## Spring 2021

 Speaker Date

## 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]
Video: [mp4]

• 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]
Video: [mp4]

• 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]
Video: [mp4]

• 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.

Slides: (coming soon)
Video: [mp4]

• 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]
Video: [mp4]

• 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.

Slides: (coming soon)
Paper: [arXiv1] [arXiv2]
Video: [mp4]

• 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]
Video: [mp4]

• 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]
Video: [mp4]

• 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]
Video: [mp4]

• 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]
Video: [mp4]

• 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.
Video: [mp4]

• 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]
Video: [mp4]

• ## 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]
Video: [mp4]

• 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]
Video: [mp4]
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]
Video: [mp4]

• 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]
Video: [mp4]

• 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]
Video: [mp4]

• 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]
Video: [mp4]

• 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]
Video: [mp4]
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]
Video: [mp4]

• 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]
Video: [mp4]
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]
Video: [mp4]
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]
Video: [mp4]
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]
Video: [mp4]

• 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]
Video: [mp4]
Paper: [arXiv]

• Aug 28
David Krieg (JKU Linz)
Order-optimal point configurations for function approximation
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.

Video: [mp4]
Slides: [pdf]