# Scientific

## Mathematical ecology: A century of progress, and challenges for the next century

The subject of mathematical ecology is one of the oldest and most exciting

in mathematical biology, and has helped in the management of natural

systems and infectious diseases. Though many problems remain in those

areas, we face new challenges today in finding ways to cooperate in

managing our Global Commons. From behavioral and evolutionary

perspectives, our societies display conflict of purpose or fitness across

levels, leading to game-theoretic problems in understanding how

cooperation emerges in Nature, and how it might be realized in dealing with

problems of the Global Commons. This lecture will attempt to weave these

topics together, tracing the evolution from earlier work to challenges for the

future.

Simon Levin is the J.S. McDonnell distinguished university professor in Ecology and Evolutionary Biology at Princeton University. He is a recipient of the National Medal of Science, the Kyoto Prize and a Robert H. MacArthur Award.

## Modularity of Calabi-Yau Varieties

Let $X$ be a Calabi-Yau variety of dimension d. We will confine ourselves to Calabi-Yauvarieties of small dimensions, e.g., $d < 3$. Dimension one Calabi–Yaus are elliptic curves, those of dimension two are $K3$ surfaces, and dimension three ones are Calabi-Yau threefolds. Geometry and physics are both very much in evidence on Calabi-Yau varieties over the field of complex numbers.

Today I will focus on Calabi-Yau varieties defined over the field $Q$ of rational numbers (or number fields), and will discuss the modularity/automorphy of Calabi-Yau varieties in the framework of the Langlands Philosophy.

In the last twenty-five years, we have witnessed tremendous advances on the modularity question for Calabi-Yau varieties. All these results rest on the modularity of the two-dimensional Galois representations associated to them. In this lecture, I will present these fascinating results. If time permits, I will discuss a future direction for the realization of the Langlands Philosophy, in particular, for Calabi-Yau threefolds.

### Biography

Noriko Yui is a professor of mathematics at Queen’s University in Kingston, Ontario. A native of Japan, Yui obtained her B.S. from Tsuda College, and her Ph.D. in Mathematics from Rutgers University in 1974 under the supervision of Richard Bumby. Known internationally, Yui has been a visiting researcher at the Max-Planck-Institute in Bonn a number of times and a Bye-Fellow at Newnham College, University of Cambridge.

Her research is based in arithmetic geometry with applications to mathematical physics and notably mirror symmetry. Currently, much of her work is focused upon the modularity of CalabiYau threefolds. Professor Yui has been the managing editor for the journal Communications in Number Theory and Physics since its inception in 2007. She has edited a number of monographs,and she has co-authored two books.

## An Introduction to Randomized Algorithms for Matrix Computations

The emergence of massive data sets, over the past twenty or so years, has led to the development of Randomized Numerical Linear Algebra. Fast and accurate randomized matrix algorithms are being designed for applications like machine learning, population genomics, astronomy, nuclear engineering, and optimal experimental design.

We give a flavour of randomized algorithms for the solution of least squares/regression problems. Along the way, we illustrate important concepts from numerical analysis (conditioning and pre-conditioning), probability (concentration inequalities), and statistics (sampling and leverage scores).

## 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.