# Mathematics

## The nonlinear eigenvalue problem: recent developments

Given a matrix-valued function F that depend nonlinearly on a single

parameter z, the basic nonlinear eigenvalue problem consists of finding complex scalars z for which F(z) is singular. Such problems arise in many areas of computational science and engineering, including acoustics, control theory, fluid mechanics and structural engineering.

In this talk we will discuss some interesting mathematical properties of

nonlinear eigenvalue problems and then present recently developed

algorithms for their numerical solution. Emphasis will be given to the linear algebra problems to be solved.

## Regularity of interfaces in phase transitions via obstacle problems

The so-called Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase change, for example ice melting to water. An important goal is to describe the structure of the interface separating the two phases. In its stationary version, the Stefan problem can be reduced to the classical obstacle problem, which consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. The aim of this talk is to give a general overview of the classical theory of the obstacle problem, and then discuss recent developments on the structure of interfaces, both in the static and the parabolic settings.

## Paradoxes of the Infinite: Classic Themes and Recent Results

In this talk, I will give a survey of recent work I have done—some published, some unpublished—on the historical, mathematical, and philosophical problems related to the assignment of "sizes" to infinite sets. I will focus in particular on infinite sets of natural numbers. The historical part of the presentation will take its start from Greek and Arabic contributions to the possibility of measuring infinite sets according to size and sketch some developments spanning the period between Galileo and Cantor. In the systematic part of the talk, I will discuss recent theories of numerosities that preserve the part-whole principle in the assignment of sizes to infinite sets of natural numbers and show how the historical and mathematical considerations yield benefits in the philosophy of mathematics. In particular, I will discuss (1) an argument by Gödel claiming that in extending counting from the finite to the infinite, the Cantorian solution is inevitable; and (2) consequences for neo-logicism.

Paolo Mancosu is the Willis S. and Marion Slusser Professor of Philosophy at the University of California, Berkeley. He has made significant contributions to the history and philosophy of mathematics and logic, especially the philosophy of mathematical practice, mathematical explanation, the history of 20th century logic, and neo-logicism. His most recent book, Abstraction and Infinity (Oxford Unversity Press, 2017), concerns the use of abstraction principles in the philosophy of mathematics. He previous books include Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford University Press, 1996), From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s (Oxford University Press, 1998), The Philosophy of Mathematical Practice (Oxford University Press, 2008), The Adventure of Reason. Interplay between Philosophy of Mathematics and Mathematical Logic: 1900–1940 (Oxford University Press, 2010), and Inside the Zhivago Storm. The Editorial Adventures of Pasternak’s Masterpiece (Feltrinelli, 2013).

## Statistical and Data Science

Statistical science has a 200-year history of advances in theory and application. Data science is a relatively newly defined area of enquiry deriving from big data. The interplay between them, and their interactions with science, are a topic of ongoing discussion among statisticians. Some thoughts on this interplay and the role of the formal use of probability will be presented.

## Multiplicative Complexity of Cryptographic Functions

A symmetric key cryptosystem is one in which the same secret key is used for both encryption and decryption. An encryption function in a block symmetric key cryptosystem is a function of both the key and a block of n bits of data, and the result would generally be n bits long. The bits can be considered to be values in GF(2), and these functions are called Boolean functions. Such an encryption function must be highly nonlinear, or the system can be broken.

One measure of the nonlinearity of a Boolean function is its multiplicative complexity, which is the number of modulo 2 multiplications (ANDs) needed to compute the function, if the only operations allowed are multiplication and addition of two values modulo 2 (AND and XOR) and adding 1 modulo 2 to a value (NOT). This talk will be a survey of some results concerning multiplicative complexity, including a comparison to some other measures of nonlinearity. Multiplicative complexity turns out to be interesting in a another way in settings such homomorphic encryption and multi-party cryptographic protocols, where it can be important that the functions being computed have low multiplicative complexity.

## Optimizing Biogas Generation Using Anaerobic Digestion

Anaerobic digestion is a complex, naturally occurring process during which organic matter is broken down into biogas and various byproducts in an oxygen-free environment. It is used for bioremediation and the production of methane which can be used to produce energy from animal waste. A system of differential equations modelling the interaction of microbial populations in a chemostat is used to describe three of the four main stages of anaerobic digestion: acidogenesis, acetogenesis, and methanogenesis. To examine the effects of the various interactions and inhibitions, we study both an inhibition-free model and a model with inhibition.

A case study illustrates the importance of including inhibition on the regions of stability. Implications for optimizing biogas production are then explored. In particular, which control parameters and changes in initial conditions the model predicts can move the system to, or from, the optimal state are then considered. An even more simplified model proposed in Bornh\”{o}ft, Hanke-Rauschenback, and Sundmacher [Nonlinear Dynamics 73, 535-549 (2013)], claimed to capture most of the qualitative dynamics of the process is then analyzed. The proof requires considering growth in the chemostat in the case of a general class of response functions including non-monotone functions when the species death rate is included.

## Inversions for reduced words

The number of inversions of a permutation is an important statistic that arises in many contexts, including as the minimum number of simple transpositions needed to express the permutation and, equivalently, as the rank function for weak Bruhat order on the symmetric group. In this talk, I’ll describe an analogous statistic on the reduced expressions for a given permutation that turns the Coxeter graph for a permutation into a ranked poset with unique maximal element. This statistic simplifies greatly when shifting our paradigm from reduced expressions to balanced tableaux, and I’ll use this simplification to give an elementary proof computing the diameter of the Coxeter graph for the long permutation. This talk is elementary and assumes no background other than passing familiarity with the symmetric group.

## The KPZ fixed point

- Read more about The KPZ fixed point
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## The Topology of Azumaya Algebras

Azumaya algebras over a commutative ring R are generalizations of central simple algebras over a field k, and both are "twisted matrix algebras". In this, they bear the same relationship to a noncommutative ring of matrices Mat_n(k) that a vector bundles (or projective modules) bear to vector spaces. That is, they are bundles of algebras. In this talk, I will show that thinking about Azumaya algebras from the algebraic-topological point of view, as bundles of algebras, is fruitful, both in producing examples of algebras with interesting properties, and in proving certain results about such algebras that are difficult to prove by direct, algebraic methods.

## Symmetry, bifurcation, and multi-agent decision-making

I will present nonlinear dynamics for distributed decision-making that derive from principles of symmetry and bifurcation. Inspired by studies of animal groups, including house-hunting honeybees and schooling fish, the nonlinear dynamics describe a group of interacting agents that can manage flexibility as well as stability in response to a changing environment.

Biography:

Naomi Ehrich Leonard is Edwin S. Wilsey Professor of Mechanical and Aerospace Engineering and associated faculty in Applied and Computational Mathematics at Princeton University. She is a MacArthur Fellow, and Fellow of the American Academy of Arts and Sciences, SIAM, IEEE, IFAC, and ASME. She received her BSE in Mechanical Engineering from Princeton University and her PhD in Electrical Engineering from the University of Maryland. Her research is in control and dynamics with application to multi-agent systems, mobile sensor networks, collective animal behavior, and human decision dynamics.