IIT Bhilai

Integer matrices with integer eigenvalues and Laplacian integral graphs

If all entries of a matrix \(A\) are integers, we call \(A\) an integer matrix. The study of such matrices lies at the intersection of linear algebra, number theory, and group theory. A particularly significant and challenging problem in this context is the characterization of integer matrices whose eigenvalues are themselves integers.

In 2009, Martin and Wong demonstrated that almost all integer matrices have no integer eigenvalues; more precisely, for any \(n \geq 2\), the probability that a random \(n\times n\) integer matrix has at least one integer eigenvalue is 0. This result naturally motivates the following questions: Under what conditions does an integer matrix have all its eigenvalues as integers? Furthermore, is there a systematic method for constructing such matrices? An important source of examples arises from graph theory. A graph is said to be Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. Thus, the Laplacian matrices of Laplacian integral graphs form a notable class of integer matrices with integer eigenvalues.

The present lecture series will focus on recent developments concerning the characterization and construction of integer matrices whose eigenvalues are all integers. The series is organized into three lectures:

1. Characterization and construction of integer matrices with integer eigenvalues: Foundational results, necessary and sufficient conditions, and explicit methods for constructing integer matrices with integer eigenvalues.
2. Spectral integral variation and constructably Laplacian integral graphs: Spectral properties of Laplacian integral graphs and constructive approaches for generating new families of such graphs.
3. Graphs with distinct integer Laplacian eigenvalues: The identification and characterization of graphs whose Laplacian eigenvalues are distinct integers, highlighting recent advances and open problems in the area.

Hyperplane arrangements: at the crossroads of combinatorics, topology, and algebra

Hyperplane arrangements have long been a fascinating and rich area of study in mathematics, bridging combinatorics, geometry, topology, and algebra. Originating in the 19th century from classical geometry and linear algebra problems, the field has since evolved into a deep and vibrant subject in its own right. In the late 20th century, pioneering contributions by mathematicians such as Thomas Zaslavsky and Richard Stanley uncovered profound combinatorial structures, initially motivated by enumerative questions like counting regions and computing characteristic polynomials of arrangements. These investigations revealed surprising connections to algebraic geometry and singularity theory. At the same time, foundational work by Vladimir Arnold and Pierre Deligne on configuration spaces and generalized braid groups expanded the subject’s scope, interweaving topology, combinatorics, group theory, and representation theory. Today, hyperplane arrangements serve as a unifying framework, with applications ranging from the topology of complexified complements to algebraic notions such as freeness and logarithmic derivations.

In this lecture series, I will survey several important milestones in the theory of hyperplane arrangements, with a particular focus on combinatorial techniques and problems that are unique to this area of mathematics. The goal is to provide participants with both historical perspective and modern tools to engage with current research in this exciting domain.

In particular, the detailed plan of three lectures is:

1. Enumerative aspects — Counting regions, Zaslavsky’s theorem, bijective techniques, and connections to deformations of reflection arrangements.
2. Topological Insights — The topology of complexified complements, configuration spaces, Artin groups, Orlik-Solomon algebras and the NBC (no broken circuit) bases.
3. Algebraic Structures — Matroids and arrangements. The concept of freeness, logarithmic derivations, and their algebraic implications.