Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Location: Unitobler, F-106
30 seats, 54m^2
Date: Tuesday, 09/Jul/2019
10:00am - 12:00pmMS151, part 1: Cluster algebras and positivity
Unitobler, F-106 
 
10:00am - 12:00pm

Cluster algebras and positivity

Chair(s): Lisa Lamberti (ETHZ, Switzerland), Khrystyna Serhiyenko (University of California, Berkeley, USA / University of Kentucky, Lexington), Lauren Williams (Harvard, USA)

Cluster algebras are commutative rings whose generators and relations can be defined in a remarkably succinct recursive fashion. Algebras of this kind, introduced by Fomin and Zelevinsky in 2000, are equipped with a powerful combinatorial structure frequently appearing in many mathematical contexts such as Lie theory, triangulations of surfaces, Teichmueller theory and beyond. Coordinate rings of Grassmannians and related invariant rings are well-studied examples of algebras of this type. One important aspect arising from the intrinsic combinatorial structure of cluster algebras is that it uncovers systematic, intriguing and complex positivity properties in these families of rings. For instance, it is expected that for each cluster algebra there is a distinguished basis, such that all elements can be expressed as a "positive" linear combination of basis vectors. Seemingly elementary claims of this type, so far proved only in certain cases, have triggered important developments in research areas at the intersection of geometry, algebra and combinatorics.

In this session, we glimpse at recent developments in this field and discuss open questions.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Toric degenerations of cluster varieties and cluster duality

Konstanze Rietsch
King’s College London

This talk will review some aspects of mirror symmetry for generalised flag varieties G/P and its interaction with the representation theory of G. For a cominuscule homogeneous space G/P there is an expression for the mirror LG model W in terms of coordinates which, by the geometric Satake correspondence in representation theory, are naturally identified with cohomology classes of G/P (joint works with Marsh, Pech, and Williams) leading to some combinatorially very attractive formulas. A relationship between the critical points of W and its tropicalisation, and representation theory (work of Judd) may also be discussed.

 

On mirror symmetry for homogeneous spaces

Lara Bossinger1, Juan Bosco Frías Medina2, Tim Magee2, Alfredo Nájera Chávez2
1Max Planck Institute for Mathematics in the Sciences, 2Instituto de Matematicas UNAM, Mexico

Cluster varieties are a particularly nice class of log Calabi-Yau varieties-- the non-compact analogue of usual Calabi-Yaus. They come in pairs (A,X), with A and X built from dual tori. The punchline of this talk will be that compactified cluster varieties are a natural progression from toric varieties. Essentially all features of toric geometry generalize to this setting in some form, and the objects studied remain simple enough to get a hold of and do calculations.

Compactifications of A and their toric degenerations were studied extensively by Gross, Hacking, Keel, and Kontsevich. These compactifications generalize the polytope construction of toric varieties-- a construction which is recovered in the central fiber of the degeneration. Compactifications of X were introduced by Fock and Goncharov and generalize the fan construction of toric varieties. Recently, Lara Bossinger, Juan Bosco Frías Medina, Alfredo Nájera Chávez, and I introduced the notion of an X-variety with coefficients, expanded upon the notion of compactified X-varieties, and for each torus in the atlas gave a toric degeneration where each fiber is a compactified X-variety with coefficients. We showed that these fibers are stratified, and each stratum is again a compactified X-varieties with coefficients. In the central fiber, we recover the toric variety associated to the fan in question, and we show that strata of the fibers degenerate to toric strata. This talk is based on arXiv:1809.08369.

 

Generalised friezes and the weak Ptolemy map

Ilke Canakci, Peter Jørgensen
Newcastle University, UK

Frieze patterns, introduced by Conway, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Frieze patterns with positive integer values are of a special interest since they are in one-to-one correspondence with triangulations of polygons by Conway--Coxeter. Remarkably, this established a connection to cluster algebras–predating them by 30 years– and to cluster categories. Several generalisations of frieze patterns are known. Joint with Jørgensen, we associated frieze patterns to dissections of polygons where the entries are over a (commutative) ring. Furthermore, we introduced an explicit combinatorial formula for the entries of these friezes by generalising the 'T-path formula' of Schiffler which was introduced to give explicit formulas for cluster variables for cluster algebras of type A.

 

Perfect matching modules for dimer algebras

Ilke Canakci1, Alastair King2, Matthew Pressland3
1Newcastle University, UK, 2University of Bath,UK, 3Universität Stuttgart, D

The theory of dimer models, or bipartite graphs on surfaces, first arose in theoretical physics, and later found diverse applications in geometry and representation theory. Recently, there has been much interest in dimer models on the disk, particularly those arising from Postnikov diagrams, and their relationship to Grassmannian cluster algebras and categories. Perfect matchings of dimer models play a central role in the theory. In joint work with İlke Çanakçı and Alastair King, we provide an algebraic viewpoint on these objects, by defining and studying a module for the dimer algebra for each perfect matching. As an application, we explain the relationship between combinatorial and homological formulae for computing Grassmannian cluster variables.

 
3:00pm - 5:00pmMS154, part 1: New developments in matroid theory
Unitobler, F-106 
 
3:00pm - 5:00pm

New developments in matroid theory

Chair(s): Alex FInk (Queen Mary), Ivan Martino (KTH), Luca Moci (Bologna)

The interactions between Matroid Theory, Algebra, Geometry, and Topology have long been deep and fruitful. Pertinent examples of such interactions include breakthrough results such as the g-Theorem of Billera, Lee and Stanley (1979); the proof that complements of finite complex reflection arrangements are aspherical by Bessis (2014); and, very recently, the proof of Rota's log-concavity conjecture by Adiprasito, Huh, and Katz (2015).

The proposed mini-symposia will focus on the new exciting development in Matroid Theory such as the role played by Bergman fans in tropical geometry, several results on matroids over a commutative ring and over an hyperfield, and the new improvement in valuated matroids and about toric arrangements. We plan to bring together researchers with diverse expertise, mostly from Europe but also from US and Japan. We are going to include a number of postdocs and junior mathematicians.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Positivity of the coefficients of G-Tutte polynomials

Tan Nhat Tran
Hokkaido

Recently, the notions of G-Tutte polynomials and G-plexifications were introduced to build a general framework for studying hyperplane, toric, q-reduced arrangements and their "Tutte-like" polynomials (Tutte, arithmetic Tutte, characteristic (quasi-)polynomials) en masse. Like the above-mentioned (quasi-)polynomials, the G-Tutte polynomials possess Deletion-Contraction and convolution formulas, but unlike them, the G-Tutte polynomials may have negative coefficients. We are currently interested in under what conditions their coefficients all are positive? In this talk, we will propose some ideas and partial answers. This talk is based on two recent joint works with Ye Liu and Masahiko Yoshinaga.

 

Enumerative aspects of G-Tutte polynomials

Masahiko Yoshinaga
Hokkaido

G-Tutte polynomial is a generalization of arithmetic Tutte polynomial. I will discuss some results on enumerative aspects of G-Tutte polynomial.

 

Abelian arrangements, matroids and group actions

Emanuele Delucchi
Fribourg (CH)

Arrangements of hyperplanes have long offered a geometric point of view on matroids - at times leading to structural advances even in the nonrealizable case. The theory of arrangements recently broadened its scope beyond the case of hyperplanes to include arrangements in the torus, in products of elliptic curves and, more generally, in Abelian Lie groups. This development spurred the search for suitable enrichments of matroid theory.

In this context, I will introduce the foundations of a theory of group actions on (semi)matroids, focussing mainly on applications to the structure of intersection posets of arrangements. I will also outline how this framework relates to (arithmetic) Tutte polynomials, arithmetic matroids and G-Tutte polynomials. A further ramification of this setup will be illustrated in A. D’Alì’s talk on generalized Stanley-Reisner rings.

The material I will present is partly drawn from joint works with Alessio D’Alì, Giacomo d’Antonio, Noriane Girard, Giovanni Paolini and Sonja Riedel.

 

Group actions on generalized Stanley-Reisner rings

Alessio D'Ali
Genova

The Stanley-Reisner correspondence, which assigns a commutative ring to each finite simplicial complex, is a useful and well-studied bridge between commutative algebra and combinatorics, yielding particularly nice results for the independence complex of a matroid. In 1987 Sergey Yuzvinsky proposed a construction that allows to see the Stanley-Reisner ring of any finite simplicial complex as the ring of global sections of a sheaf of rings on a poset. Motivated by applications in the theory of Abelian arrangements, E. Delucchi and I extend Yuzvinsky's construction to the case of (possibly infinite) finite-length simplicial posets. We show that this generalization behaves well with respect to quotients of simplicial complexes and posets by group actions such as those introduced in E. Delucchi's talk.

 

Date: Wednesday, 10/Jul/2019
10:00am - 12:00pmMS151, part 2: Cluster algebras and positivity
Unitobler, F-106 
 
10:00am - 12:00pm

Cluster algebras and positivity

Chair(s): Lisa Lamberti (ETHZ, Switzerland), Khrystyna Serhiyenko (University of California, Berkeley, USA / University of Kentucky, Lexington), Lauren Williams (Harvard, USA)

Cluster algebras are commutative rings whose generators and relations can be defined in a remarkably succinct recursive fashion. Algebras of this kind, introduced by Fomin and Zelevinsky in 2000, are equipped with a powerful combinatorial structure frequently appearing in many mathematical contexts such as Lie theory, triangulations of surfaces, Teichmueller theory and beyond. Coordinate rings of Grassmannians and related invariant rings are well-studied examples of algebras of this type. One important aspect arising from the intrinsic combinatorial structure of cluster algebras is that it uncovers systematic, intriguing and complex positivity properties in these families of rings. For instance, it is expected that for each cluster algebra there is a distinguished basis, such that all elements can be expressed as a "positive" linear combination of basis vectors. Seemingly elementary claims of this type, so far proved only in certain cases, have triggered important developments in research areas at the intersection of geometry, algebra and combinatorics.

In this session, we glimpse at recent developments in this field and discuss open questions.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Combinatorics of cluster structures in Schubert varieties

Khrystyna Serhiyenko1, Melissa Sherman-Bennett2, Lauren Williams3
1University of California, Berkeley, USA / University of Kentucky, Lexington, 2University of California, Berkeley, USA, 3Harvard, USA

The (affine cone over the) Grassmannian is a prototypical example of a variety with cluster structure. Scott (2006) gave a combinatorial description of this cluster algebra in terms of Postnikov's plabic graphs. It has been conjectured essentially since Scott's result that Schubert varieties also have a cluster structure with a description in terms of plabic graphs. I will discuss recent work with K. Serhiyenko and L. Williams proving this conjecture. The proof uses a result of Leclerc, who shows that many Richardson varieties in the full flag variety have cluster structure using cluster-category methods, and a construction of Karpman to build plabic graphs for each Schubert variety.

 

Cluster tilting modules for mesh algebras

Karin Erdmann1, Sira Gratz2, Lisa Lamberti3
1University of Oxford, UK, 2University of Glasgow, UK, 3ETHZ, Switzerland

Mesh algebras are a class of finite-dimensional algebras which generalize naturally preprojective algebras. In this talk, I describe cluster tilting modules for mesh algebras of Dynkin type and discuss possible relations to skew-symmetrizable cluster algebras structures in certain coordinate rings. This is joint work with K. Erdmann and S. Gratz.

 

Strings, snake graphs and the cluster expansion formulas

Ilke Canakci1, Vincent Pilaud2, Nathan Reading3, Sibylle Schroll4
1Newcastle University, 2École polytechnique, 3NCSU Campus, 4University of Leicester, UK

Snake graphs arise from cluster algebras associated to triangulations of marked oriented surfaces in the work of Musiker, Schiffler and Williams in the context of Laurent expansion formulas. In this talk we will show a correspondence between snake graphs and combinatorial objects called strings. String combinatorics has first arising in the context of the classification of indecomposable modules in a large class of tame algebras, the so-called special biserial algebras. We will show how this new interpretation of snake graphs in terms of strings leads to an alternative cluster expansion formula for cluster algebras arising from triangulations of surfaces. This is joint work with Ilke Canakci as well as joint work in progress with Nathan Reading and Vincent Pilaud.

 

Friezes and Grassmannian cluster structures

Karin Baur1, Eleonore Faber2, Sira Gratz3, Khrystyna Serhiyenko4, Gordana Todorov5
1Universität Graz / University of Leeds, Austria / UK, 2University of Leeds, UK, 3University of Glasgow, UK, 4University of California, Berkeley, USA / University of Kentucky, Lexington, 5Northeastern University, Boston, USA

In this talk, I will show how to obtain SL_k-friezes using Plücker coordinates by taking certain subcategories of the Grassmannian cluster categories. These are cluster structures associated to the Grassmannians of k-spaces in n-space. Many of these friezes arise from specialising a cluster to 1. We use Iyama-Yoshino reduction to reduce the rank of such friezes.

This is joint work with E. Faber, S. Gratz, G. Todorov, K. Serhyenko. https://arxiv.org/abs/1810.10562

 
3:00pm - 5:00pmMS154, part 2: New developments in matroid theory
Unitobler, F-106 
 
3:00pm - 5:00pm

New developments in matroid theory

Chair(s): Alex FInk (Queen Mary), Ivan Martino (Northeastern University, United States of America), Luca Moci (Bologna)

The interactions between Matroid Theory, Algebra, Geometry, and Topology have long been deep and fruitful. Pertinent examples of such interactions include breakthrough results such as the g-Theorem of Billera, Lee and Stanley (1979); the proof that complements of finite complex reflection arrangements are aspherical by Bessis (2014); and, very recently, the proof of Rota's log-concavity conjecture by Adiprasito, Huh, and Katz (2015).

The proposed mini-symposia will focus on the new exciting development in Matroid Theory such as the role played by Bergman fans in tropical geometry, several results on matroids over a commutative ring and over an hyperfield, and the new improvement in valuated matroids and about toric arrangements. We plan to bring together researchers with diverse expertise, mostly from Europe but also from US and Japan. We are going to include a number of postdocs and junior mathematicians.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Cohomology rings of projective models of toric arrangements

Giovanni Gaiffi
PIsa

I will describe, by providing generators, relations and examples, the cohomology rings of projective models of toric arrangements (joint work with Corrado De Concini).

 

Arithmetic matroids, posets and cohomology of toric arrangements

Roberto Pagaria
Pisa

Matroids are cryptomorphic to geometric lattices and from an oriented matroid can be build an Orlik-Solomon algebra. This algebra is the cohomology algebra of the complement of an arrangement of hyperplanes or of pseudospheres, hence the Tutte polynomial specializes to the Poincaré polynomial of the complement.

Recent works introduced arithmetic matroids and studied their relations with the cohomology algebra of toric arrangements. We will discuss the relation between arithmetic matroids and posets of layers of toric arrangements. This study leads to a construction - from the poset of layers - of a "toric Orlik-Solomon algebra" isomorphic to the cohomology algebra of the complement of the toric arrangement. Indeed, the Poincaré polynomial of the complement is a specialization of the arithmetic Tutte polynomial.

 

Categories of matroids, Hopf algebras, and Hall algebras

Jaiung Jun
SUNY Binghamton

In their recent paper, Baker and Bowler introduced the notion of matroids over partial hyperstructures which unifies various generalizations (including oriented, valuated, and phase matroids). One can generalize the notion of minors and direct sums (of matroids) to the case of matroids over partial hyperstructures. In particular, this allows one to generalize the matroid-minor Hopf algebra to this setup. We then investigate the category of (ordinary) matroids, showing that the matroid-minor Hopf algebra is dual to the Hall algebra associated to the category of matroids. This is joint work with Chris Eppolito and Matt Szczesny.

 

Date: Thursday, 11/Jul/2019
10:00am - 12:00pmMS164, part 1: Algebra, geometry, and combinatorics of subspace packings
Unitobler, F-106 
 
10:00am - 12:00pm

Algebra, geometry, and combinatorics of subspace packings

Chair(s): Emily Jeannette King (University of Bremen, Germany), Dustin Mixon (Ohio State University)

Frame theory studies special vector arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown alongside the emergence of new methods in signal processing. Modern advances in frame theory involve techniques from algebraic geometry, semidefinite programming, algebraic and geometric combinatorics, and representation theory. This minisymposium will explore a multitude of these algebraic, geometric, and combinatorial developments in frame theory.
The theme of the first session is "Systems with non-abelian group symmetry."

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Algebra, Geometry, and Combinatorics of Subspace Packings: Gabor-Steiner Equiangular Tight Frames

Emily King
University of Bremen

Desirable traits of subspace arrangements in applications like signal processing and quantum information theory include having large geometric spread between any two subspaces and yielding a resolution of the identity. Methods from algebraic graph theory, real algebraic geometry, symplectic geometry, combinatorial design theory, semidefinite programming, and more can be used to design and characterize such subspace packings. This talk will serve as an introduction to the minisymposium “Algebra, Geometry, and Combinatorics of Subspace Packings.” Gabor-Steiner equiangular tight frames, which are covariant under the Weyl-Heisenberg group and have properties described by different combinatorial designs will also be discussed.

 

Group frames, full spark, and other topics

Romanos-Diogenes Malikiosis
Aristotle University of Thessaloniki

A group frame is the orbit of a vector in a vector space of dimension N under the action of a (projective) linear representation of a finite group. Such a frame satisfies the full spark property, if every selection of N vectors from the frame constitutes a basis.

We will examine whether certain families of group frames satisfy the full spark property, extending the results by the speaker for the Weyl-Heisenberg group (i.e. Gabor frames) and by Oussa-Sheehan for the dihedral case. If time allows, we will also mention other topics as well, for example equiangularity.

This is joint work with Vignon Oussa.

 

Equiangular tight frames from nonabeilan groups

John Jasper
South Dakota State University

Several applications in signal processing require lines through the origin of a finite-dimensional Hilbert space with the property that the smallest interior angle is as large as possible. Packings that achieve equality in the Welch bound are known as equiangular tight frames (ETFs). Since optimal packings often exhibit symmetry, it is natural to expect such packings to be related to groups. Indeed, a popular type of ETFs are the so-called harmonic ETFs, that is, ETFs that arise from the action of an abelian group on a single vector. On the other hand, perhaps the most famous open problem in this area is Zauner's conjecture, which asks for an ETF from the action of the Heisenberg group, which is nonabelian. The theory of harmonic ETFs is fairly well understood as it is equivalent to well-studied objects known as difference sets. The theory of ETFs generated by nonabelian groups is much more mysterious. In this talk we will discuss this theory and present a construction of the first infinite family of ETFs arising from nonabelian groups.

 

SIC-POVM existence and the Stark conjectures

Gene Kopp
University of Bristol

The existence of a configuration of equiangular lines in d-dimensional complex Hilbert space of cardinality achieving the theoretical upper bound of d^2 is known only for finitely many dimensions d. Such configurations have been studied extensively in the context of quantum information theory, in which they are known as symmetric informationally complete positive operator-valued measures (SIC-POVMs).

We give an explicit conjectural construction of SIC-POVMs in an infinite family of dimensions. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields. Moreover, in the same family, we prove a conditional result stating that SIC-POVMs exist under a strong algebraic hypothesis about units in a certain number field. The talk will include a worked example in dimension d=5 and an overview of some number-theoretic background necessary to understand the main results.

 
3:00pm - 5:00pmMS154, part 3: New developments in matroid theory
Unitobler, F-106 
 
3:00pm - 5:00pm

New developments in matroid theory

Chair(s): Alex FInk (Queen Mary), Ivan Martino (Northeastern University, United States of America), Luca Moci (Bologna)

The interactions between Matroid Theory, Algebra, Geometry, and Topology have long been deep and fruitful. Pertinent examples of such interactions include breakthrough results such as the g-Theorem of Billera, Lee and Stanley (1979); the proof that complements of finite complex reflection arrangements are aspherical by Bessis (2014); and, very recently, the proof of Rota's log-concavity conjecture by Adiprasito, Huh, and Katz (2015).

The proposed mini-symposia will focus on the new exciting development in Matroid Theory such as the role played by Bergman fans in tropical geometry, several results on matroids over a commutative ring and over an hyperfield, and the new improvement in valuated matroids and about toric arrangements. We plan to bring together researchers with diverse expertise, mostly from Europe but also from US and Japan. We are going to include a number of postdocs and junior mathematicians.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Characterizing quotients of positroids

Anastasia Chavez
UC Berkeley

We characterize quotients of specific families of positroids. Positroids are a special class of representable matroids introduced by Postnikov in the study of the nonnegative part of the Grassmannian. Postnikov defined several combinatorial objects that index positroids. In this paper, we make use of two of these objects to combinatorially characterize when certain positroids are quotients. Furthermore, we conjecture a general rule for quotients among arbitrary positroids on the same ground set.

 

Algebraic matroids and flocks

Rudi Pendavingh
TU Eindhoven

Let $K$ be a field, $E$ a finite set, and $Xsubseteq K^E$ an algebraic variety. Then $M(X)$ is the matroid on ground set $E$ in which a set $Fsubseteq E$ is independent if and only if the projection ${(x_i: i in F): xin X}$ is a dominant subset of $K^F$. In general it is difficult to decide if a given matroid $M$ is algebraic over $K$, that is if $M=M(X)$ for some variety $Xsubseteq K^E$.

We have recently found that if the field $K$ has positive characteristic $p$, then the variety $X$ determines further structure on $M(X)$ which comprises information on the generic tangent spaces of $X$ as well as a family of closely related varieties. This additional structure can either be cast as a matroid over a hyperfield, or as a {em flock}, which is essentially a labelling of the cells of a tropical linear space by linear subspaces of $K^E$.

We show how this gives useful necessary conditions on the algebraicity of matroids.

This is joint work with Guus Bollen and Jan Draisma.

 

Tropical Ideals

Jeffrey Herschel Giansiracusa
Swansea

The scheme-theoretic approach to tropical geometry has motivated the study of tropical ideals, which are sequences of (valuated) matroids $M_i$ on the monomials of a polynomial ring that form an ideal in the sense that $x_j M_i subset M_{i+1}$. While the class of arbitrary ideals can behave very badly, tropical ideals exhibit many nice properties, while also presenting many new features, challenges, and mysteries. There are realizable tropical ideals, meaning that they are formed by tropicalizing classical ideals, and there are non-realizable tropical ideals. Three interesting questions are:

1. What invariants of a classical ideal are encoded in its associated tropical ideal?
2. How does the tropicalization of an ideal change as the ideal changes (moving within the Hilbert scheme)?
3. How can one construct non-realizable tropical ideals?

In this talk I will discuss examples, progress on each of these questions.

 

Date: Friday, 12/Jul/2019
10:00am - 12:00pmMS164, part 2: Algebra, geometry, and combinatorics of subspace packings
Unitobler, F-106 
 
10:00am - 12:00pm

Algebra, geometry, and combinatorics of subspace packings

Chair(s): Emily Jeannette King (University of Bremen, Germany), Dustin Mixon (Ohio State University)

Frame theory studies special vector arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown alongside the emergence of new methods in signal processing. Modern advances in frame theory involve techniques from algebraic geometry, semidefiniteprogramming, algebraic and geometric combinatorics, and representation theory. This minisymposium will explore a multitude of these algebraic, geometric, and combinatorial developments in frame theory.

The theme of the second session is "Equiangular lines."

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Equiangular tight frames from group divisible designs

Matthew Fickus
Air Force Inst. of Technology

An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem for many choices of parameters. We discuss how many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs.

 

Using Biangular Gabor Frames to Construct Equiangular Tight Frames

Mark Magsino
Ohio State University

Biangular Gabor Frames can be described by a system of real polynomial equations of many variables. Numerical computations suggests this system is usually one dimensional. Furthermore, a vector containing all ones leads to a biangular Gabor frame. This gives the idea of following the curve to an equiangular tight frame somewhere along this curve. We present numerical results of finding equiangular tight frames using this method.

 

Doubly transitive lines: Symmetry implies optimality

Joseph Iverson
Iowa State University

Since the work of Tóth on regular figures, it has been widely observed that optimal solutions to packing problems frequently display extraordinary symmetries. For instance, spheres centered on points in the Leech lattice give an optimal packing in 24 dimensions, while lines through through antipodal vertices of an icosahedron give an optimal packing in two-dimensional projective space. In this talk, we demonstrate an extreme case of this phenomenon for line packings: symmetry can be a sufficient condition for optimality. Specifically, consider n lines spanning a space of dimension d < n. If the lines have a doubly transitive automorphism group, then they are optimally packed in projective space. In fact, unit norm representatives for the lines reach equality in the Welch bound to create an equiangular tight frame. We will explain this phenomenon, and then discuss progress toward a classification of all doubly transitive lines.

This is joint work with Dustin G. Mixon.

 

Equiangular lines in $\mathbb R^{17}$ and the characteristic polynomial of a Seidel matrix

Gary Greaves
Nanyang Technological University

A system of lines through the origin of $mathbb R^d$ for which the angle between any pair of lines is a constant is called equiangular. A Seidel matrix, which can be interpreted as a variation of the adjacency matrix of a graph, is a tool for studying equiangular line systems. In this talk we present our recent improvement on the upper bound for the cardinality of an equiangular line system in $mathbb R^{17}$. A crucial ingredient for this improvement is a new restriction on the characteristic polynomial of a Seidel matrix.

This talk is based on joint work with Pavlo Yatsyna.

 
3:00pm - 5:00pmMS154, part 4: New developments in matroid theory
Unitobler, F-106 
 
3:00pm - 5:00pm

New developments in matroid theory

Chair(s): Alex FInk (Queen Mary), Ivan Martino (Northeastern University, United States of America), Luca Moci (Bologna)

The interactions between Matroid Theory, Algebra, Geometry, and Topology have long been deep and fruitful. Pertinent examples of such interactions include breakthrough results such as the g-Theorem of Billera, Lee and Stanley (1979); the proof that complements of finite complex reflection arrangements are aspherical by Bessis (2014); and, very recently, the proof of Rota's log-concavity conjecture by Adiprasito, Huh, and Katz (2015).

The proposed mini-symposia will focus on the new exciting development in Matroid Theory such as the role played by Bergman fans in tropical geometry, several results on matroids over a commutative ring and over an hyperfield, and the new improvement in valuated matroids and about toric arrangements. We plan to bring together researchers with diverse expertise, mostly from Europe but also from US and Japan. We are going to include a number of postdocs and junior mathematicians.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Gain matroids and their applications

Viktoria Kasznitzky
Eötvös Loránd University

Let (G=(V,E)) be an undirected graph and let (k) and (ell) be two integers. (G) is said to be emph{((k,ell))-sparse} if (|E'|leq k|V'|-ell) holds for every subgraph (G'=(V',E')) of (G). The edge sets of the ((k,ell))-sparse subgraphs form a matroid on the edge set of a graph (H) called the emph{((k,ell)) count matroid}.

Gain matroids are a generalisation of count matroids. Let (Gamma) be a group. Now assign an element of (Gamma) (a emph{gain}) and a reference direction to every edge in (E). The emph{gain} of a (not necessarily directed) closed walk is defined as the group element obtained by the multiplication of the gains of its edges where the inverse gain should be used for the edges used in the reverse direction. The gain group corresponding to the edge set of a connected subgraph is the set of gains of closed walks starting at one of its vertices, (v). (After some observations (v) can be dropped from the definition and it can be extended to arbitrary subgraphs.)

 

Matroid threshold hypergraphs

José Alejandro Samper
Miami

In this talk we introduce the notion of a matroid threshold hypergraph: a collection of bases of a matroid obtained by capping the total weight of the bases under given a function of the ground set. Focusing on the uniform matroid yields the classical theory of threshold hypergraphs. In this talk we will motivate the definition, explain a few their interesting properties and speculate about the uses of the theory.

 

Whitney Numbers for Cones

Galen Dorpalen-Barry
Minnesota

An arrangement of hyperplanes dissects space into connected components called chambers. A nonempty intersection of halfspaces from the arrangement will be called a cone. The number of chambers of the arrangement lying within the cone is counted by a theorem of Zaslavsky, as a sum of certain nonnegative integers that we will call the cone's "Whitney numbers of the 1st kind". For cones inside the reflection arrangement of type A (the braid arrangement), cones correspond to posets, chambers in the cone correspond to linear extensions of the poset, and these Whitney numbers refine the number of linear extensions. We present some basic facts about these Whitney numbers, and interpret them for two families of posets.

 

Date: Saturday, 13/Jul/2019
10:00am - 12:00pmMS164, part 3: Algebra, geometry, and combinatorics of subspace packings
Unitobler, F-106 
 
10:00am - 12:00pm

Algebra, geometry, and combinatorics of subspace packings

Chair(s): Emily Jeannette King (University of Bremen, Germany), Dustin Mixon (Ohio State University)

Frame theory studies special vector arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown alongside the emergence of new methods in signal processing. Modern advances in frame theory involve techniques from algebraic geometry, semidefinite programming, algebraic and geometric combinatorics, and representation theory. This minisymposium will explore a multitude of these algebraic, geometric, and combinatorial developments in frame theory.

The theme of the third session is "Numerical methods in line configurations and spectral decompositions."

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

k-point semidefinite programming bounds for equiangular lines

Fabrício Machado
Universidade de São Paulo

The problem of equiangular lines asks for the maximum number of lines in the n-dimensional Euclidean space with a fixed common angle. By selecting a unit vector along each line we get a spherical code with inner-products a and -a for some fixed 0 < a < 1 and by making a graph with these vectors as vertices and defining an edge whenever the inner-product is -a, this problem also becomes interesting from the algebraic graph theory point of view.

Parallel to this, techniques from semidefinite programming have been successfully applied to many problems in discrete geometry. The problem is modeled with an infinite graph and methods such as the Lovász theta-number used to upper bound the independence number of a finite graph are extended to this infinite setting. A key step in this approach is the use of symmetries from the problem to simplify the formulation and make it solvable by computer.

In this work we define a hierarchy of semidefinite programs specially suited for symmetry reduction techniques. Let S^{n-1} denote the unit sphere in R^n. In the setting of spherical codes, the symmetry group is the orthogonal group O(n) and we consider kernels S^{n-1} x S^{n-1} -> R invariant with respect to the stabilizer subgroup of a set of k points. We show that these kernels can be represented with matrices with size depending on k and not on n and use this characterization to compute new bounds for the maximum number of equiangular lines with fixed common angle.

Joint work with D. de Laat (MIT), F.M. de Oliveira Filho (TU Delft), and F. Vallentin (Universität zu Köln).

 

Using quantum information techniques to find the number of mutually unbiased bases in any given dimension

Marcin Pawłoski
University of Gdansk

Quantum information has seen a very rapid development in the recent years mostly because it provides very abstract and, at the same time, intuitive view of the quantum theory. It makes it easier for us to find new applications in a wide range of fields from metrology, through cryptography to pure mathematics. In this talk I will start by briefly explaining why and how this happens and then spend the most of the time by presenting an example: our recent result in which we were able to link the number of mutually unbiased bases (MUBs) in any given dimension with a success probability of a certain information-processing task. Next I will present how to bound this probability with a hierarchy of semi-definite programmes (SDPs) and report on how we applied it in practice and where did it lead us.

 

Fourier expansions of discrepancy kernels

Martin Ehler
Universität Wien

Many geometrical and combinatorial problems can be formulated as the minimization of specific kernel functions over compact subsets. To derive efficient numerical schemes, we study the spectral decomposition of discrepancy kernels. In particular, we obtain the kernels' Fourier expansions for several distinct examples.

 

Detection of Ambiguities in Linear Arrays in Signal Processing

Frederic Matter
TU Damstadt

This talk considers the detection of ambiguities that arise in the reconstruction of signals that are measured by a linear array which is given by a set of sensors located on a line in the plane.

In this case, signals are related to measurements by a linear equation system, whose matrix depends on the directions-of-arrival (DOA) of the signals and the positions of the sensors.

It can happen that certain DOAs together with certain sensor positions yield a measurement matrix that does not have full rank. This means that for a corresponding measurement the underlying original signal cannot be uniquely identified, and an ambiguity is said to arise.

In order to detect which DOAs produce ambiguities, we consider quadratic submatrices of the measurement matrix. Using Young tableaux and roots of unity, determining rank-deficient quadratic submatrices can be reduced to a mixed-integer program, whose solutions correspond to roots of the so-called Schur polynomial and thus to ambiguities. We demonstrate this approach using examples.

 
3:00pm - 5:00pmMS164, part 4: Algebra, geometry, and combinatorics of subspace packings
Unitobler, F-106 
 
3:00pm - 5:00pm

Algebra, geometry, and combinatorics of subspace packings

Chair(s): Emily Jeannette King (University of Bremen, Germany), Dustin Mixon (Ohio State University)

Frame theory studies special vector arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown alongside the emergence of new methods in signal processing. Modern advances in frame theory involve techniques from algebraic geometry, semidefinite programming, algebraic and geometric combinatorics, and representation theory. This minisymposium will explore a multitude of these algebraic, geometric, and combinatorial developments in frame theory.

The theme of the fourth session is "Symplectic and real algebraic geometry in frame theory."

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Symplectic Geometry and Frame Theory

Clayton Shonkwiler
Colorado State University

Geometric tools are increasingly important in the study of (finite) frames, which are simply redundant bases that are useful in signal processing and other applications where robustness to noise and erasures are important. Symplectic geometry was developed as the right general setting for Hamiltonian mechanics and in practice is often quite closely related to complex geometry: for example, smooth projective varieties are always symplectic manifolds. This is a promising, though mostly unexplored, collection of tools to be applied to the theory of frames in complex vector spaces.

The slogan is that any property which can be characterized as a level set of a moment map is likely to be amenable to symplectic techniques. In particular, unit-norm tight frames – which are particularly useful for applications – arise as the level set of a natural Hamiltonian group action on the set of complex matrices of a given size.

In this talk I will describe how symplectic tools can be used to generalize the frame homotopy theorem of Cahill–Mixon–Strawn and to give new insight into the Paulsen problem.

 

Symplectic Geometry, Optimization and Applications to Frame Theory

Tom Needham
Ohio State University

In recent work with Clayton Shonkwiler, we show that any space of complex frames (considered up to global rotations) with a prescribed frame operator can naturally be endowed with an extra geometric structure called a symplectic form. The goal of this talk is to explain how classical results from symplectic geometry can be used to provide theoretical guarantees for the convergence of optimization algorithms arising in frame theory. More specifically, spaces of frames with prescribed frame operator admit torus actions which are compatible with the symplectic structure (the torus actions are Hamiltonian). A result of Duistermaat says that gradient flows of certain functionals associated to Hamiltonian actions have no spurious local minima. We will discuss applications of this framework to the Paulsen problem from frame theory.

 

The optimal packing of eight points in the real projective plane

Hans Parshall
Ohio State University

How can we arrange $n$ lines through the origin in three-dimensional Euclidean space in order to maximize the minimum angle between pairs of lines? Conway, Hardin and Sloane (1996) produced numerical line packings for $n leq 55$ that they conjectured to be optimal in this sense, but until now only the cases $n leq 7$ have been solved. We will discuss the resolution, joint with Dustin Mixon, of the case $n = 8$. Drawing inspiration from recent work on the Tammes problem, we proceed by enumerating potential contact graphs for an optimal configuration and eliminating those that violate various combinatorial and geometric constraints. The contact graph of the putatively optimal numerical packing of Conway, Hardin and Sloane is the only graph that survives, and we convert this numerical packing to an exact packing through cylindrical algebraic decomposition. We will further describe some potential improvements to our approach that could yield more exact optimal packings.

 

Spherical configurations with few angles

William J. Martin
Worcester Polytechnic Institute

Let X be a spherical code in d-dimensional space. The degree of X is the number of inner products <u,v> that occur as u and v range over pairs of distinct elements from X. We are interested in spherical codes of small degree that arise from, or give rise to, association schemes. We will discuss equiangular lines, real mutually unbiased bases, work of Kodalen on "linked simplices" and joint work with Kodalen on "orthogonal projective doubles". A set of k full-dimensional simplices on the unit sphere is said to be "linked" if only two possible angles occur between vectors in distinct simplices. Given a graph G with vertex set V, a set L of lines through the origin in d-dimensional space is an "orthogonal projective double" of G if there is a bijection V --> L that maps adjacent pairs of vertices to orthogonal pairs of lines and non-adjacent pairs to lines forming some fixed angle between zero and 90 degrees. There is one aspect of this study involving elementary algebraic geometry. The ideal of X is the set of polynomials in d variables that vanish on each point in X and our goal is to determine, for each of the families mentioned above, a generating set for this ideal consisting of polynomials all having lowest possible total degree. This talk is based in part on joint work with my student Brian Kodalen and is supported by the US National Science Foundation.