The term Irregularities of Distribution (often replaced with Geometric Discrepancy) has been introduced by Klaus Roth in 1954 and refers to the question of how to choose a set of $N$ sampling points which can be used to approximate all the sets in a given reasonably large family inside the unit square. In this talk we consider a planar convex body $C$ and we prove several analogs of Roth's theorem. When $\partial C$ is $\mathcal{C}^{2}$ regardless of curvature, we prove that for every set $\mathcal{P}_{N} $ of $N$ points in $\mathbb{T}^{2}$ we have the sharp bound \[ \left\{\int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( \lambda C+t\right) \right) -\lambda^{2}N\left\vert C\right\vert \right\vert ^{2}~dtd\lambda\right\}^{1/2}\geqslant cN^{1/4}\;. \] When $\partial C$ is only piecewise $\mathcal{C}^{2}$ and is not a polygon we prove the sharp bound \[ \left\{\int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( \lambda C+t\right) \right) -\lambda ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtd\lambda\right\}^{1/2}\geqslant cN^{1/5}. \] We also give a whole range of intermediate sharp results between $N^{1/5}$ and $N^{1/4}$. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of $C$.

I will discuss a fundamental model for wireless communications (called Noncoherent Communications) that has a very simple mathematical explanation and modelling. It turns out that in order to achieve optimal communication rate one must choose a finite collection of points in the complex Grassmannian. In this talk I will present the problem from scratch, explaining the setting, the model, the road to the mathematical problem of point distribution… and a new approach to this last mathematical problem, based on Riemannian optimization, which outperforms all known methods to the date for choosing well distributed codes in these classical spaces. This is joint work with a team of engineers, credits will be given during the talk.

We explain a construction of an efficient net on the sphere in a high-dimensional space, and draw some applications in random matrix theory (partly joint with Tikhomirov and Vershynin). One of the steps in our construction is also relevant for estimating minimal dispersion in the cube (joint with Litvak).

In this talk, based on recent joint work with Vesa Kaarnioja, Yoshihito Kazashi, Frances Kuo and Fabio Nobile, I describe a fast method for high-dimensional approximation on the torus. A typical approximation scheme for a given function $f(\mathbf{t})$ involves choosing a set of points $\mathbf{t}_1, \ldots, \mathbf{t}_n$ at which function values are to be given as inputs, and a methodology for constructing a more or less smooth approximation $f_n(\mathbf{t})$. We shall see that lattice points have advantages, but also limitations, as sample points. The method to be described is based on so-called kernels and lattice points. It appears to offer considerable promise for practical high-dimensional approximation.

Tight frames in Euclidean spaces are widely used convenient generalizations of orthonormal bases. A particularly nice class of such frames is generated as orbits under irreducible actions of finite groups of orthogonal matrices: these are called irreducible group frames. Integer spans of rational irreducible group frames form Euclidean lattices with some very nice geometric properties, called strongly eutactic lattices. We discuss this construction, focusing on an especially interesting infinite family in arbitrarily large dimensions, which comes from vertex transitive graphs. We demonstrate several examples of such lattices from graphs that exhibit some rather fascinating properties. This is joint work with Deanna Needell, Josiah Park and Yuxin Xin.

Improving a result of Brown and Steinerberger, we present a Berry-Esseen type smoothing inequality for the quadratic Wasserstein metric on compact Riemannian manifolds, which estimates the distance between two probability measures in terms of their Fourier transforms. The inequality is sharp, and has a wide range of applications in probability theory and number theory. We discuss sharp convergence rates of the empirical measure of an i.i.d. or stationary weakly dependent sample, complementing recent results of Bobkov and Ledoux on the unit cube, and Ambrosio, Stra and Trevisan on compact manifolds. We also estimate the convergence rate of random walks on compact groups to the Haar measure, and establish the functional CLT and the functional LIL for additive functionals. On compact semisimple Lie groups these hold even without a spectral gap assumption. As an application to finite point sets arising in number theory, we show that a classical construction of Lubotzky, Phillips and Sarnak on SU(2) and SO(3) achieves optimal rate in the quadratic Wasserstein metric.

The Embedded-Atom Model (EAM) provides a phenomenological description of atomic arrangements in metallic systems. It consists of a configurational energy depending on atomic positions and featuring the interplay of two-body atomic interactions and nonlocal effects due to the corresponding electronic clouds. In this talk, I will present minimality results for this system among lattices in dimensions 2 and 3 as well as other aspects of the problem. This is a joint work with Manuel Friedrich (University of Erlangen) and Ulisse Stefanelli (University of Vienna).

The classical Erdős-Turán inequality on the distribution of roots for complex polynomials can be equivalently stated in a potential theoretic formulation, that is, if the logarithmic potential generated by a probability measure on the unit circle is close to 0, then this probability measure is close to the uniform distribution. We generalize this classical inequality from $d=1$ to higher dimensions $d>1$ with the class of Riesz potentials which includes the logarithmic potential as a special case. In order to quantify how close a probability measure is to the uniform distribution in a general space, we use Wasserstein-infinity distance as a canonical extension of the concept of discrepancy. Then we give a compact description of this distance. Then for every dimension $d$, we prove inequalities bounding the Wasserstein-infinity distance between a probability measure $\rho$ and the uniform distribution by the $L^p$-norm of the Riesz potentials generated by $\rho$. Our inequalities are proven to be sharp up to the constants for singular Riesz potentials. Our results indicate that the phenomenon discovered by Erdős and Turán about polynomials is much more universal than it seems. Finally we apply these inequalities to prove stability theorems for energy minimizers, which provides a complementary perspective on the recent construction of energy minimizers with clustering behavior.

We consider the sharp next-order asymptotics problems for: (1) the minimum energy for optimal N-point configurations; (2) the N-Marginal Optimal Transport; and (3) the Jellium problem for N-point configurations, in all three cases with Riesz costs with inverse power-law long-range interactions. The first problem describes the ground state of a Coulomb or Riesz gas, the second appears as a semiclassical limit of DFT energy, modelling a quantum version of the same system (and is called Uniform Electron Gas in the physics literature), and the third describes charges in a uniform negative background, a rough model for electrons in a metal. Recently the second-order terms in the large-N asymptotic expansions for power s in dimension d were shown for: (1) for $\text{max}(0,d-2)\le s< d$ (remaining open outside this range prior to our paper, as previous methods break down); and for (2) for $0< s< d$. The asymptotics expansion for (3) has long been known for $s=d-2$, but it has been otherwise open until now.

In the present work, we extend the sharp asymptotics for: 1) to $0< s<\text{max}(0,d-2)$; and for 3) to $0< s<d$. Our paper's unified proof for these sharp asymptotics for $0< s<d$ is based on a new and robust screening procedure, which allowed a series of improvements on the existing theory. Our methods and results are extendable to other potentials with long-range and short-range interaction.

Moreover, we show here for the first time that for inverse-power-law interactions with power $0<s<d$, the second-order terms for these three problems are equal. For the Coulomb cost in $d=3$, our result was the first to verify the physicists' long-standing conjecture regarding the equality of the second-order terms for Jellium and Uniform Electron Gas. Moreover, if the crystallization hypothesis in $d=3$ holds, which is an extension of Abrikosov's conjecture originally formulated in $d=2$, then our result is the first to verify the physicists' conjectured 1.4442 lower bound on the famous Lieb-Oxford constant. Our work rigorously confirms some of the predictions formulated by physicists, regarding the optimal value of the Uniform Electron Gas second-order asymptotic term.

Additionally, we show that on the whole range $s\in(0,d)$, the Uniform Electron Gas second-order constant is continuous in $s$.

Given any full rank lattice and a natural number $N$, we regard the point set given by the scaled lattice intersected with the unit square under the Lambert map to the unit sphere, and show that its spherical cap discrepancy is at most of order $N$, with leading coefficient given explicitly and depending on the lattice only. The proof is established using a lemma that bounds the amount of intersections of certain curves with fundamental domains that tile $\mathbb R^2$, and even allows for local perturbations of the lattice without affecting the bound, proving to be stable for numerical applications. A special case yields the smallest constant for the leading term of the cap discrepancy for deterministic algorithms up to date.

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.

(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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

We will discuss the dispersion of a point set, which is simply the volume of the largest box not intersecting the given point set. We shall present several recent result about this notion, including estimates of its high-dimensional asymptotic and deterministic constructions. If time permits, we sketch the most important parts of the proofs.

We will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalization of Venkatesh's lattice packing result of 2013. In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rationals. This improves the best known lower bounds on lattice packing problem in many dimensions. The talk will cover previously known bounds, an overview of the new bounds and a live numerical simulation of Siegel's mean value theorem.

In the seminar we provide upper and lower bounds on the minimal spherical dispersion. In particular, we see that the inverse of the minimal spherical dispersion behaves linearly in the dimension. We also talk about upper and lower bounds of the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere.

The content of the talk is partially based on https://arxiv.org/abs/2103.11701.

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Surprisingly, a certain weighted least squares recovery operator which uses random samples from a distribution, depending on the spectral properties of the corresponding embedding, leads to near optimal results in several relevant situations. As an application we obtain new recovery guarantees for Sobolev type spaces related to Jacobi type differential operators on the one hand and classical multivariate periodic Sobolev type spaces with general smoothness weight on the other hand. By applying a recently introduced sub-sampling technique related to Weaver's conjecture, we further reduce the sampling budget and improve on bounds for the corresponding sampling numbers.
This is a joint work with Tino Ullrich.

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.

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.

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.

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.)

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.

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 $N^{-\frac 12}\sqrt{\log N}$.

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.

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.

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

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

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.

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.

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 $5$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

(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.

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.

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.

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.

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).

We find upper and lower estimate for the discrete energies whose Legendre-Fourier coefficients decrease to zero approximately as power functions.

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.

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).

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.

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.

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.

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.