Abstract

Optimal configurations can typically be realized as minimizers of some cost function. While in general the function can have many local minima, making it challenging to search for minimizers numerically, some cost functions which have natural interpretations in geometric invariant theory and symplectic geometry are surprisingly simple to optimize despite being non-convex. The goal in this talk is to explain some of the geometric context, and then to illustrate this approach with some applications, including to equal-norm Parseval frames, tight fusion frames, and normal matrices. This is joint work with Tom Needham and partially with Dustin Mixon and Soledad Villar.

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This talk will provide an overview of recent work describing the topology of various spaces of structured configurations of vectors; for example, spaces of unit norm tight frames or spaces of normal matrices with additional constraints. Our work is largely based on the observation that these spaces arise naturally from the perspective of symplectic geometry, allowing for the application of powerful results from that field. I will explain the general ideas that we use from symplectic geometry, and illustrate them on concrete spaces of interest. I will also describe recent results on frames on manifolds — that is, structured collections of vector fields — where the topological questions become much more subtle.

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We discuss the problem of optimally arranging 2d lines (1-dimensional subspaces) through the origin of C^d. Specifically, we conjecture that the Welch can always be attained in this setting, and we present new explicit constructions of equiangular tight frames that perform this feat.

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In this talk a new extension of known semidefinite and linear programming upper bounds for spherical codes will be presented and consider a version of this bound for distance graphs. The main result can be applied for the distance distribution of a spherical code, and it will be shown that this method can work effectively. In particular, I get a shorter solution to the kissing number problem in dimension 4. I will consider reasonable approaches for a solutions of two long standing open problems: the uniqueness of maximum kissing arrangements in 4 dimensions and the 24-cell conjecture. In a recent preprint by de Laat -- Leijenhorst -- de Muinck Keizer claimed a solution to the uniqueness problem. We will also discuss their approach.

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In this talk, we'll explore several new constructions of strongly regular graphs. Specifically, we will show how some known optimal line packings can be coerced into generating new families of SRGs. We will also introduce a new construction of optimal line packings, yielding additional infinite families of SRGs.

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We describe work towards a construction of d^2 equiangular lines in d-dimensional complex Hilbert space (equivalently, a SIC-POVM) from special values of complex analytic functions. A leading role is played by certain function-valued cocycles for subgroups of SL_2(Z), which arise in both number theory and quantum field theory. The remaining obstacles are two unproven conjectures and the presence of a non-continuous Galois automorphism in the construction. Despite these obstacles, the construction has been implemented to produce new examples of SIC-POVMs. This talk covers joint work with Marcus Appleby and Steven Flammia and may also cover joint work with Jeffrey Lagarias.

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After introducing the concepts of minimal projections and maximal projection constants, we highlight some reformulated expressions for the maximal relative projection constants which are more amenable to numerical computations. From here, we derive upper estimates for the maximal relative projection constants that turn into equalities when and only when equiangular tight frames exist. As shown by Deregowska and Lewandowska, a similar equivalence holds between m-th maximal absolute projection constants and maximal equiangular tight frames in dimension m, hence essentially rephrasing (the weak form of) Zauner's conjecture in terms of the values of the maximal absolute projection constants in the complex setting. In the real setting, where maximal equiangular tight frames can fail to exist, we present supporting evidence for our speculated value of the fifth maximal absolute projection constant, exploiting in particular the notion of real mutually unbiased equiangular tight frames.

Abstract

Every equi-isoclinic tight fusion frame (EITFF) is a type of optimal Grassmannian code, and yields a collection of isometries with minimal block coherence. We discuss two recent projects involving EITFFs. The first is joint work with John Jasper, Joey Iverson and Dustin Mixon. We characterize when an EITFF arises as the orbit of a single subspace under the action of a finite abelian group of unitary operators. This yields new EITFFs, and generalizes the well-known equivalence between harmonic equiangular tight frames and difference sets. The second is joint work with Enrique Gomez-Leos and Joey Iverson. We generalize some classical results of Lemmens and Seidel to the complex setting, fully characterizing the existence of EITFFs whose subspaces have dimension equal to exactly half of that of the ambient space. We moreover show that all such EITFFs are necessarily highly symmetric.

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We examine the problem of minimizing the continuum nonlocal, nonconvex variational problem that arises from modeling a large number of pairwise interacting particles in the presence of thermal noise (i.e., molecular dynamics). Determining global minima (ground states) to these functionals is important as they characterize the structure of matter, self-assembly, and phase transitions in materials. Determining global minima is, however, in general difficult. We will derive linear programming (LP) lower bounds in the spirit of Cohn and Kumar via a dual approach as convex relaxations over closed subsets of probability measures. We will then present solutions to the LP bounds for several interaction kernels inspired by molecular dynamics – the Morse potential and an Onsagar potential from liquid crystals. We will also discuss several discrete cases where the minimizers are provably sharp and counter examples where the LP bound fails to be a tight lower bound.

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This talk is concerned with achieving optimal coherence for highly redundant real or complex unit-norm frames. When the number of vectors in a frame becomes too large to admit equiangular arrangements, other geometric optimality criteria need to be identified. The key idea for these results is iterating the embedding technique by Conway, Hardin and Sloane. As a consequence of their work, a quadratic mapping embeds equiangular lines into a simplex in a real Euclidean space. Here, higher degree polynomial maps embed highly redundant unit-norm frames to simplices in high-dimensional Euclidean spaces. This talk focuses on the lowest degree maps and extends earlier work with John Haas on the real case.

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Plasma can get very unstable: Minor perturbations to equilibrium states can potentially instigate rapid growth, resulting in substantial disruptions of the equilibrium. Deploying external electric field to suppress such instabilities is a desirable demand in numerous practical applications, but also a very challenging task. We use the Vlasov-Poisson equation as a fundamental model that simulates plasma dynamics, and aim at designing external fields to suppress the Two-Stream and Bump-on-Tail instabilities, two celebrated unstable equilibrium in plasma physics. Specifically, we introduce a comprehensive framework that employs linear stability analysis to engineer an external field aimed at counteracting poles within the complex plane, thereby averting solutions that tend toward infinity. Additionally, we note that an external field designed to naturally oppose the electric field can lead the plasma towards a free-streaming flow characterized by exponential decay. Our demonstrations reveal that both methodologies yield computationally favorable outcomes. Furthermore, we establish that the initial approach, when applied with a particular counteracting function, inherently gives rise to the second approach.

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In 1955, Rankin resolved the classical problem of determining the maximum number of unit vectors in R^r with no pairwise inner product exceeding alpha for all alpha ≤ 0 and more recently, Bukh and Cox asked about what happens when alpha is slightly bigger than 0. In this talk, we will answer their question by showing that the maximum is (2 + o(1))r for all 0 ≤ alpha ≪ r^(-2/3), where the exponent -2/3 is best possible. As a consequence, we obtain an upper bound on the size of a q-ary code with block length r and distance (1 - 1/q)r - o(r^(1/3)), which is tight up to the multiplicative factor 2(1 - 1/q) + o(1) for any prime power q and infinitely many r. When q = 2, this resolves a conjecture of Tietäväinen from 1980 in a strong form. Time permitting, we will mention how our result translates to set-coloring Ramsey numbers via a recently discovered connection.

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Motivated by the asymptotic optimality of van der Corput point sets and Fibonacci lattices with respect to discrepancy we study more general point sets constructed from permutations. In this direction, generalizing a result already observed by Hinrichs, Kritzinger and Pillichshammer for van der Corput point sets and rational lattices, we show that the periodic and extremal L2-discrepancy of permutation sets are related to each other by a precise equality. This result can be generalized to arbitrary dimensions when working with point sets constructed from latin hypercubes. We also obtain asymptotically tight bounds on the optimal periodic L2 discrepancy of latin hypercubes for dimensions d>2.

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I will show how to construct computationally simple sequences S in the d-dimensional hypercube with base b. For d = 1 these will be (generalized) van der Corput sequences. Further, I will introduce the notion of f -subadditivity and use it to define a very general notion of discrepancy function D which serves as an umbrella term that covers the Lp -discrepancy, Wasserstein p-distance, and many more methods to compare empirical measures to an underlying base measure. I will use these concepts to prove novel bounds of the Lp discrepancy of van der Corput sequences in terms of digit sums. This talk is based on the paper with the same title: https://arxiv.org/abs/2308.13297

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We present a new algorithm that computes packings on the Stiefel manifold for multi-input multi-output coherent communications. This algorithm builds upon an algorithm by Alvarez-Vizoso et al. for Grassmannian packings and leverages the Riemannian geometry of the Stiefel manifold by optimizing with gradient descent on the Steifel manifold. We apply this algorithm to generate new Steifel packings and compare them to other packings for coherent communications.

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The Paley--Wiener Theorem is a classical result about the stability of basis in Banach spaces claiming that if a sequence is close to a basis, then this sequence is a basis. Similar results are extended to frames in Hilbert spaces. As the extension of finite frames for $\mathbb{R}^d$, probabilistic frames are probability measures on $\mathbb{R}^d$ with finite second moments and the support of which span $\mathbb{R}^d$. This paper generalizes the Paley--Wiener theorem to the probabilistic frame setting. We claim that if a probability measure is close to a probabilistic frame in some sense, this probability measure is also a probabilistic frame.

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It was previously shown by the authors that the discrete potentials of almost all known sharp codes attain universal lower bounds for polarization (PULB) for spherical $\tau$-designs, where “universal” is meant in the sense of applying to a large class of potentials that includes absolutely monotone functions of inner products and in the sense that the computational parameters of the bound are invariant with respect to the potential. In this talk we characterize the sets of universal minima D for some of these sharp codes $C$ found in the Leech lattice and establish a duality relationship, namely that the normalized discrete potentials of $C$ and $D$ have the same minimum value and the sets $C$ and $D$ are each others minima sets (up to antipodal symmetrization in some cases). The extremal duality is obtained by utilizing the natural embedding of the PULB pair codes in the Leech lattice and its properties, which simplifies the analysis significantly. In the process we discover a new universally optimal code in $\mathbb{RP}^{21}$ with $1408$ points.

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We develop “linear programming” bounds for the energy of lattice periodic configurations in Euclidean space and prove the optimality of two A2-periodic configurations in the plane (one with four generators and and with six generators) for all energy functionals with rapid decay arising from completely monotone functions of distance squared; i.e., these are universally optimal (in the language of Cohn and Kumar) amongst all A2 periodic configurations with the same number of generators. This is joint work with Nate Tenpas.

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One way of finding an "optimal" point configuration is to determine one that maximizes the sum of pairwise distances between distinct points. Specifically in the case of Euclidean distance on the sphere, such point sets are uniformly distributed, minimize the quadratic spherical cap discrepancy (which is equivalent to a certain worst case error estimate) over all point sets of the same cardinality. However, for other metrics on other spaces, such as geodesic distances on spheres or projective spaces, maximizing the sum of distances may not result in uniformly distributed point sets. We will discuss what is known in these settings, as well as recent progress in determining optimizers of the more general Geodesic Riesz energies on these spaces.

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We will discuss some simple perspectives on Fourier uncertainty from the point of view of exact signal recovery. The main theme is that one always has a stronger uncertainty principle in the presence of non-trivial Fourier restriction estimates. Arithmetic aspects of the problem will be discussed as well.

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(Joint with R. Ben-Av, X. Chen, G. Dula, K. Okoudjou) Some recent papers discuss a method of generation of complex line configurations from association schemes. If the scheme is Schurian, subordinate to a group G, then G is an automorphism subgroup of the configuration. However, G then acts on the points, without re-phasing. In this work we propose an extension to weighted Schemes, which introduces G actions that involve re-phasing. Our extension may also accommodate algebraic configurations, in which G is also allowed to act by Galois transformations. This construction involves the study of the cohomology groups of G and of certain subgroups, and is closely related to arithmetic invariants, like the Brauer group.

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We describe an algorithm for generating all the possible PIW(m, n, k) - integer m × n Weighing matrices of weight k up to Hadamard equivalence. Our method relies on properties of a specific matrix ordering, and the results can used for classification of isomorphism classes of integer matrices. Our method is efficient on a personal computer for small size matrices, up to m ≤ n = 12, and k ≤ 50. As a by product we also improved the nsoks [12] algorithm to find all possible representations of an integer k as a sum of n integer squares. We have implemented our algorithm in Sagemath and as an example we provide a complete classification for n = m = 7 and k = 25. Our list of IW(7, 25) can serve as a step towards finding the open classical weighing matrix W(35, 25).

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This talk explores a subcase of the Heil-Ramanathan-Topiwala (HRT) conjecture, which proposes that a set of any finite time-frequency shifts of a nonzero square-integrable function is linearly independent. We identify and discuss certain sufficient conditions, focusing on the rational dimension of a particular vector space and the size of the zero set of the Zak transform under which the conjecture remains valid. A notable implication of our main result is the successful resolution of a case of the HRT subconjecture, originally proposed by Chris Heil.

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In this presentation, we will address the motivation behind studying the eigenvalue distribution of spatio-frequency limiting operators and provide an overview of the recent developments in this area.

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The p-frame potential for a pair of unit vectors x, y is defined as |<x,y>|^p. The general problem is to determine a minimizing point configuration of a given size for a given p. In the talk I will give a brief overview of this problem and its analog for measures and show several new results for 0<p<2. In particular, I will explain why measures whose support is an orthonormal basis are local minimizers of the p-frame energy and use this result to resolve the conjecture of Ben Av, Chen, Goldberger, Kang, and Okoudjou.

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We will discuss some optimization problems originating in physics. Non-isotropic interactions depending not only on the distances between interacting particles, but also on their relative positions will be of special interest. This presentation is partly based on the joint work with J. Batle and O. Ciftja.

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Joint work with: A. Anderson, D. Bilyk, B. Borda, J. Brauchart, M. Dostert, R. Matzke, T. Stepaniuk We study Riesz, Green and logarithmic energy on two-point homogeneous spaces. More precisely we consider the real, the complex, the quaternionic and the Cayley projective spaces. For each of these spaces we provide upper estimates for the mentioned energies using determinantal point processes. Moreover, we determine lower bounds for these energies. Furthermore, we extend the notion of hyperuniformity to the projective spaces and study the connection between energy and the Wasserstein distance.

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