## Summer 2021

 Speaker Affiliation Title Date Jun 2 Austin Anderson and Alex White Florida State Asymptotics of Best Packing and Best Covering Jun 9 Alan Legg Purdue Fort Wayne Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges Jun 16 Assaf Goldberger Tel Aviv U Configurations, Automorphisms and Cohomology Jun 30 Robert McCann UToronto Maximizing the sum of angles between pairs of lines in Euclidean space Jul 7 Giuseppe Negro U of Birmingham Intermittent symmetry breaking for the maximizers to the Agmon-Hörmander estimate on the sphere Jul 14 Alexander McDonald U of Rochester Volumes spanned by k-point configurations in $\mathbb R^d$ Jul 21 Jonathan Passant U of Rochester Configurations and Erdős style distance problems Jul 28 Fátima Lizarte U of Cantabria A sequence of well conditioned polynomials Aug 4 Jordi Marzo U of Barcelona Quadrature rules, Riesz energies, discrepancies and elliptic polynomials Aug 18 Damir Ferizović TU Graz The spherical cap discrepancy of HEALPix points Aug 25 Aicke Hinrichs JKU Linz Dispersion - a survey of recent results and applications Sep 1 Friedrich Pillichshammer JKU Linz $L_2$ star, extreme and periodic discrepancy

• June 2
Austin Anderson and Alex White (Florida State)
Asymptotics of Best Packing and Best Covering
We discuss recent progress on asymptotics for the dual problems of best packing and best covering in Euclidean space. For future investigations, we highlight their relation to large parameter limits of minimal Riesz s-energy and Riesz s-polarization, respectively. Next, we address a weak-separation argument for coverings and its flexibility as compared to similar arguments for polarization and packing. Finally, we examine how a recent non-existence proof for the asymptotics of best packing on dependent fractals is adapted to both constrained and unconstrained covering- the second case owing largely to weak separation of coverings. This is joint work with Oleksandr Vlasiuk and Alexander Reznikov of Florida State University.

Slides: [pdf]

• June 9
Alan Legg (Purdue U Fort Wayne)
Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges
(Joint work with Peter Dragnev) We consider the problem of finding the equilibrium measure on the unit sphere in R^3 using logarithmic potentials, in the presence of external fields made up of a finite number of point charges on the sphere. For any such external field, the complement of the equilibrium measure turns out to be the stereographic preimage from the plane of a union of classical quadrature domains.

Paper: [arXiv]   [Journal]
Slides: [pdf]

• June 16
Assaf Goldberger (Tel Aviv U)
Configurations, Automorphisms and Cohomology
Point configurations on finite dimensional real or complex spaces, typically on the unit sphere, are important in Physics, Coding Theory, Classical and Quantum Information Theory, Geometry, Number Theory and more. The Automorphism group a of point configuration is a tool to study it, and to generate new ones. In this talk we will show how to generate automorphism groups from group-theoretic considerations, and how to construct configurations that satisfy the group. The main tool in use is Group Cohomology. We show that there is a spectral sequence which captures all possible solutions and all obstructions to the construction of a solution. In addition this sequence captures the Galois structure of algebraic configurations. Galois structures were discovered recently in the case of Zauner SIC-POVMs. This point of view can be generalized to a much broader framework, e.g. higher tensors as replacements of the Gramian matrix, perfect squares are seen to be dual to configuration Gramians when one uses homology instead of cohomology, and there are some connections to Number Theory, such as the theory of Brauer Groups. Other applications are the generation of Hadamard and Weighing matrices. We will discuss these extensions as time permits. This is joint work with Giora Dula.

Slides: [pdf]

• June 30
Robert McCann (University of Toronto)
Maximizing the sum of angles between pairs of lines in Euclidean space
Choose $N$ unoriented lines through the origin of $\mathbb R^{d+1}$. Suppose each pair of lines repel each other with a force {whose strength is} independent of the (acute) angle between them, so that they prefer to be orthogonal to each other. However, unless $N \le d+1$, it is impossible for all pairs of lines to be orthogonal. What then are their stable configurations? An unsolved conjecture of Fejes Tóth (1959) asserts that the lines should be equidistributed as evenly as possible over a standard basis in $\mathbb R^{d+1}$. By modifying the force to make it increase as a power of the distance, we show the analogous claim to be true for all positive powers if we are only interested in local stability, and for sufficiently large powers if we require global stability.

These results represent joint work with Tongseok Lim (of Purdue's Krannert School of Management).

Papers: [arXiv1]   [arXiv2]
Slides: [pdf]

• July 7
Giuseppe Negro (University of Birmingham)
Intermittent symmetry breaking for the maximizers to the Agmon-Hörmander estimate on the sphere
The $L^2$ norm of a function on Euclidean space equals the $L^2$ norm of its Fourier transform; this is the theorem of Plancherel. This is true for functions, but it fails for measures, such as densities on a sphere. In 1976, Agmon and Hörmander observed that it is possible to recover a kind of Plancherel theorem in this case, by localizing on balls; this turns out to be the most basic example of a "Fourier restriction estimate", relevant both to analysis and to PDE. In this talk, we will explicitly determine the densities that maximize such estimate, discovering that they break the rotational symmetry depending on the radius of the localizing ball. This is joint work with Diogo Oliveira e Silva.

Paper: [arXiv1]
Slides: [pdf]

• July 14
Alexander McDonald (University of Rochester)
Volumes spanned by k-point configurations in $\mathbb R^d$
We consider a Falconer type problem concerning volumes determined by point configurations in $\mathbb R^d$, and prove that a set with sufficiently large Hausdorff dimension determines a positive measure worth of volumes. The strategy for proving the result is to study the group action of the special linear group on the space of configurations.

Paper: [arXiv]
Slides: [pdf]

• July 21
Jonathan Passant (University of Rochester)
Configurations and Erdős style distance problems
I will discuss point large configurations in real space and how incidence geometry results of Guth-Katz and Rudnev can help generalise the results of Solymosi-Tardos and Rudnev on the number of congruent and similar triangles.

Paper: [arXiv]
Slides: [pdf]

• July 28
Fátima Lizarte (University of Cantabria)
A sequence of well conditioned polynomials
In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials $P_N$ of degree $N$ with condition number bounded above by $N$. In this talk, we show the origin of this problem, previous knowledge until this work, its relation to other interesting mathematical problems such as Smale's 7th problem, and our main result obtained: a simple and direct answer to the Shub and Smale problem for $N=4M^2$, with $M$ a positive integer, as well as comments on its proof.

This is a joint work with Carlos Beltrán.

Slides: [pdf]

• Aug 4
Jordi Marzo (University of Barcelona)
Quadrature rules, Riesz energies, discrepancies and elliptic polynomials
I will talk about the relation between optimal quadratures, Riesz (or logarithmic) energies and minimal discrepancy configurations. In particular I will discuss the use of the zeros of elliptic (or Kostlan-Shub-Smale) polynomials, among other configurations, as quadrature nodes for Sobolev spaces on the sphere. There will be many open problems throughout the talk.

Slides: [pdf]

• Aug 18
Damir Ferizović (TU Graz)
The spherical cap discrepancy of HEALPix points
In this talk I will present an algorithm well known in the Astrophysics and Cosmology community: HEALPix, short for "Hierarchical, Equal Area and iso-Latitude Pixelation," which divides the two dimensional sphere $\mathbb{S}^2$ into 12 rectangular shapes (base pixel) of equal area, and allows for further subdivision of each pixel into four smaller, equal area subpixel mimicking the simplicity of the unit square in many ways. This algorithm, introduced by Górski et al., also comes with a projection to the plane that up to a constant preserves area. HEALPix also distributes $N$-many points on $\mathbb{S}^2$ by placing them at centers of pixel of the current level of subdivision, i.e. first $N=12$, then $N=12\cdot 4$, $N=12\cdot 4^2, \ldots, N=12\cdot 4^k$, etc. The spherical cap discrepancy of these points will be proven to be of order $N^{-1/2}$, via recycling methods introduced by Aistleitner, Brauchart and Dick. This is a joint work with Julian Hofstadler and Michelle Mastrianni.

Slides: [pdf]

• Aug 25
Aicke Hinrichs (JKU Linz)
Dispersion - a survey of recent results and applications
The dispersion of a point set, which is the volume of the largest axis-parallel box in the unit cube that does not intersect the point set, is an alternative to the discrepancy as a measure for certain (uniform) distribution properties. The computation of the dispersion, or even the best possible dispersion, in dimension two has a long history in computational geometry and computational complexity theory. Given the prominence of the problem, it is quite surprising that, until recently, very little was known about the size of the largest empty box in higher dimensions. This changed in the last five years. In this survey talk we focus on recent developments and new applications of dispersion outside the area of computational geometry.

Slides: [pdf]

• Sep 1
Friedrich Pillichshammer (JKU Linz)
$L_2$ star, extreme and periodic discrepancy
This talk is devoted to three notions of discrepancies with respect to the $L_2$ norm and a variety of test sets. The $L_2$ star discrepancy uses as test sets the class of axis-parallel boxes anchored in the origin, the $L_2$ extreme discrepancy uses arbitrary axis-parallel boxes and the $L_2$ periodic discrepancy uses so-called periodic intervals which range over the whole torus. All three geometrical notions of $L_2$-discrepancy can be interpreted as worst-case error for quasi-Monte Carlo integration in corresponding function spaces. We compare these notions of discrepancy, discuss some relationships and present results for typical QMC point sets such as lattice point sets and digital nets. We turn our attention also to the dependence on the dimension $d$ and examine whether these $L_2$ discrepancies satisfy some tractability properties or suffer from the curse of dimensionality.

• ## Spring 2021

 Speaker Affiliation Title Date Feb 3 David Garcı́a-Zelada Aix-Marseille U A large deviation principle for empirical measures Feb 10 Arno Kuijlaars KU Leuven The spherical ensemble with external sources Feb 17 Alex Iosevich U of Rochester Finite point configurations and frame theory Feb 24 Kasso Okoudjou Tufts U Completeness of Weyl-Heisenberg POVMs Mar 3 Mircea Petrache PUC Chile Sharp isoperimetric inequality, discrete PDEs and semidiscrete optimal transport Mar 10 Yeli Niu U of Alberta Discretization of integrals on compact metric measure spaces Mar 17 Xuemei Chen UNC Wilmington Frame Design Using Projective Riesz Energy Mar 24 Ruiwen Shu U of Maryland Dynamics of Particles on a Curve with Pairwise Hyper-singular Repulsion Mar 31 Oleg Musin U of Texas Rio Grande Valley Majorization, discrete energy on spheres and f-designs Apr 7 Woden Kusner U of Georgia Measuring chirality with the wind Apr 14 Peter Dragnev Purdue Fort Wayne Bounds for Spherical Codes: The Levenshtein Framework Lifted Apr 21 Doug Hardin Vanderbilt U Asymptotics of periodic minimal discrete energy problems Apr 28 Shujie Kang UT Arlington On the rank of non-informationally complete Gabor POVMs May 5 Mario Ullrich JKU Linz Random matrices and approximation using function values May 12 William Chen Macquarie U The Veech 2-circle problem and non-integrable flat dynamical systems May 19 Johann Brauchart TU Graz Weighted $L^2$-Norms of Gegenbauer Polynomials — and more!

• Feb 3
A large deviation principle for empirical measures
The main object of this talk will be a model of $n$ interacting particles at equilibrium. I will describe its macroscopic behavior as $n$ grows to infinity by showing a Laplace principle or, equivalently, a large deviation principle. This implies, in some cases, an almost sure convergence to a deterministic probability measure. Among the main motivating examples we may find Coulomb gases on Riemannian manifolds, the eigenvalue distribution of Gaussian random matrices and the roots of Gaussian random polynomials. This talk is based on arXiv:1703.02680.

Slides: [pdf]
Paper: [arXiv]

• Feb 10
Arno Kuijlaars (KU Leuven)
The spherical ensemble with external sources
The talk is based on joint work with Juan Criado del Rey.
We study a model of a large number of points on the unit sphere under the influence of a finite number of fixed repelling charges. In the large n limit the points fill a region that is known as the droplet. For small external charges the droplet is the complement of the union of a number of spherical caps, one around each of the external charges. When the external charges grow, the spherical caps will start to overlap and the droplet ondergoes a non-trivial deformation.
We explicitly describe the transition for the case of equal external charges that are symmetrically located around the north pole. In our approach we first identify a motherbody that, due to the symmetry in the problem, will be located on a number of meridians connecting the north and south poles. After projecting onto the complex plane, and undoing the symmetry, we characterize the motherbody by means of the solution of a vector equilibrium problem from logarithmic potential theory.

Slides: [pdf]
Paper: [arXiv]

• Feb 17
Alex Iosevich (U of Rochester)
Finite point configurations and frame theory
We are going to discuss some recent and not so recent applications of analytic and combinatorial results on finite point configurations to problems of existence of exponential and Gabor frames and bases.

Slides: [pdf]

• Feb 24
Kasso Okoudjou (Tufts U)
Completeness of Weyl-Heisenberg POVMs
The finite Gabor (or Weyl-Heisenberg) system generated by a unit-norm vector $g\in \mathbb{C}^d$ is the set of vectors $$\big\{g_{k,\ell}=e^{2\pi i k\cdot}g(\cdot - \ell)\big\}_{k, \ell =0}^{d-1}.$$ It is know that every such system forms a finite unit norm tight frame (FUNTF) for $\mathbb{C}^d$, i.e., $$d^3 \|x\|^2=\sum_{k, \ell=0}^{d-1}|\langle x, g_{k,\ell}\rangle |^2\quad \forall \, x\in \mathbb{C}^d.$$ Furthermore, the Zauner conjecture asserts that for each $d\geq 2$, there exist unit-norm vectors $g \in \mathbb{C}^d$ such that this FUNTF is equiangular, that is, $|\langle g, g_{k, \ell}\rangle |^2= \tfrac{1}{d+1}.$ Assuming the existence of a unit-vector $g$ that positively answers Zauner's conjecture, one can show that the set of rank-one matrices $$\big\{\pi_{k,\ell}=\langle \cdot, g_{k,\ell}\rangle g_{k, \ell}\big\}_{k \ell=0}^{d-1}$$ is complete in the space of $d\times d$ matrices. Consequently, $\big\{\pi_{k,\ell}\big\}_{k \ell=0}^{d-1}$ forms a symmetric informationally complete positive operator-valued measure (SIC-POVM). In fact, it is known that given a unit-norm vector $g\in \mathbb{C}^d$, the POVM $\big\{\pi_{k,\ell}\big\}_{k \ell=0}^{d-1}$ is informationally complete (IC) if and only if $\langle g, g_{k, \ell} \rangle \neq 0$ for all $(k ,\ell)\neq (0,0)$. In this talk, we give a different proof of the characterization of the IC-POVMs. We then focus on investigating non-informationally complete POVMs. We will present some preliminary results pertaining to the dimensions of the linear spaces spanned by these rank-one matrices. (This talk is based on on-going joint work with S. Kang and A. Goldberger.)

• Mar 3
Mircea Petrache (PUC Chile)
Sharp isoperimetric inequality, discrete PDEs and semidiscrete optimal transport
Consider the following basic model of finite crystal cluster formation: in a periodic graph G with vertices in R^d (representing possible molecular bonds) a subset (of atoms) must be chosen, so that the total number of bonds between a point in X and one outside X is minimized. These bonds form the edge-perimeter of X, denoted \partial X. If the graph is periodic and locally finite, any X satisfies an inequality of the form |X|^{d-1} \leq C |partial X|^d, where the optimal C depends on the graph. How can we determine the structure of sets X realizing equality in the above, based on the geometry and of G? If we take the continuum limit of G, then the classical Wulff shape theory describes optimal limit shapes, and at least two proofs of isoperimetric inequality apply, one based on PDE and calibration ideas, and the other based on Optimal Transport ideas. We focus on using the heuristic coming from the continuum analogue, to answer the above question in some cases, in the discrete case. This approach highlights the tight connection between discrete PDEs and semidiscrete Optimal Transport, and a link to the Minkowski theorem for convex polyhedra.

Paper: [arXiv1] [arXiv2]
Slides: [pdf]

• Mar 10 at 9:30am CST/10:30am EST/4:30pm CET
Yeli Niu (U of Alberta)
Discretization of integrals on compact metric measure spaces
Let $\mu$ be a Borel probability measure on a compact path-connected metric space $(X, \rho)$ for which there exist constants $c,\beta\ge 1$ such that $\mu(B) \ge c r^{\beta}$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $\Phi:[0,\infty)\to\mathbb R$ that are piecewise within a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $\rho$ and the measure $\mu$ that for each positive integer $N\ge 2$, and each $g\in L^\infty(X, d\mu)$ with $\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N }\in X$ and real numbers $\lambda_1, \ldots, \lambda_{ N }$ such that for any $x\in X$, \begin{align*} & \left| \int_X \Phi (\rho (x, y)) g(y) \,\text{d} \mu (y) - \sum_{j = 1}^{ N } \lambda_j \Phi (\rho (x, y_j)) \right| \leqslant C N^{- \frac{1}{2} - \frac{3}{2\beta}} \sqrt{\log N}, \end{align*} where the constant $C>0$ is independent of $N$ and $g$. In the case when $X$ is the unit sphere $\mathbb{S}^{d}$ of $\mathbb R^{d+1}$ with the ususal geodesic distance, we also prove that the constant $C$ here is independent of the dimension $d$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $N^{-\frac 12}\sqrt{\log N}$.

Paper: [arXiv]
Slides: [pdf]

• Mar 17
Xuemei Chen (UNC Wilmington)
Frame Design Using Projective Riesz Energy
Tight and well-separated frames are desirable in many signal processing applications. We introduce a projective Riesz kernel for the unit sphere and investigate properties of N-point energy minimizing configurations for such a kernel. We show that these minimizing configurations, for N sufficiently large, form frames that are well-separated (have low coherence) and are nearly tight. We will also show some numerical experiments. This is joint work with Doug Hardin and Ed Saff.

Paper: [arXiv]
Slides: [pdf]

• Mar 24
Ruiwen Shu (U of Maryland)
Dynamics of Particles on a Curve with Pairwise Hyper-singular Repulsion
We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $\mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz $s$-energy with $s>1$. We show that regardless of their initial positions, for all $N$ and time $t$ large, their normalized Riesz $s$-energy will be close to the $N$-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.

Paper: [arXiv]
Slides: [pdf]

• Mar 31
Oleg Musin (U of Texas Rio Grande Valley)
Majorization, discrete energy on spheres and f-designs
We consider the majorization (Karamata) inequality and minimums of the majorization (M-sets) for f-energy potentials of m-point configurations in a sphere. We discuss the optimality of regular simplexes, describe M-sets with a small number of points, define spherical f-designs and study their properties. Then we consider relations between the notions of f-designs and M-sets, $\tau$-designs, and two-distance sets

Paper: [arXiv]
Slides: [pdf]

• Apr 7
Woden Kusner (U of Georgia)
Measuring chirality with the wind
The question of measuring "handedness" is of some significance in both mathematics and in the real world. Propellors and screws, proteins and DNA, in fact *almost everything* is chiral. Can we quantify chirality? Or can we perhaps answer the question:
"Are your shoes more left-or-right handed than a potato?"
We can begin with the hydrodynamic principle that chiral objects rotate when placed in a collimated flow (or wind). This intuition naturally leads to a trace-free tensorial chirality measure for space curves and surfaces, with a clear physical interpretation measuring twist. As a consequence, the "average handedness" of an object with respect to this measure will always be 0. This also strongly suggests that a posited construction of Lord Kelvin--the isotropic helicoid--can not exist.
joint with Giovanni Dietler, Rob Kusner, Eric Rawdon and Piotr Szymczak

Paper: [arXiv]
Slides: [pdf]

• Apr 14
Peter Dragnev (Purdue Fort Wayne)
Bounds for Spherical Codes: The Levenshtein Framework Lifted
Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein’s framework to obtain lower bounds for the minimum henergy of spherical codes of prescribed dimension and cardinality, and upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. These bounds are universal in the sense that they hold for a large class of potentials h and in the sense of Levenshtein. Moreover, codes attaining the bounds are universally optimal in the sense of Cohn-Kumar. Referring to Levenshtein bounds and the energy bounds of the authors as “first level”, our results can be considered as “next level” universal bounds as they have the same general nature and imply necessary and sufficient conditions for their local and global optimality. For this purpose, we introduce the notion of Universal Lower Bound space (ULB-space), a space that satisfies certain quadrature and interpolation properties. While there are numerous cases for which our method applies, we will emphasize the model examples of 24 points (24-cell) and 120 points (600-cell) on $\mathbb{S}^3$. In particular, we provide a new proof that the 600-cell is universally optimal, and in so doing, we derive optimality of the 600-cell on a class larger than the absolutely monotone potentials considered by Cohn-Kumar.

Slides: [pdf]

• Apr 21
Doug Hardin (Vanderbilt U)
Asymptotics of periodic minimal discrete energy problems
For $s>0$ and a lattice $L$ in $R^d$, we consider the asymptotics of $N$-point configurations minimizing the $L$-periodic Riesz $s$-energy as the number of points $N$ goes to infinity. In particular, we focus on the case $0<s<d$ of long-range potentials where we establish that the minimal energy $E_s(L,N)$ is of the form $E_s(L,N)=C_0 N^2 + C_1 N^{1+s/d} +o(N^{1+s/d})$ as $N\to \infty$ for constants $C_0$ and $C_1$ depending only on $s$, $d$, and the covolume of $L$. This is joint work with Ed Saff, Brian Simanek, and Yujian Su.

Slides: [pdf]

• Apr 28
Shujie Kang (UT Arlington)
On the rank of non-informationally complete Gabor POVMs
We investigate Positive Operator Valued Measures (POVMs) generated by Gabor frames in $\mathbb{C}^d$. A complete (Gabor) POVM is one that spans the space $\mathbb{C}^{d^{2}}$ of $d\times d$ matrices. It turns out that being a complete Gabor POVM is a generic property. As a result, the focus of this talk will be on non-complete Gabor POVMs. We will describe the possible ranks of these Gabor POVMs, and derive various consequences for the underlying Gabor frames. In particular, we will give details in dimensions $4$ and $5$.

• May 5
Mario Ullrich (JKU Linz)
Random matrices and approximation using function values
We consider $L_2$-approximation of functions using linear algorithms and want to compare the power of function values with the power of arbitrary linear information. Under mild assumptions on the class of functions, we show that the minimal worst-case errors based on function values decay at almost the same rate as those with arbitrary info, if the latter decay fast enough. Our results are to some extent best possible and, in special cases, improve upon well-studied point constructions, like sparse grids, which were previously assumed to be optimal. The proof is based on deep results on large random matrices, including the recent solution of the Kadison-Singer problem, and reveals that (classical) least-squares methods might be surprisingly powerful in a general setting.

Slides: [pdf]

• May 12
William Chen (Macquarie U)
The Veech 2-circle problem and non-integrable flat dynamical systems
We are motivated by an interesting problem studied more than 50 years ago by Veech and which can be considered a parity, or mod 2, version of the classical equidistribution problem concerning the irrational rotation sequence. The Veech discrete 2-circle problem can also be visualized as a continuous flat dynamical system, in the form of 1-direction geodesic flow on a surface obtained by modifying the surface comprising two side-by-side unit squares by the inclusion of barriers and gates on the vertical edges, with appropriate modification of the edge identifications. A famous result of Gutkin and Veech says that 1-direction geodesic flow on any flat finite polysquare translation surface exhibits optimal behavior, in the form of an elegant uniform-periodic dichotomy. Here the modified surface in question is no longer such a surface, and there are vastly different outcomes depending on the values of certain parameters.

Slides: [pdf]

• May 19
Johann Brauchart (TU Graz)
Weighted $L^2$-Norms of Gegenbauer Polynomials — and more!
I discuss integrals of the form $$\int_{-1}^1(C_n^{(\lambda)}(x))^2(1-x)^\alpha (1+x)^\beta\, dx,$$ where $C_n^{(\lambda)}$ denotes the Gegenbauer-polynomial of index $\lambda>0$ and $\alpha,\beta>-1$. Such integrals for orthogonal polynomials involving, in particular, a "wrong" weight function appear in physics applications and point distribution problems. I present exact formulas for the integrals and their generating functions, and give asymptotic formulas as $n\to\infty$. This is joint work with Peter Grabner also from TU Graz.

Paper: [arXiv]
Slides: [pdf]

• ## Fall 2020

 Speaker Affiliation Title Date Sep 16 Dmitriy Bilyk U of Minnesota Stolarsky principle: generalizations, extensions, and applications Sep 23 Alexander Reznikov Florida State Minimal discrete energy on fractals Sep 30 Stefan Steinerberger U of Washington Optimal Transport and Point Distributions on the Torus Oct 2 Stefan Steinerberger U of Washington Optimal Transport and Point Distributions on Manifolds Oct 7 Oleksandr Vlasiuk Florida State Asymptotic properties of short-range interaction functionals Oct 14 Peter Grabner TU Graz Fourier-Eigenfunctions and Modular Forms Oct 21 Michelle Mastrianni U of Minnesota Bounds for Star-Discrepancy with Dependence on the Dimension Oct 28 Carlos Beltrán U of Cantabria Smale’s motivation in describing the 7th problem of his list Nov 4 Adrian Ebert RICAM Construction of (polynomial) lattice rules by smoothness-independent component-by-component digit-by-digit constructions Nov 11 Alexey Glazyrin U of Texas Rio Grande Valley Mapping to the space of spherical harmonics Dec 2 Alexander Barg U of Maryland Stolarsky's invariance principle for the Hamming space Dec 9 Paul Leopardi Australian National U Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces Dec 16 Laurent Bétermin U of Vienna Theta functions, ionic crystal energies and optimal lattices

• Sep 16
Dmitriy Bilyk (U of Minnesota)
Stolarsky principle: generalizations, extensions, and applications
In 1973 Kenneth Stolarsky proved a remarkable identity, which connected two classical quantities, which measure the quality of point distributions on the sphere: the $L^2$ spherical cap discrepancy and the pairwise sum of Euclidean distances between points. This fact, which came to be known as the Stolarsky Invariance Principle, established a certain duality between problems of discrepancy theory on one hand, and distance geometry or energy optimization on the other, and allowed one to transfer methods and results of one field to the other. Since then numerous versions, extensions, and generalizations of this principle have been found, leading to connections between various notions of discrepancy and discrete energies in different settings and to a number of applications to various problems of discrete geometry. In this talk we shall survey known work on the Stolarsky principle, as well as some related problems.

Slides: [pdf]

• Sep 23
Alexander Reznikov (Florida State)
Minimal discrete energy on fractals
We will survey some old and new results on the existence of asymptotic behavior of minimal discrete Riesz energy of many particles located in a fractal set. Unlike in the case of a rectifiable set, when the asymptotic behavior always exists, we will show that on a large class of somewhat "balanced" fractals the energy (and best-packing) does not have any asymptotic behavior.

Slides: [pdf]

• Sep 30
Stefan Steinerberger (U of Washington)
Optimal Transport and Point Distributions on the Torus
There are lots of ways of measuring the regularity of a set of points on the Torus.  I'll introduce a fundamental notion from Optimal Transport, the Wasserstein distance, as another such measure. It corresponds quite literally over what distance one has to spread the points to be evenly distributed, it has a natural physical intuition (the notion itself was derived in Economics modeling transport) and is naturally related to other notions such as discrepancy or Zinterhof's diaphony.  Classical Fourier Analysis allows us to bound this transport distance via exponential sums which are well studied; this allows us to revisit many classical constructions and get transport bounds basically for free. We'll finish by revisiting a classical problem from numerical integration from this new angle.  There will be many open problems throughout the talk.

Slides: [pdf]

• Oct 2
Stefan Steinerberger (U of Washington)
Optimal Transport and Point Distributions on Manifolds
We'll go somewhat deeper into the connection between the Wasserstein distance and notions from potential theory: in particular, how the classical Green function can be used to derive bounds on Wasserstein transport on general manifolds. On the sphere, our results simplify and the Riesz energy appears in a nice form. We conclude with a fundamental new idea: the Wasserstein Uncertainty Principle which says that if it's terribly easy to buy milk wherever you are, then there must be many supermarkets -- the precise form of this isoperimetric principle is not known and, despite being purely geometric, it would have immediate impact on some PDE problems.

• Oct 7
Oleksandr Vlasiuk (Florida State)
Asymptotic properties of short-range interaction functionals
Short-range interactions, such as the hypersingular Riesz energies, are known to be amenable to asymptotic analysis, which allows to obtain for them the distribution of minimizers and asymptotics of the minima. We extract the properties making such analysis possible into a standalone framework. This allows us to give a unified treatment of hypersingular Riesz energies and optimal quantizers. We further obtain new results about the scale-invariant nearest neighbor interactions, such as the $k$-nearest neighbor truncated Riesz energy. The suggested approach has applications to common methods for generating distributions with prescribed density: Riesz energies, centroidal Voronoi tessellations, and popular meshing algorithms due to Persson-Strang and Shimada-Gossard. It naturally generalizes from 2-body to $k$-body interactions.
Based on joint work with Douglas Hardin and Ed Saff.

Slides: [pdf]

• Oct 14
Peter Grabner (TU Graz)
Fourier-Eigenfunctions and Modular Forms
Eigenfunctions of the Fourier-transform play a major role in Viazovska's proof of the best packing of the $E_8$ lattice in dimension 8 and the subsequent determination of the Leech lattice as best packing configuration dimension 24 by Cohn, Kumar, Miller, Radchenko, and Viazovska. In joint work with A. Feigenbaum and D. Hardin we have shown that the constructions as used for these results are unique; we could shed more light on the underlying modular and quasimodular forms and determine linear recurrence relations and differential equations characterising these forms.

Paper: [arXiv1] [arXiv2]

• Oct 21
Michelle Mastrianni (U of Minnesota)
Bounds for Star-Discrepancy with Dependence on the Dimension
The question of how the star-discrepancy (with respect to corners) of an n-point set in the d-dimensional unit cube depends on the dimension d was studied in 2001 by Heinrich, Novak, Wasilkowski and Wozniakowski. They established an upper bound that depends only polynomially on d/n. The proof makes use of the fact that the set of corners in the d-dimensional unit cube is a VC-class, and employs a result by Talagrand (1994) that uses a partitioning scheme to study the tails of the supremum of a Gaussian process under certain conditions that are always satisfied by VC-classes. In 2011, Aistleitner produced a simpler proof of this upper bound using a direct dyadic partitioning argument and explicitly computed the constant; an improvement on the constant was given by Pasing and Weiss in 2018. The best lower bound was achieved by Hinrichs (2003), who built upon the ideas of using VC-inequalities to achieve a lower bound with polynomial behavior in d/n as well. In this talk I will introduce the notion of VC dimension and discuss how it is employed in the above proofs, and outline how the direct partitioning argument for the upper bound uses the same underlying ideas about where the bulk of the contribution to the tails arises.

Slides: [pdf]

• Oct 28 at 9am CDT (Chicago), 10am EDT (New York), 3pm CET (Paris, Berlin)
Carlos Beltrán (U of Cantabria)
Smale’s motivation in describing the 7th problem of his list
In 1993, Mike Shub and Steve Smale posed a question that would be later included in Smale’s list as 7th problem. Although this last problem has became so famous, the exact reason for its form and the consequences that its solution would have for the initial goal are not so well known in the mathematician community. In this seminar, I will describe the thrilling story of these origins: where the problem came from, would it still be useful for that task, and what is left to do. I will probably talk a lot and show very few formulas, and I will also present some open problems.

• Nov 4
Construction of (polynomial) lattice rules by smoothness-independent component-by-component digit-by-digit constructions
In this talk, we introduce component-by-component digit-by-digit algorithms (CBC-DBD) for the construction of (polynomial) lattice rules in weighted Korobov/Walsh spaces with prescribed decay of the involved series coefficients and associated smoothness $\alpha > 1$. The presented methods are extensions of a construction algorithm established by Korobov in [1] to the modern quasi-Monte Carlo (QMC) setting. We show that the introduced CBC-DBD algorithms construct QMC rules with $N = 2^n$ points which achieve the almost optimal worst-case error convergence rates in the studied function spaces. Due to the used quality functions, the algorithms can construct good (polynomial) lattice rules independent of the smoothness α of the respective function class. Furthermore, we derive suitable conditions on the weights under which the mentioned error bounds are independent of the dimension. The presented algorithms can be implemented in a fast manner such that the construction only requires $O(sN \ln N )$ operations, where $N = 2^n$ is the number of lattice points and s denotes the dimension. We stress that these fast constructions achieve this complexity without the use of fast Fourier transformations (FFTs), as in, e.g., [2]. We present extensive numerical results which confirm our theoretical findings.
Joint research with Peter Kritzer, Dirk Nuyens, Onyekachi Osisiogu, and Tetiana Stepaniuk.
[1] N.M. Korobov. On the computation of optimal coefficients. Dokl. Akad. Nauk SSSR, 26:590–593. 1982.
[2] D. Nuyens, R. Cools. Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. J. Complexity 22(1), 4–28. 2006.

Slides: [pdf]

• Nov 11
Alexey Glazyrin (UT Rio Grande Valley)
Mapping to the space of spherical harmonics
For a variety of problems for point configurations in spheres, the space of spherical harmonics plays an important role. In this talk, we will discuss maps from point configurations to the space of spherical harmonics. Such maps can be used for finding bounds on packings, energy bounds, and constructing new configurations. We will explain classical results from this perspective and prove several new bounds. Also we will show a new short proof for the kissing number problem in dimension 3.

Slides: [pdf]

• Dec 2
Alexander Barg (U of Maryland)
Stolarsky's invariance principle for the Hamming space
Stolarsky's invariance principle has enjoyed considerable attention in the literature in the last decade. In this talk we study an analog of Stolarsky's identity in finite metric spaces with an emphasis on the Hamming space. We prove several bounds on the spherical discrepancy of binary codes and identify some discrepancy minimizing configurations. We also comment on the connection between the problem of minimizing the discrepancy and the general question of locating minimum-energy configurations in the space. The talk is based on arXiv:2005.12995 and arXiv:2007.09721 (joint with Maxim Skriganov).

Slides: [pdf]

• Dec 9
Paul Leopardi (Australian National U)
Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces
The algorithm devised by Feige and Schechtman for partitioning higher dimensional spheres into regions of equal measure and small diameter is combined with David and Christ's construction of dyadic cubes to yield a partition algorithm suitable to any connected Ahlfors regular metric measure space of finite measure. This is joint work with Giacomo Gigante of the University of Bergamo.

Slides: [pdf]
Paper: [arXiv] G. Gigante and P. Leopardi, "Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces", Discrete and Computational Geometry, 57 (2), 2017, pp. 419–430.

• Dec 16
Laurent Bétermin (U of Vienna)
Theta functions, ionic crystal energies and optimal lattices
The determination of minimizing structures for pairwise interaction energies is a very challenging crystallization problem. The goal of this talk is to present recent optimality results among charges and lattice structures obtained with Markus Faulhuber (University of Vienna) and Hans Knüpfer (University of Heidelberg). The central object of these works is the heat kernel associated to a lattice, also called lattice theta function. Several connections will be showed between interaction energies and theta functions in order to study the following problems: - Born’s Conjecture: how to distribute charges on a fixed lattice in order to minimize the associated Coulombian energy? In the simple cubic case, Max Born conjectured that the alternation of charges +1 and -1 (i.e. the rock-salt structure of NaCl) is optimal. The proof of this conjecture obtained with Hans Knüpfer will be briefly discussed as well as its generalization to other lattices and energies. - stability of the rock-salt structure: what could be conditions on interaction potentials such that the minimal energies among charges and lattices has a rock-salt structure? Many results, both theoretical and numerical and obtained with Markus Faulhuber and Hans Knüpfer, will be presented. - maximality of the triangular lattices among lattices with alternation of charges: we will present this new universal optimality among lattices obtained with Markus Faulhuber.

Slides: [pdf]

• ## Summer 2020

 Speaker Affiliation Title Date Jun 3 Ujué Etayo TU Graz Astounding connections of the logarithmic energy on the sphere Jun 10 Josiah Park Georgia Tech Optimal measures for three-point energies and semidefinite programming Jun 17 Maria Dostert EPFL Semidefinite programming bounds for the average kissing number Jun 24 Philippe Moustrou The Arctic University of Norway Exact semidefinite programming bounds for packing problems Jul 1 David de Laat TU Delft High-dimensional sphere packing and the modular bootstrap Jul 8 Matthew de Courcy-Ireland EPFL Lubotzky-Phillips-Sarnak points on a sphere Jul 17 Mateus Sousa LMU München Uncertainty principles, interpolation formulas and packing problems Jul 22 Giuseppe Negro U of Birmingham Sharp estimates for the wave equation via the Penrose transform Jul 29 Tetiana Stepaniuk U of Lübeck Estimates for the discrete energies on the sphere Jul 31 Mathias Sonnleitner JKU Linz Uniform distribution on the sphere and the isotropic discrepancy of lattice point sets Aug 5 Oscar Quesada IMPA Developments on the Fourier sign uncertainty principle Aug 12 Louis Brown Yale Positive-definite functions, exponential sums and the greedy algorithm: a curious phenomenon Aug 14 Julian Hofstadler JKU Linz On a subsequence of random points Aug 19 Felipe Gonçalves U of Bonn Sign uncertainty Aug 28 David Krieg JKU Linz Order-optimal point configurations for function approximation

• Jun 3
Ujué Etayo (TU Graz)
Astounding connections of the logarithmic energy on the sphere
During this talk we will present different problems that are somehow related to the following one: find the minimum value of the logarithmic energy of a set of N points on the sphere of dimension 2. This late problem has been studied for years, a computational version of it can be found as Problem Number 7 of Steve Smale list "Mathematical Problems for the Next Century". This computational version of the problem was proposed after Smale and Shub found out a beautiful relation between minimizers of the logarithmic energy and well conditioned polynomials. Working on this relation, we are able to relate these two concepts to yet a new one: a sharp Bombieri type inequality for univariate polynomials. The problem can also be rewritten as a facility location problem, as proved by Beltrán, since the logarithmic energy is just a normalization of the Green function for the Laplacian on the sphere.

Slides: [pdf]

• Jun 10
Josiah Park (Georgia Tech)
Optimal measures for three-point energies and semidefinite programming
Given a potential function of three vector arguments, $$f(x,y,z)$$, which is $$O(n)$$-invariant, $$f(Qx,Qy,Qz)=f(x,y,z)$$ for all $$Q$$ orthogonal, we find that surface measure minimizes those interaction energies of the form $$\int\int\int f(x,y,z) d\mu(x)d\mu(y)d\mu(z)$$ over the sphere whenever the potential function satisfies a positive definiteness criteria. We use semidefinite programming bounds to determine optimizing probability measures for other energies. This latter approach builds on previous use of such bounds in the discrete setting by Bachoc-Vallentin, Cohn-Woo, and Musin, and is successful for kernels which can be shown to have expansions in a particular basis, for instance certain symmetric polynomials in inner products $$u=\langle x,y \rangle$$, $$v=\langle y,z\rangle$$, and $$t=\langle z, x \rangle$$. For other symmetric kernels we pose conjectures on the behavior of optimizers, partially inferred through numerical studies. This talk is based on joint work with Dmitriy Bilyk, Damir Ferizovic, Alexey Glazyrin, Ryan Matzke, and Oleksandr Vlasiuk.

• Jun 17
Maria Dostert (EPFL)
Semidefinite programming bounds for the average kissing number
The average kissing number of $\mathbb R^n$ is the supremum of the average degrees of contact graphs of packings of finitely many balls (of any radii) in $\mathbb R^n.$ In this talk I will provide an upper bound for the average kissing number based on semidefinite programming that improves previous bounds in dimensions $3,\ldots, 9$. A very simple upper bound for the average kissing number is twice the kissing number; in dimensions $6,\ldots, 9$ our new bound is the first to improve on this simple upper bound. This is a joined work with Alexander Kolpakov and Fernando Mário de Oliveira Filho.

Slides: [pdf]
Paper: [arXiv]

• Jun 24
Philippe Moustrou (The Arctic University of Norway)
Exact semidefinite programming bounds for packing problems
(Joint work with Maria Dostert and David de Laat.)
In the first part of the talk, we present how semidefinite programming methods can provide upper bounds for various geometric packing problems, such as kissing numbers, spherical codes, or packings of spheres into a larger sphere. When these bounds are sharp, they give additional information on optimal configurations, that may lead to prove the uniqueness of such packings. For example, we show that the lattice $E_8$ is the unique solution for the kissing number problem on the hemisphere in dimension 8.
However, semidefinite programming solvers provide approximate solutions, and some additional work is required to turn them into an exact solution, giving a certificate that the bound is sharp. In the second part of the talk, we explain how, via our rounding procedure, we can obtain an exact rational solution of semidefinite program from an approximate solution in floating point given by the solver.

Slides: [pdf]

• Jul 1
David de Laat (TU Delft)
High-dimensional sphere packing and the modular bootstrap
Recently, Hartman, Mazáč, and Rastelli discovered a connection between the Cohn-Elkies bound for sphere packing and problems in the modular bootstrap. In this talk I will explain this connection and discuss our numerical study into high dimensional sphere packing and the corresponding problems in the modular bootstrap. The numerical results indicate an exponential improvement over the Kabatianskii-Levenshtein bound. I will also discuss implied kissing numbers and how these relate to improvements over the Cohn-Elkies bound.
Joint work with Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman, and Amirhossein Tajdini.

Slides: [pdf]

• Jul 8
Matthew de Courcy-Ireland (EPFL)
Lubotzky-Phillips-Sarnak points on a sphere
We will discuss work of Lubotzky-Phillips-Sarnak on special configurations of points on the two-dimensional sphere: what these points achieve, the sense in which it is optimal, and aspects of the construction that are specific to the sphere.

Slides: [pdf]

• Jul 17
Mateus Sousa (LMU München)
Uncertainty principles, interpolation formulas and packing problems
In this talk we will discuss how certain uncertainty principles and interpolation formulas are connected to packing problems and talk about some recent developments on these fronts.

Slides: [pdf]

• Jul 22
Giuseppe Negro (U of Birmingham)
Sharp estimates for the wave equation via the Penrose transform
In 2004, Foschi found the best constant, and the extremizing functions, for the Strichartz inequality for the wave equation with data in the Sobolev space $\dot{H}^{1/2}\times \dot{H}^{-1/2}(\mathbb{R}^3)$. He also formulated a conjecture, concerning the extremizers to this Strichartz inequality in all spatial dimensions $d\ge 2$. We disprove such conjecture for even $d$, but we provide evidence to support it for odd $d$. The proofs use the conformal compactification of the Minkowski space-time given by the Penrose transform. Part of this talk is based on joint work with Felipe Gonçalves (Univ. Bonn).

Slides: [pdf]
Paper: [arXiv1] [arXiv2]

• Jul 29
Tetiana Stepaniuk (U of Lübeck)
Estimates for the discrete energies on the sphere
We find upper and lower estimate for the discrete energies whose Legendre-Fourier coefficients decrease to zero approximately as power functions.

Slides: [pdf]

• Jul 31
Mathias Sonnleitner (JKU Linz)
Uniform distribution on the sphere and the isotropic discrepancy of lattice point sets
Aistleitner, Brauchart and Dick showed in 2012 how the spherical cap discrepancy of mapped point sets may be estimated in terms of their isotropic discrepancy. We provide a characterization of the isotropic discrepancy of lattice point sets in terms of the spectral test, the inverse length of the shortest vector in the corresponding dual lattice. This is used to give a lower bound on the discrepancy in question. The talk is based on joint work with F. Pillichshammer.

Slides: [pdf]
Paper: [arXiv]

• Aug 5
Developments on the Fourier sign uncertainty principle
Can we control the signs of a function and its Fourier transform, simultaneously, in an arbitrary way?
An uncertainty principle in Fourier analysis is the answer to this type of question. They lie at the heart of Fourier optimization problems, such as the Cohn-Elkies linear program for sphere packings. We will discuss some answers to this question from a new perspective, and why it might be relevant for problems in diophantine geometry and optimal configurations. (Joint work with Emanuel Carneiro).

Slides: [pdf]
Paper: [arXiv]

• Aug 12
Louis Brown (Yale)
Positive-definite functions, exponential sums and the greedy algorithm: a curious phenomenon
We describe a curious dynamical system that results in sequences of real numbers in $[0,1]$ with seemingly remarkable properties. Let the even function $f:\mathbb{T} \rightarrow \mathbb{R}$ satisfy $\widehat{f}(k) \geq c|k|^{-2}$ and define a sequence via $x_n = \arg\min_x \sum_{k=1}^{n-1}{f(x-x_k)}$.
Such greedy sequences seem to be astonishingly regularly distributed in various ways.  We explore this, and generalize the algorithm (and results on it) to higher-dimensional manifolds, where the setting is even nicer.

Slides: [pdf]
Paper: [arXiv]

• Aug 14
On a subsequence of random points
We want to study the ideas of R. Dwivedi, O. N. Feldheim, O. Guri-Gurevich and A. Ramdas from their paper 'Online thinning in reducing discrepancy', where they give a criterion for choosing points of a random sequence. This technique, called thinning, shall improve the distribution of random points, and we also want to discuss their attempt to create thinned samples with small discrepancy.

Slides: [pdf]

• Aug 19
Felipe Gonçalves (U of Bonn)
Sign uncertainty
We will talk about recent developments of the sign uncertainty principle and its relation with sphere packing bounds and spherical designs. This is joint work with J. P. Ramos and D. Oliveira e Silva.

Slides: [pdf]
We show that independent and uniformly distributed sampling points are as good as optimal sampling points for the approximation (and integration) of functions from the Sobolev space $W_p^s(\Omega)$ on domains $\Omega\subset \mathbb{R}^d$ in the $L_q(\Omega)$-norm whenever $q<p$, where we take $q=1$ if we only want to compute the integral. In the case $q\ge p$ there is a loss of a logarithmic factor. More generally, we characterize the quality of arbitrary sampling points $P\subset \Omega$ via the $L_\gamma(\Omega)$-norm of the distance function ${\rm dist}(\cdot,P)$, where $\gamma=s(1/q-1/p)_+^{-1}$. This improves upon previous characterizations based on the covering radius of $P$. This is joint work with M. Sonnleitner.