Office 2.111, Alan Turing Building, M13 9PL, Manchester ยท
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?

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.

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 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.

- M. Fasi, G. M. Negri Porzio.
*"Determinants of Normalized Bohemian Upper Hessemberg Matrices"*, under review. - N. J. Higham, G. M. Negri Porzio, and F. Tisseur.
*"An Updated Set of Nonlinear Eigenvalue Problems"*, MIMS EPrint 2019.5, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, Mar. 2019. 12 pp.

- Lead demonstrator for MATH10202 Linear Algebra A.
- Lead demonstrator for MATH19872 Mathematics 0D2 .
- Lead demonstrator for MATH19842 Mathematics 0F2.

- Demonstrator for the MATH10001 Mathematical Workshop computer lab session.
- Lead demonstrator for MATH19861 Mathematics 0N1.
- Lead demonstrator for MATH29681 Mathematics 2E1.

- Teaching assistant in MATH37012 Markov Processes
- Teaching assistant in MATH20602 Numerical Analysis

- Organizer of the SIAM UKIE National Student Chapter Conference 2019
*Alan Turing Building, 10–11 June 2019* - Advances in Numerical Linear Algebra: Celebrating the Centenary of the Birth of James H. Wilkinson
*Alan Turing Building, Manchester, 29–30 May 2019*

- Due Giorni di Algebra Linear Numerica
*La Sapienza University, Rome, 18–19 February 2019*

**Talk**: "A Contour Integral Approach for the Solution of Nonlinear Eigenvalue Problems"

- Invited speaker at the Bath Numerical Analysis Seminars
*The University of Bath, 02 November 2018*

**Talk**: "A Contour Integral Approach for the Solution of Nonlinear Eigenvalue Problems"

- Bohemian Matrices and Applications
*Alan Turing Building, Manchester, 20–22 June 2018*

- SIAM UKIE National Student Chapter Conference 2018
*University of Bath, 18–19 June 2018*

**Poster**: "A contour Integral Eigensolver for Dense Nonlinear Eigenvalue Problem" - Organizer of the Manchester SIAM–IMA Student CHapter 2018
*Alan Turing Building, 20 April 2018*

When I am not in the office, you can find me practicing almost any kind of sport activities. I love everything with a racquet and I used to play tennis quite competitevely when I was in high school. Now I am happy when the Manchester sun comes out (or tries to) and I can hit a couple of balls with my friends. In winter time, I try to go back in Italy and sky as much as I can.

When I travel or I want to spend some time alone, I go around looking for nice shots with my camera. Here on Flickr you can find some of them, even though I have not been updating my profile for some time.

Two shots taken by night in Pisa. On the left, River Arno in flood ; on the right, the fireworks for the Luminara 2014 .

Here below you can find some of my favorite links around the web:

- NLA Group webpage
- Nick's Higham Blog
- Some Mathematical humor (Warning: you may lose your friends!).
- If I had known this blog post when I was younger, I would have known how to answer to many people around me
- I used to be one of the admins of Poisson, the maths student server at the Mathematics Department of Pisa (webpage in Italian).