# Gian Maria Negri Porzio

Office 2.111, Alan Turing Building, M13 9PL, Manchester · gianmaria.negriporzio@manchester.ac.uk

I am a PhD student in the Numerical Linear Algebra Group of the School of Mathematics at the University of Manchester. My PhD is supervised by Prof. Françoise Tisseur and co-supervised by Prof. Nick Higham. My main research interests are nonlinear eigenvalue problems, rational approximation of nonlinear functions, tropical linear algebra and Bohemian matrices.

I am also the Vice President of the Manchester SIAM–IMA Student Chapter. As a SIAM and IMA affiliated, we encourage the promotion of applied mathematics to students. If you are a SIAM and IMA member, you can join our chapter for free and get all the benefits that the membership grants. If you are not in Manchester, why not joining one of the other Student Chapters?

## Research

### Nonlinear Eigenvalue Problems

The standard eigenvalue problem consists in finding all the eigenpairs $(\lambda, v) \in \mathbb{C}\times \mathbb{C}^{n}$ $(A-\lambda I)v = 0$ where $A$ is a $n \times n$ matrix. I am interested in a generalization of this concept. Instead of considering a single matrix, I work with a matrix valued function (usually analytic or meromorphic) $F(z)$ and I look for the eigenpairs $(\lambda, v) \in \mathbb{C}\times \mathbb{C}^{n}$ such that $F(\lambda)v = 0.$ Nonlinear eigenvalue problems are a fondamental tool to model many real life applications. In recent years many researchers have been focusing on this large class of problems, but at the moment there is no clear best method to solve them. I mainly focus on contour integration, exploiting the core idea that $f(A) = \frac{1}{2\pi i}\int_\Gamma f(z)(zI - A)^{-1}\,dz,$ if $f(z)$ is analytic on and inside the closed contour $\Gamma$ that contains the spectrum of $A$.

I have also contributed to the newest (4.0) release of the NLEVP Library . It is a MATLAB library where the users can find more than 70 nonlinear eigenvalue problems, from quadratic to rational or pure nonlinear, in order to use them in their papers as numerical examples.

### Tropical linear algebra

A tropical polynomial is a formal expression of the form $p(x) = \bigoplus_{j=1}^{n}p_{j}\otimes z^{j} = \max_{0\leq j \leq n}(p_{j} + jz), \qquad a_j \in \mathbb{R}\cup \{-\infty\}.$ One can define the roots of these mathematical objects, called the tropical roots. I am interested in exploiting the tropical roots to extract information about the initial polynomial. This approach may be used in different applications and it is also a fascinating theoretical problem.

### Bohemian matrices

Bohemian matrices are sets of matrices in which the entries are sample from a finite (usually small) set of integers. The term was coined by Robert Corless and Steven Thornthon and it is an euphony for "BOunded HEight Matrix of Integers". Even though Bohemian matrices arise in many situations, the notation and the study is still on an early stage. We still need to unveil many structures that lie behind such simple objects. More details can be found on Bohemian website or on Nick Higham's blog post .

Two plots of the eigenvalues of two subclasses of Bohemian matrices. Credits: Bohemian matrices.