# AG21 Prize Spotlight

Congratulations to Paul Breiding who will receive the SIAM Activity Group on Algebraic Geometry (SIAG/AG) Early Career Prize at the virtual SIAM Conference on Applied Algebraic Geometry. Additional information about Breiding, including a Q&A, can be found below.

## Paul Breiding

Paul Breiding is the 2021 recipient of the SIAM Activity Group on Algebraic Geometry Early Career Prize. The award will be presented at the SIAM Conference on Applied Algebraic Geometry (AG21), held in a virtual format August 16 – 20, 2021. Breiding will give a talk at the conference titled “Tensors, Numerical Analysis, and Algebraic Geometry” on August 18, 2021.

The prize is awarded to Breiding for his extraordinary work on geometric, statistical, and algorithmic aspects of tensors and their applications.

The SIAM Activity Group on Algebraic Geometry awards this prize every two years to one outstanding early career researcher in the field of algebraic geometry, and its applications, for distinguished contributions to the field in the three calendar years prior to the year of the award. The contributions must be contained in a paper or papers in English in peer-reviewed journals.

Paul Breiding is head of the Emmy-Noether-Research-Group “Numerical and Probabilistic Nonlinear Algebra” at the Max-Planck-Institute for Mathematics in the Sciences in Leipzig. Born in 1988, he got his Master’s degree at the Georg-August-Universität Göttingen in 2013 and completed his Ph.D. at the Technische Universität Berlin in 2017. He is a fellow of the Junge Akademie Mainz. His research interest is in nonlinear algebra and its connections to numerical analysis and probability.

Q: Why are you excited to receive the award of the SIAG/AG Early Career Prize?

**A:** I feel humbled to be selected for the SIAG/AG Early Career Prize. It makes me proud that I am allowed to work within a group of such brilliant researchers. Furthermore, I am particularly happy to receive the prize as I have recently become a parent. From this perspective, I consider the prize as a sign that doing research in mathematics and having children is compatible, and I am thankful to the SIAM Activity Group on Algebraic Geometry for creating a working atmosphere, which makes this possible.

Q: Could you tell us a bit about the research that won you the prize?

**A:** I received the prize for my work on geometric, statistical, and algorithmic aspects of tensors and their applications. Tensors are n-dimensional arrays. They are higher-dimensional analogues of matrices. I am interested in the numerical and statistical properties of their decompositions, and I study them using tools from algebraic geometry. This is because many interesting problems in the context of tensor decompositions arise as the solution of a systems of polynomial equations, which are one of the central objects studied in algebraic geometry.

Part of the nomination, which won me the SIAG/AG Early Career Prize, was my paper "How many eigenvalues of a random tensor are real?" (the title is a reference to the article "How many eigenvalues of a random matrix are real?" by Edelman, Kostlan and Shub). In my paper, I consider symmetric tensors whose entries are standard Gaussian random variables. While the eigenvalues of a symmetric matrix are all real, for symmetric tensors they can be complex. The main result in my article is an exact formula for the expected number of real eigenvalues of a Gaussian symmetric tensor.

A component of the proof is the computation of an exact formula for the expected absolute value of the characteristic polynomial of a matrix from the Gaussian Orthogonal Ensemble.

Q: What does your work mean to the public?

**A: **In my research I combine methods from different fields: algebraic geometry, differential geometry, convex geometry, numerical analysis, probability to name a few. I think the meaning of my research is that those fields are not separated, and that they can be beneficial for each other. Traditionally, many fields in mathematical research are either sorted into "pure" or "applied" mathematics. With my work I want to show that we can find both new and important research directions when transcending this boundary.

Q: What does being a member of SIAM mean to you?

**A:** Being a SIAM member for me means that I can be part of an amazing scientific community. I perceive this community as very inviting and open, which has allowed me to grow as a scientist. Especially when I was a Ph.D. student, this atmosphere helped me a lot in finding the courage to ask questions. However, this is not to say we shouldn’t make efforts to increase diversity and inclusion in our group. But my impression is that SIAM provides a framework for reaching out further towards this goal. In addition, SIAM promotes research across different disciplines, which has always been the backbone of my interdisciplinary approach explained above.