Scientific

Geometric aspects of arithmetic statistics - 1 of 2

Speaker: 
Jordan Ellenberg
Date: 
Mon, Jun 10, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
Arithmetic statistics asks about the distribution of objects arising “randomly” in number-theoretic context. How many A_5-extensions of Q are there whose discriminant is a random squarefree integer? How big is the Selmer group fo a random quadratic twist of an elliptic curve over Q? How likely is the the maximal extension of Q unramified away from a random set of 3 primes to be an extension of infinite degree? Traditionally we ask these questions over Q or other number fields, but they make sense, and are expected to have broadly similar answers, when we replace Q with the function field of a curve over a finite field F_q. Having done this, one suddenly finds oneself studying the geometric properties of relevant moduli spaces over F_q, and under ideal circumstances, the topology of the corresponding moduli spaces over the complex numbers. Under even more ideal circumstances, results in topology can be used to derive theorems in arithmetic statistics over function fields. I’ll give an overview of this story, including some recent results in which the arrow goes the other way and analytic number theory can be used to prove results about the geometry of moduli spaces.

 


This is the first lecture in a two part series: part 2.

Public Goods, from Biofilms to Societies.

Speaker: 
Simon Levin
Date: 
Sat, Jun 15, 2019
Location: 
PIMS, University of Victoria
Conference: 
Levin Fest
Abstract: 
Ecological and economic systems are alike in that individual agents compete for limited resources, evolve their behaviors in response to interactions with others, and form exploitative as well as cooperative interactions as a result. In these complex adaptive systems, macroscopic properties like the flow patterns of resources like nutrients and capital emerge from large numbers of microscopic interactions, and feed back to affect individual behaviors. In this talk, I will explore some common features of these systems, especially as they involve the evolution of cooperation in dealing with public goods, common pool resources and collective movement. I will describe examples from bacteria and slime molds to vertebrate groups to insurance arrangements in human societies and international agreements on environmental issues.

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

Speaker: 
Simon Levin
Date: 
Sat, Jun 15, 2019
Location: 
PIMS, University of Victoria
Conference: 
Levin Fest
Abstract: 
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

Speaker: 
Noriko Yui
Date: 
Thu, Apr 25, 2019
Location: 
PIMS, University of Saskatchewan
Conference: 
Hugh C. Morris Lecture
Abstract: 
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

Speaker: 
Ilse C.F. Ipsen
Date: 
Thu, Mar 14, 2019
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
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

Speaker: 
Françoise Tisseur
Date: 
Thu, Feb 14, 2019
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
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

Speaker: 
Alessio Figalli
Date: 
Fri, Feb 8, 2019
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Distinguished Colloquium
Abstract: 
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

Speaker: 
Paolo Mancosu
Date: 
Wed, Dec 5, 2018
Location: 
PIMS, University of British Columbia
Conference: 
The Calgary Mathematics & Philosophy Lectures
Abstract: 
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

Speaker: 
Nancy Reid
Date: 
Thu, Jan 17, 2019
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
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

Speaker: 
Joan Boyer
Date: 
Thu, Oct 18, 2018
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
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.
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