weiss, Birkhäuser (2025).
[C5] L. Brandolini, B. Gariboldi, G. Gigante, A. Monguzzi, Single radius spherical cap discrepancy on compact two-point homogeneous spaces, Math. Z. 308 (2024).
C6] B. Gariboldi, G. Gigante, Almost positive kernels on compact Riemannian manifolds, Math. Z. 302(2), 783–801 (2022).
[C7] H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, Vol. 84, AMS (1994).
[C8] E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press (1971).