Scientific

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.

Optimizing Biogas Generation Using Anaerobic Digestion

Speaker: 
Gail Wolkowicz
Date: 
Tue, Nov 27, 2018
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
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

Speaker: 
Sami Assaf
Date: 
Fri, Nov 9, 2018
Location: 
PIMS, University of British Columbia
Conference: 
Discrete Math Seminar
Abstract: 
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

Speaker: 
Jeremy Quastel
Date: 
Fri, Oct 19, 2018
Location: 
PIMS, University of British Columbia
Conference: 
CRM-Fields-PIMS Prize Lecture
Abstract: 
The (1d) KPZ universality class contains random growth models, directed random polymers, stochastic Hamilton-Jacobi equations (e.g. the eponymous Kardar-Parisi-Zhang equation). It is characterized by unusual scale of fluctuations, some of which appeared earlier in random matrix theory, and which depend on the initial data. The explanation is that on large scales everything approaches a special scaling invariant Markov process, the KPZ fixed point. It is obtained by solving one model in the class, TASEP, and passing to the limit. Both TASEP and the KPZ fixed point turn out to have a novel structure: "stochastic integrable systems" (Joint work with Konstantin Matetski and Daniel Remenik).

The Topology of Azumaya Algebras

Speaker: 
Ben Williams
Date: 
Fri, Oct 12, 2018
Location: 
PIMS, University of British Columbia
Conference: 
UBC-PIMS Mathematical Sciences Faculty Award
Abstract: 
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

Speaker: 
Naomi Leonard
Date: 
Mon, Oct 1, 2018
Location: 
PIMS, University of British Columbia
Conference: 
IAM-PIMS Distinguished Colloquium Series
Abstract: 
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.

Integers in many-body quantum physics

Speaker: 
Sven Bachmann
Date: 
Fri, Sep 28, 2018
Location: 
PIMS, University of British Columbia
Conference: 
UBC Math Department Colloquium
Abstract: 
Although integers are ubiquitous in quantum physics, they have different mathematical origins. In this colloquium, I will give a glimpse of how integers arise as either topological invariants or as analytic indices, as is the case in the so-called quantum Hall effect. I will explain the difficulties arising in extending well-known arguments when one relaxes the approximation that the particles effectively do not interact with each other in matter. Recent advances have made such realistic generalizations possible.

Birational geometry and algebraic cycles

Speaker: 
Burt Totaro
Date: 
Fri, Sep 14, 2018
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-UBC Math Department Colloquium
Abstract: 
A fundamental problem of algebraic geometry is to determine which algebraic varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. We discuss the history of the problem. Some dramatic progress in the past 5 years uses a new tool in this context: the Chow group of algebraic cycles.

Using mathematics to fight cancer

Speaker: 
Ami Radunskaya
Date: 
Mon, Aug 13, 2018
Location: 
PIMS, University of British Columbia
Conference: 
Diversity in Mathematics
Abstract: 

What can mathematics tell us about the treatment of cancer? In this talk I will present some of work that I have done in the modeling of tumor growth and treatment over the last fifteen years.

Cancer is a myriad of individual diseases, with the common feature that an individual's own cells have become malignant. Thus, the treatment of cancer poses great challenges, since an attack must be mounted against cells that are nearly identical to normal cells. Mathematical models that describe tumor growth in tissue, the immune response, and the administration of different therapies can suggest treatment strategies that optimize treatment efficacy and minimize negative side-effects.

However, the inherent complexity of the immune system and the spatial heterogeneity of human tissue gives rise to mathematical models that pose unique challenges for the mathematician. In this talk I will give a few examples of how doctors, immunologists, and mathematicians can work together to understand the development of the disease and to design effective treatments.

This talk is part of the PIMS Diversity in Mathematics Summer School and is intended for a general audience: no knowledge of biology or advanced mathematics will be assumed.

Biography

A California native, Professor Radunskaya received her Ph.D. in Mathematics from Stanford University. She has been a faculty member in the Math Department Pomona College since 1994.

In her research, she specializes in ergodic theory, dynamical systems, and applications to various "real-world" problems. Some current research projects involve mathematical models of cancer immunotherapy, developing strategies for targeted drug delivery to the brain, and studying stochastic perturbations of dynamical systems.

Prior to her academic career, Professor Radunskaya worked extensively as a cellist and composer. Her music, described as "techno-clectic", combines traditional forms with improvisation, acoustic sounds with electronic, computer-generated, and found sounds, and abstract structures with narrative visual and sonic elements.

Contrary to popular belief, Professor Radunskaya thinks that anyone can succeed in mathematics, and she has committed herself to increasing the participation of women and underrepresented groups in the mathematical sciences.

She is currently the President of the Association for Women in Mathematics, and co-directs the EDGE (Enhancing Diversity in Graduate Education) program, which won a "Mathematics Program that Makes a Difference" award from the American Mathematics Society in 2007, and a Presidential Award for Excellence in Science, Mathematics and Engineering Mentoring (PAESMEM) in 2015.

Professor Radunskaya was recently been elected as a Fellow of the American Math Society, and she is the recipient of several awards, including a WIG teaching award in 2012, and the 2017 AAAS Mentor award.

Class Numbers of Certain Quadratic Fields

Speaker: 
Kalyan Chakraborty
Date: 
Thu, Jul 5, 2018
Location: 
PIMS, University of Calgary
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
Class number of a number field is one of the fundamental and mysterious objects in algebraic number theory and related topics. I will discuss the class numbers of some quadratic fields. More precisely, I will discuss some results concerning the divisibility of the class numbers of certain families of real (respectively, imaginary) quadratic fields in both qualitative and quantitative aspects. I will also look at the 3-rank of the ideal class groups of certain imaginary quadratic fields. The talk will be based on some recent works done along with my collaborators.
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