# Mini-Symposia

**Combinatorial Matrix Theory**: *Minerva Catral (Xavier University, OH) and Louis Deaett (Quinnipiac University)*

Connections between linear algebra and combinatorics have yielded rich interactions between the two areas and have led to a great deal of interesting and important mathematics. A matrix can be viewed through a combinatorial lens in a variety of ways, for example via a description such as a matrix pattern that retains only discrete information from the matrix, e.g., the signs of the entries, or by using a graph or directed graph to describe various constraints on the matrix. One may then ask what information such a combinatorial description can carry about the operator-theoretic properties of the matrix, such as its rank or its spectrum. Moreover, classes of matrices with discrete entries, such as Hadamard and alternating sign matrices, form an important area of study in combinatorics, and one in which the role of linear algebra is naturally central. This mini-symposium will feature recent advances and questions of current interest in these areas.

**Compressed sensing and matrix completion**: *Simon Foucart (Texas A&M University) and Namrata Vaswani (Iowa State University)*

Compressed Sensing has recently had a tremendous impact in science and engineering, because it revealed the theoretical possibility of acquiring structured high-dimensional objects using much less information than previously expected, and because it also provided practical procedures to perform the reconstruction based on the limited information available. The foundations of the field rely on an elegant mathematical theory with linear algebra at its core. The standard compressed sensing problem consists in solving underdetermined linear systems whose solutions are known to possess an a priori structure such as sparsity. There are several extensions of the standard problem, e.g. when sparse vectors are replaced by low-rank matrices which must be completed from the knowledge of only a few of their entries. A motivating application is found in the Netflix problem, where the matrix of movie ratings has to be reconstructed based on only a few ratings by each user. The goal of the mini-symposium is to highlight interplays between mathematics in general, and linear algebra in particular, with other fields (engineering, computer science, and statistics) that have shaped the theory of compressive sensing and low-rank matrix recovery.

**Distance problems in linear algebra, dynamical systems and control**: *Elias Jarlebring (KTH Royal Institute of Technology, Sweden) and Wim Michiels (KU Leuven, Belgium)*

Matrix distance problems are at the intersection of numerical linear algebra, systems and control, and optimization. From an application point of view, distance problems form, for instance, the core of methods for assessing and optimization the transient behavior of dynamical systems, their stability robustness, as well as performance measures, expressed in terms of H-2- H-infinity criteria. Many of the recent theoretical and computational advances are based on eigenvalue perturbation theory and eigenvalue optimization algorithms. The aim of the minisymposium is to bring together experts on the topic, who will highlight several recent developments in the area of distance problems. They will pay special attention to exploiting structure and sparsity on the uncertainty, inferred from the underlying application, and on exploiting the structure of the perturbed eigenvalue problem, e.g., a nonlinearity in the eigenvalue parameter.

**Distances on networks and its applications**: * Angeles Carmona (Universitat Politécnica de Catalunya), Andres M. Encinas (Universitat Politécnica de Catalunya), and Margarida Mitjana (Universitat Politécnica de Catalunya)*

In contrast with the geodesic distance, the resistance distance takes into account all paths between vertices. The standard method to compute the resistance distance of a network is via the Moore-Penrose inverse of the combinatorial Laplacian of the network. From the matrix point of view, this is equivalent to defining the effective resistance associated with an irreducible, symmetric and diagonally dominant M-matrix. Since the matrix associated with a positive semidefinite Schrödinger operator L_q is an irreducible, symmetric M-matrix, we can assign an effective resistance with respect to a non-negative parameter and a weight to any irreducible, symmetric M-matrix. The mini-symposium aims to provide an overview the latest developments in the area and its applications, present current research and stimulate new ideas and collaboration.

** Linear Algebra Aspects of Association Schemes**: * Allen Herman (Regina) and Bangteng Xu (Eastern Kentucky)*

Research on association schemes has continued to evolve in the last decade, led by advances in both the combinatorial and representation-theoretic directions that rely heavily on linear algebraic techniques. This mini-symposia will highlight the applications of linear algebra in this field, featuring talks by invited speakers on characterizations of table algebras of low rank, on new characterizations of P- and Q-polynomial association schemes, on new applications of Terwilliger algebras of association schemes, and on new descriptions of the algebraic properties of distance regular and strongly regular digraphs.

** Linear Algebra and Geometry**: *Gabriel Larotonda (Universidad Nacional de General Sarmiento and Instituto Argentino de Matemática (CONICET), Argentina) and Alejandro Varela (Universidad Nacional de General Sarmiento and Instituto Argentino de Matemática (CONICET), Argentina)*

Geometry has proved to be an excellent tool to model known structures in linear algebra but also to provide new problems in the field. Contributions in either of these directions will be part of this mini-symposium. The lectures will be centered around the ideas of metric and differential geometry in the context of matrices, such as Grassmann manifolds, the space of positive invertible matrices, the unitary group and the general linear group. Subjects such as metrics, geodesics, metric preservers and linear connections are related in this context to matrix inequalities and characterizations of special classes of matrices.

**Linear Algebra and Mathematical Biology**: *Julien Arino (University of Manitoba, Canada) and Natalia Komarova (University of California Irvine)*

Mathematical biology and linear algebra have a rich history of interactions. Early on, the relationship was somewhat one-sided, with mathematical biology using linear algebra mostly as a tool. For instance, population biology models involving more than one population or stage were naturally formulated using systems of difference equations, leading, for example, to the Leslie matrix. Nonlinear systems of differential equations are often studied using a technique called linearisation, which requires to understand the localisation of the spectrum of the associated Jacobian matrix. With the introduction of more complex models, the interactions among individuals or molecules on networks in diverse contexts required more complex mathematics, utilizing the intricate connections between graph theory and matrix theory. As a consequence, more recently, the relationship between biology and linear algebra has become more symbiotic: mathematical biology produces problems that are interesting in their own right to linear algebraists. This minisymposium brings together researchers interested in a wide variety of applications of linear algebra in the context of mathematical biology. Presentations will more than cover the span of topics mentioned earlier and will provide a glimpse into an area that offers interesting theoretical challenges.

**Linear Algebra and Positivity with Applications to Data Science**: *Dominique Guillot (University of Delaware), Apoorva Khare (Stanford University), and Bala Rajaratnam (Stanford University)*

Making sense of vast amounts of data has become one of the great challenges of the 21st century. This mini-symposium will highlight how recent advances in analysis (in particular positivity and related topics), linear algebra, algebraic geometry, and statistics provide new tools to analyze data in areas such as covariance estimation and the theory of graphical models.

**Linear Algebra and Quantum Information Science**: *Chi-Kwong Li (College of William and Mary), Yiu Tung Poon (Iowa State University), and Raymond Nung-Sing Sze (The Hong Kong Polytechnic University)*

Quantum information science is a highly interdisciplinary research area. The common language of mathematics provides a foundation on which quantum information scientists from different backgrounds can communicate ideas. The purpose of this mini-symposium is to provide an opportunity for researchers to present their results and/or raise problems in the area of linear algebra related to quantum information.

**Linear Algebra Education**:* Rachel Quinlan (National University of Ireland Galway, Ireland) and Megan Wawro (Virginia Polytechnic Institute and State University)*

Linear algebra is at the heart of a university mathematics curriculum, and a deep understanding of the subject is central to a student’s success in mathematics and mathematics-intensive degrees. In accordance with the ILAS 2017 theme of Connections, the broad theme of the Linear Algebra Education mini-symposium is to explore the connections between teaching and research within undergraduate linear algebra. Thus, we invite experts from around the world to present insights they have developed through their own pedagogy or through their research on teaching and learning in linear algebra. Topics may include, but are not limited to: the role of visualization in learning concepts, advances in the use of technology in teaching and learning, the role of proofs in learning concepts, the importance of exposure to real world applications and suggested examples, curricular resources for student-centered instruction, and insights regarding how students make sense of particular concepts as they learn.

**Matrices, Tensors and Manifold Optimization**:* Daniel Kressner (Ecole Polytechnique Fédérale de Lausanne, Switzerland) and Bart Vandereycken (University of Geneva, Switzerland)*

Optimization on manifolds and tensor decompositions are popular research topics in numerical linear algebra. The purpose of this mini-symposium is to bring together recent advances in both fields that highlight geometrically inspired algorithms and analyses for rank-structured matrix problems. Example topics include global recovery guarantees for matrix completion and phase retrieval, optimization on bounded rank matrices, continuous low-rank decompositions, manifolds for shape analysis, and computational complexity of manifold optimization.

**Matrix Analysis: Inequalities, Means, and Majorization**:* Fumio Hiai (Tohoku University, Japan) and Yongdo Lim (Sungkyunkwan University, Korea)*

A core concept in Matrix Analysis is order structure constituting different notions of positive semidefiniteness order (or Loewner order), spectral and majorization order for eigenvalues, etc. Correspondingly, different types of matrix/operator inequalities have become a big research field with wide applications. In particular, various majorization results have played vital roles in matrix norm inequalities for unitarily invariant norms. A subject strongly connected to these order relations is theory of matrix/operator means, whose multivariate extensions have recently been developed extensively. In particular, the extension of the geometric mean in Riemannian geometry approach is remarkable. This mini-symposium is aimed to exchange new developments and ideas in topics including matrix (norm) inequalities, matrix/operator means, and majorizations for matrices.

**Matrix Polynomials**: *Froilán Dopico (Universidad Carlos III de Madrid, Spain) and Paul Van Dooren (Catholic University of Louvain, Belgium)*

Matrix polynomials is a classical topic in Matrix Analysis that has received a lot of attention in the last decade. This renewed interest comes mainly from the necessity of improving the algorithms for computing the eigenvalues of Matrix Polynomials, both in terms of efficiency and accuracy, from the many applications where Matrix Polynomials arise, and from the key role that Matrix Polynomials play in solving more general nonlinear eigenvalue problems. As a consequence of these numerical and practical necessities many classical concepts in the theory of Matrix Polynomials have been revised, simplified, expanded, and set in more applied perspectives, as well as many new lines of theoretical research have been initiated and analyzed. This Mini-Symposium aims to give an overview of recent activity on Matrix Polynomials from the point of view of theory, applications, and numerical practice.

** Matrix techniques in operator algebra theory**:* Vern Paulsen (University of Waterloo) and Hugo Woerdeman (Drexel University)*

This mini-symposium is intended to highlight the interplay between Operator Algebra and Matrix Theory. Clearly, a problem in Operator Theory can first be explored in the finite dimensional setting, and thus becomes a matrix theory problem. But, the interplay goes well beyond this. A recent example is the 1959 Kadison-Singer conjecture, where a reformulation as the Paving conjecture was crucial to its solution. The Paving conjecture is one stated in matrix terms, and a clever use of interlacing families of polynomials led to its solution in 2013. Another intriguing conjecture, due to Crouzeix, found its origin in numerical linear algebra, but its solution will have an impact on Operator Algebra theory. These are just two examples illustrating the interplay. The mini-symposium will provide more examples of this interplay, and may also include the latest developments of the Crouzeix conjecture.

**The Nonnegative Inverse Eigenvalue Problem**:* Charles R. Johnson (The College of William and Mary) and Pietro Paparella (University of Washington Bothell)*

The longstanding nonnegative inverse eigenvalue problem (NIEP) is to characterize the spectra of (entrywise) nonnegative matrices. The NIEP is unsolved when order greater than five and remains one of the premier unsolved problem in matrix analysis. The real NIEP and symmetric NIEP are important variants that are also unsolved for orders greater than five. The NIEP has been open since 1949 and has drawn the attention of numerous researchers from many distinct areas of mathematics. The purpose of this CMS is to bring together various international researchers working on the NIEP and its variants in order to disseminate recent advances and connections to other branches of mathematics. We also welcome talks on the single-eigenvalue NIEP (stochastic/doubly stochastic regions).

**Krylov and filtering methods for eigenvalue problems**:* Jared Aurentz (Instituto de Ciencias Matemáticas, UAM, Madrid) and Karl Meerbergen (KU Leuven, Belgium)*

The algebraic eigenvalue problem is at the heart of many applications in science and engineering. The reliable and accurate solution of large scale problems is often a difficult task. We focus on recent advances in Krylov methods and rational filtering for large scale eigenvalue problems.

** Numerical Ranges**:* Patrick X. Rault (University of Arizona South) and Ilya Spitkovsky (NYU Abu Dhabi)*

In this mini-symposium we will bring together many of the world's leading researchers on the numerical range of a matrix, also known as the field of values of a matrix. Topics will include the recently introduced Gau-Wu invariant, the Crouzeix Conjecture, and an analogue of numerical ranges over finite fields.

**Random matrix theory for networks**: *Dustin Mixon (Air Force Institute of Technology) and Rachel Ward (University of Texas at Austin)*

Increasingly, methods of high-dimensional probability and heuristics from statistical physics are being used in harmony with techniques from manifold optimization to derive statistical guarantees for scalable algorithms in the analysis of big data with latent manifold structure. Applications include dimension reduction, network analysis, neural networks, phase retrieval, computer vision, and sparse signal recovery. This mini-symposium aims to bring together mathematicians and engineers from academia and industry at the forefront of this effort, albeit from various perspectives, in hopes that new collaborations and insights will result.

** Recent Advancements in Numerical Methods for Eigenvalue Computation**:* James Vogel (Purdue University), Xin Ye (Purdue University), and Jianlin Xia (Purdue University)*

This mini-symposium presents recent novel techniques for numerical computation of eigenvalues, singular values, and spectral decompositions. These include contour-integral based eigensolvers, divide-and-conquer methods, and structured Lanczos methods that are very efficient and reliable for modern large-scale computing applications. These methods help to highlight the “connections” between linear algebra and other areas of mathematics; such as complex analysis, graph theory, and probability theory.

** Representation Theory**:* Jonas Hartwig (Iowa State University)*

The goal of the mini-symposium is to bring together a cross-section of the community in representation theory of algebras in finite and infinite-dimensional vector spaces to discuss recent and ongoing developments. Topics include rings, algebras, modules, Lie algebras, quantum groups, quivers, combinatorics, module categories, and applications.

**Solving Matrix Equations**: * Qing-Wen Wang (Shanghai University) and Yang Zhang (University of Manitoba)*

Matrix equations are widely used in many scientific areas such as engineering, physics, system and control theory, computer science, and information science. The research on solvability conditions and structural representations of solutions to matrix equations is of central importance in matrix algebra as well as in general mathematics. The topics in this mini-symposium focus mainly on the solvability conditions, special and general solutions, structural representations to some matrix algebraic equations over various algebraic structures including fields, division rings, regular rings and so on. Moreover, efficient computing algorithms and their applications in system, control theory, and quantum computing will be discussed and explored. This mini-symposium will provide an important opportunity for experts in linear algebra, matrix theory, ring theory and computer science to exchange ideas, to share new results, and to work in collaboration.

**Spectral Graph Theory**:* Nair Abreu (Universidade Federal do Rio de Janeiro, Brazil) and Leonardo de Lima (Federal Center of Technological Education Celso Suckow da Fonseca, Brazil)*

Spectral Graph Theory studies interactions of graphs and matrices. Topics of interest include, but are not limited to, the following: (i) characteristic polynomials, eigenvalues and eigenvectors of matrices related to graphs such as the adjacency matrix, the Laplacian and normalized Laplacian matrices, the signless Laplacian matrix, Seidel matrix and distance matrices; (ii) relations between the spectrum and the structure of a graph; (iii) cospectral graphs and graphs characterized by their spectra; (iv) the usage of spectral techniques to prove graph-theoretical results; (v) applications of spectral graph theory in combinatorics, chemistry, physics, computer science, engineering and other areas are welcome. Recent developments in spectral graph theory and an opportunity to exchange new ideas are expected as outcomes of this mini-symposium.

**Toeplitz Matrices and Riemann Hilbert Problems**:* György Pal Gehér (University of Reading, Reading, United Kingdom) and Jani Virtanen (University of Reading, Reading, United Kingdom) *

Toeplitz matrices and Riemann-Hilbert problems are currently of great interest in many parts of mathematics and its applications. They demonstrate a fruitful interplay between operator theory, complex analysis and linear algebra, and have found many applications in mathematical physics, in particular in random matrix theory. This mini-symposium will bring together both young researchers and specialists in these areas. The main aim of the mini-symposium is to facilitate an exchange of ideas between researchers who apply the Riemann-Hilbert problem to integrable systems, orthogonal polynomials and random matrices, and those who work in the spectral theory of Toeplitz matrices and operators on function spaces, to review the current state of the field and, most importantly, to discuss plans for the future.

**Zero Forcing: Its Variations and Applications**:* Daniela Ferrero (Texas State University), Mary Flagg (University of St. Thomas), and Michael Young (Iowa State University)*

Zero forcing was introduced as a graph theoretic tool for obtaining an upper bound on the maximum nullity of real symmetric matrices whose nonzero pattern of off-diagonal entries is defined by a given graph. The minimum rank problem was motivated by the inverse eigenvalue problem. Independently, zero forcing was introduced by mathematical physicists studying control of quantum systems. Zero forcing has yielded many interesting results in linear algebra. Even more connections have been discovered, modeling problems in disciplines as diverse as theoretical biology and electrical engineering. The goal of our mini-symposium is to provide a forum for researchers in different variations and applications of zero forcing to share ideas and learn of new techniques that will transfer across applications to advance the state of the art in many directions.