People

Research Area: Analysis, applied-mathematics

Research Profile: Nonlinear partial differential equations, mathematical physics.
I work in the areas of elliptic partial differential equations, the calculus of variations, and mathematical physics. Elliptic PDEs often arise as stationary equilibria in physical problems or in describing curved surfaces in differential geometry. The calculus of variations is concerned with extrema (critical points) of functions defined on infinite dimensional spaces. For example, solutions to the Dirichlet problem minimize an associated integral among all functions with the given boundary data (“Dirichlet’s Principle”). This observation, known to Gauss and Riemann, introduced variational methods as a tool in studying elliptic PDEs. Today we use a combination of classical variational techniques, real and functional analysis, and topology to study existence, multiplicity, smoothness, stability, and other qualitative properties of solutions to PDEs. Of particular interest are those problems (arising in physics and geometry) where minimizing sequences may not converge, due to the natural symmetries of the problem.

Research Area: Actuarial and Financial Mathematics

Research Profile: Actuarial Science
My research area is in actuarial science, with focus on: dependence modeling, loss reserving, capital allocation and predictive modeling in non-life insurance and ratemaking/pricing.
The objective of my research is to propose hierarchical statistical approaches to model dependence within and between risks in non-life insurance in general, and in a loss reserving context in particular. One of the most critical problems in Property & Casualty (P&C) insurance is to determine an appropriate reserve for incurred but unpaid losses. These provisions generally comprise most of the liabilities of a non-life insurance company. The global provisions are often determined under an assumption of independence between the lines of business. However, most risks are related to each other in practice, and this correlation needs to be taken into account.

Research Area: probability-statistics

Research Profile: Order statistics, distribution theory.
My research interests are in order statistics, outliers, distribution theory, statistical inference, records, multivariate analysis, and industrial statistics. Order statistics arise naturally in many problems such as life-testing and reliability experiments, outlier detection, robust inference, etc. My work in this area covers both theoretical (like bounds, approximations, recurrence relations, characterizations, etc.) as well as applied (like linear inference, acceptance sampling plans, tolerance limits, prediction, etc.) aspects. Another major area of interest to me is statistical distribution theory. Statistical distributions play a fundamental role in many applied problems and form the basis for model building and parametric statistical inference. My contributions to this area have been through proposing new distributions, developing inference techniques for several distributions, and deriving characterizations of distributions. My most recent work concerns order statistics under non-standard conditions. Work in this direction is carried out by using permanents (of some special matrices) and their properties.

Research Area: geometry-topology

Research Profile: Gauge theory and low-dimensional topology.
My research concerns the topology and geometry of character varieties and moduli spaces of flat connections, and applications to invariants of knots, links, and 3-manifolds.

Research Area: mathematical-biology

Research Profile: mathematical-biology.
I work on a wide variety of problems in theoretical ecology and evolution. Mathematical interests: the use of “moment equations” (equations for the mean, variance, and second-order spatial statistics such as spatial correlation) to understand the qualitative dynamics of biological communities (e.g. hosts and parasites, competing plant species)interacting in continuous spatial arenas. Biological interests: host/parasite interactions, broadly defined to include any “intimate association” among natural enemies and their victims: viruses, bacteria, and classical macroparasites. I work on the general dynamics of infectious disease epidemics, and particularly on the evolution of virulence. Statistical interests: fitting of parameters for ecological and evolutionary models, especially in a maximum likelihood settings. My recent work has focused on improvement and dissemination of tools for fitting generalized linear mixed models.

Research Area: applied-mathematics, Fluids & Turbulence

Research Profile: applied-mathematics, Fluids & Turbluence

applied-mathematics, Fluids & Turbluence I am an applied mathematician interested primarily in problems combining the theoretical and computational fluid dynamics with the theory of optimization and control. In addition, I also have a broad interest in applied-mathematics and scientific computing. My research is a blend of fundamental investigations in core problems of hydrodynamics and more applied studies relevant for industrial fluid mechanics. I focus on the use of advanced mathematical methods to develop efficient computational algorithms and models useful for studying problems in fluid mechanics. My administrative responsibilities include directing McMaster’s School of Computational Science and Engineering.

Research Area: Analysis, applied-mathematics

Research Profile:Nonlinear partial differential equations, interface dynamics.
Interface dynamics is a study of the qualitative behaviour of solutions to certain nonlinear partial differential equations known as singular reaction-diffusion systems. These systems are usually models developed by material scientists and mathematicians in order to understand the properties of interfaces or phase boundaries in alloys, polymers or glasses. Under stress, the alloy divides into several regions where the orientation of the molecules differs. The boundary between these regions evolve in time in order to decrease surface tension. The mathematical models developed to study this evolution lead to geometrical problems for the interfaces. An important example is mean curvature evolution, where the normal velocity of the interface is given by its mean curvature. A formal asymptotic tool called the method of matched asymptotic expansion is often used to predict the behaviour for the evolution of the interfaces. Once this formal study is done, the problem is then to develop analytical and geometrical tools to verify rigorously the expected behaviour. Since the models involve nonlinear partial differential equations, new tools are often needed to successfully complete the work.

Research Area: probability-statistics

Research Profile: Computational statistics
My research focuses on the applied uses of computationally intensive methods in statistics. One area in which computaional power is particularly useful is in the use of Monte Carlo approximations which can replace intractable calculations or asymptotic argeuments. The use of these methods, however, require reliable software and good diagnostics for when there may be problems with the underling assumptions. In my doctoral work I examined diagnostics for Markov chain Monte Carlo methods which are particularly important in Bayesian statistics. Since then I have worked on the bootstrap and other resampling methods. A suite of functions that I have developed to implement many of these methods is used worldwide. These functions allow for the application of resampling methods in many fields. Much remains to be done, however. I am currently working on applied graphical diagnostics for bootstrap failure. I am also examining new areas of application for these methods and improvements on existing methods. My latest area of research is in Monte Carlo implementations of the EM algorithm for likelihood inference when more standard methods cannot be used.

Research Area: probability-statistics

Research Profile: Statistical Inference, order statistics, outliers
I am interested in the diverse applications of order statistics to statistical inference and outlier accomodation. Some of my research involves order statistics from independent and non-identically distributed (I.NI.D) random variables. I.NI.D. random variables are a generalization of the multiple-outlier slippage model, and therefore have applications to outlier accomodation. Order statistics also have applications in classical inference problems; for example I have developed some hypothesis tests based on order statistics and inverse sampling which provide a more efficient alternative to the standard chi-square test for homogeneity. I have also developed a set of Maple programs that allow one to compute series approximations for moments of order statistics. Another area of interest to me is waiting time problems. My contribution to this area has been to develop new methods, some of which are based on uniform random variables and generating functions, which enable one to solve a wide variety of very complex waiting time problems. I have also been involved in some consulting projects, one of which requires testing for independence between virus and respiratory disease data using time series analysis.

Research Area: Probabiility and Statistics

Research Area: Algebra & Number Theory

Reserach Profile:  Algebra and Number Theory
My background is in number theory and algebra. Currently I solve functional equations; examples of these are:
f(x + y) = f(x) + f(y) (Cauchy) g(xy) + g(1/x) + g(1/y) = 0 (Kepler-Napier)
D(xy x) = xy D(x) + x D(y) x + D(x) yx (Herstein-Kaplansky)
F(x, y) + F(x + y, z) = F(x, y + z) + F(y, z) (Cocycle)
The methods I use are, in the main, algebraic: I do not consider continuity, analyticity and so on of solutions.

Research Area: Combinatorics, Optimization

Research Area: applied-mathematics

Research Profile: Partial differential equations, solitons

My research concentrates in the areas of applied nonlinear partial differential equations, spectral analysis, nonlinear waves (solitons), mathematical modeling in nonlinear optics, photonic crystals and Bose-Einstein condensates.

Research Area: mathematical-biology

Research Area: applied-mathematics, mathematical-biology

Research Profile: Epidemiology, Ecology, Evolutionary Game Theory
Biological systems involve intricate interactions on many spatial and temporal scales. Mathematical models often make it possible to identify mechanisms behind complex biological processes and to predict the outcomes of environmental changes. My current research can be classified according to the time scale over which biological changes occur. Short time scales: Population dynamics of ecological and epidemiological systems. This work, which has implications for conservation of endangered species and eradication of infectious diseases, makes use of modern bifurcation theory. Analytical investigations of discrete maps and differential equations are usually supplemented by extensive numerical studies. Long time scales: Evolutionary dynamics of behavioural traits. This work, which is mainly based on game theoretical analysis, clarifies the adaptive significance of animal behaviour, ranging from cooperation and parental care to foraging and cannabilism.

Research Area: applied-mathematics, probability-statistics

Reserach Profile: Diffusion theory, Electromagnetic scattering and propagation in random media, Geometric phases in quantum theory and coherent states, MRI physics, Population dynamics, Radar scattering, Stochastic differential equations, Stochastic quantization.

I specialize in probability theory, specifically in combinatorial stochastic processes. This means studying random objects and processes on combinatorial spaces, such as random graphs, as well as studying continuum scaling limits or analogues of these objects. Much of my research concerns exchangeability and continuum random trees.

Research Area: Biostatistics

Research Area: Financial Mathematics

Research Profile: Financial Mathematics, Mathematical Physics, Information Geometry 
My main research interests lie in the area of mathematical finance – the application of stochastic analysis and probability theory to the understanding and modelling of financial markets. Specific projects I am involved with include numerical and theoretical optimal portfolio selection and modelling of positive interest rates. I am also interested in classical and quantum information geometry. My work in this area ranges from the construction of infinite dimensional manifolds of probabilities to the applications of such geometric objects to models in fluid dynamics and quantum information.”

Research Area: geometry-topology

Research Profile: Algebraic and geometric topology 
One of the unifying principles in geometry is that complex systems, such as configurations of planets and stars can often be understood by means of their symmetries. Familiar symmetries include the rotations or reflections of solids in space and the Lorentz transformations of space- time. Discrete invariants and groups of symmetry of continuous motions are studied in algebraic topology, while geometric topology is concerned with the properties of differential manifolds, or higher-dimensional surfaces. Topology developed rapidly in the period between 1950 and 1980 and is now one of the most active fields of mathematics. Symmetries of differentiable manifolds are related to algebra and number theory through group theory, and to partial differential equations and analysis through differential forms. My recent work deals with finite group actions on 4-dimensional manifolds and their connections to gauge theory, and infinite discrete group actions on high-dimensional manifolds. Both areas provide many opportunities for interesting research problems.

Research Area: Mathematical Logic

Research Profile: Model theory and mathematical logic. 
My work has been primarily in model theory and to some degree in set theory. Model theory is the study of the construction and classification of structures inside some specific class of mathematical objects. It has both pure and applied sides which interact heavily. In applied model theory one takes some mathematical object in a given language, say the real numbers in the language of fields with exponentiation, and very carefully analyzes the sub-sets which are definable. This type of elimination theory has several applications in number theory and real algebraic geometry. An example of my work in this area is the first article listed below where we study certain model theoretic conditions in the context of varieties of algebras.

Pure or abstract model theory deals with several issues. The first, which sounds vaguely philosophical, is the question, “Is it possible to know if two structures are not isomorphic?” This question is at the heart of classification theory. On the more practical side is stability theory which starts out by severely restricting the classes of structures one looks at so as to have a robust dimension theory available. Luckily common mathematical objects such as modules and algebraic groups are examples of stable structures.

Research Area: Mathematical Logic

Research Profile: Model theoretic algebra 
I am interested in applying the techniques of model theory, which studies algebraic structures in a very general setting, to the examination of particular algebraic structures, especially valued fields.

Research Area: probability-statistics

Research Profile:Stochastic models, genetics and medicine, probability bounds 
Stochastic processes are models for describing phenomena which exhibit random fluctuations in their behaviour. Although individual predictions cannot be made, there is remarkable regularity over time which is then exploited in an effort to explain and predict natural phenomena.

My work has been mainly in the following areas:

• population genetics: to develop tools for making inferences about the evolutionary forces in the past which have led to the observed genetic variations among species. In May 1999, I gave a graduate course on this topic at the Fields Institute for Research in Mathematical Sciences. These will be published by the American Mathematical Society.
• combinatorics: to understand the models leading to various related combinatorical structures arising in genetics and ecology.
• large deviations
• probability bounds: to determine optimal upper and lower bounds on the probability of an event using only limited information.
• bootstrap: statistical analyis of channel and bundle power data from Ontario Power Generation CANDU reactors.

In my teaching I have recently begun to experiment with Excel for my undergraduate courses. This represents a departure from the traditional use of specialized statistical software. I have written the Excel Manual which accompanies Introduction to the Practice of Statistics by David Moore and George McCabe. This text is the most widely_used statistics text in North America.

Research Area: applied-mathematics

Reserach Area: Statistics and data mining

Research Profile: Statistics and data mining

Dr. Jeganathan’s research focuses on developing statistical and computational methods to analyze multi-domain data, especially addressing statistical challenges in microbiome multi-omic and spatial multi-omic data analysis. Current research includes multi-table integration, preprocessing and transformation of high-throughput sequencing data, visualization, hierarchical modeling, Bayesian statistics, statistical inference, block bootstrap method, data mining, and approximation theory in statistics.

Research Area: applied-mathematics, Fluids & Turbulence

Research Profile: 
applied-mathematics, Turbulent fluid flow, wavelet methods My research focuses on the theory and computation of fluid turbulence, with a special interest in dynamically adaptive numerical methods based on the wavelet transform. There are numerous problems that remain unresolved in the theory of turbulence, despite more than 100 years of research on the subject. A complete and precise theory of turbulence would be useful in areas as diverse as aerodynamics, combustion, urban pollution modelling, weather prediction and climate modelling. Although we are still far from being able to formulate such a theory, much progress has been made in the last few decades. The aim of my research is to combine several recent discoveries in order to develop a new approach to turbulence modelling. These discoveries include wavelet transforms (which are used to compress data and solve partial differential equations), penalisation methods (which can be used with any numerical method to simulate complex geometries, such as an airplane), and coherent vortices (flow structures that control turbulence dynamics). Current research projects include adaptive multiscale climate modelling, fluid-structure interaction, compressive sampling and the role of turbulence in star formation.

Research Area: Algebra & Number Theory

Research Profile: Algebraic Number Theory, Algebraic K-Theory
Some highlights in classical algebraic number theory are the development of class field theory, the study of L-functions and Zeta-functions, and their relationship to class groups and unit groups of number fields. New powerful tools have been invented since the 1960s and 1970s, in particular Iwasawa-theory, ätale cohomology, and last not least, algebraic K-theory, whose applications go far beyond the classical framework and lead into arithmetic algebraic geometry. My current research focuses on applications of this machinery to classical problems in number theory (e.g. Leopoldt’s Conjecture, Vandiver’s Conjecture), on generalizations of classical results about class groups and units to higher algebraic K-groups, and on their calculation by means of special values of L-functions.

I develop and study universal deep learning models capable of leveraging geometric structures in mathematical finance and data science problems.

Research Area: Geometry and topology, mathematical physics

My research focusses on geometric problems in general relativity and related problems in Riemannian geometry associated to asymptotically flat and hyperbolic manifolds. In particular I study black hole solutions to Einstein’s equations in four and higher dimensions. Such geometries are of intrinsic interest in the context of Lorentzian and Riemannian geometry as well as in developments in theoretical physics.

Research Area: geometry-topology, Mathematics Education

Research Profile: Differential Geometry, Applications, and Mathematics Education 
My research dealt with some aspects of the curvature of a Riemannian manifold, in particular the way various curvature assumptions influence the topological properties (pinching theorems). I had also employed the tools of Riemannian geometry (especially foliations on Riemannian manifolds) in order to study the properties of orbits of vector fields coming from various control theory problems. In collaboration with Ernst Ruh and Maung Min-Oo, I obtained new parametrization of the space of multivariate normal distributions (and a new distance formula). My new research directions consist of applying math to problems in medicine and biology (in collaboration with colleagues and grad students from Health Sciences), and also mathematics education. Besides investigating my own teaching practice, I have been studying transition from secondary to tertiary mathematics, as well as the ways mathematics textbooks promote learning of mathematics.

Research Area: Financial Mathematics

Research Profile: Financial Mathematics 
My research interests concern the mathematical modelling and analysis of problems in the area of quantitative finance. Of particular interest to me are problems that arise in credit risk, including issues of correlation, counterparty exposure, and capital assessment and allocation.

Research Area: Computational Statistics

Research Profile: Computational Statistics
Dr. McNicholas’ research focuses on computational statistics, and he is at the cutting edge of international research on mixture model-based clustering and classification. Current research includes work on big data featuring outlying or spurious points, with a focus on classification, clustering, dimension reduction and discriminant analysis. Another important aspect of Dr. McNicholas? current research is work on non-Gaussian mixture models, which present a useful alternative to the Gaussian mixture model. Work on clustering categorical data and data of mixed type is ongoing. Applications of Dr. McNicholas? research are readily found in several fields, including bioinformatics, sensometrics, and psychometrics.

Research Area: geometry-topology

Research Profile: Geometry and Topology 
More specifically, I compute topological invariants, such as equivariant cohomology theories, of spaces with such structure. Symplectic geometry is the mathematical framework of classical physics; hyperkahler manifolds are symplectic manifolds wiht extra structure, are of particular recent interest due to their connections to theoretical physics. I am mainly concerned with the theory of symmetries of manifolds with these structures, as encoded by a Hamiltonian Lie group action, i.e. there exists a moment map on M encoding the action by Hamiltonian flows. Such group actions on symplectic and hyperkahler manifolds arise naturally in the context of physics, representation theory, and algebraic geometry. To a Hamiltonian space, one associates a symplectic (hyperkahler) quotient, which inherits a symplectic (hyperkahler) structure from the original manifold. The main theme of my recent research is the study of the topology and equivariant topology of these quotients, in particular the computation of their cohomology and complex K-theory rings.

Research Area: geometry-topology

Research Profile: Differential Geometry, Geometric Analysis, Finance
Currently, I am working on the following problems:
1. Properties of Large Scale Networks and Graphs
2. Efficient Manifold Learning Algorithms
3. Asymptotic properties of Heat Kernels
4. Implementing geometric ideas in Probability and Statistics

Research Area: probability-statistics

Research Area: geometry-topology

Research Profile: Topology and geometry
Topology and geometry Geometry is the study of shapes and spaces. Most people are aware of the standard objects of Euclidean geometry: lines, circles, polygons, and of familiar notions such as angles, parallel lines, and congruent figures. In its modern form geometry has a much wider scope, reaching into higher dimensions and encompassing a broad range of current ideas. The subject of Topology is concerned with those features of geometry which remain unchanged after twisting, stretching or other deformations of a geometrical space. It includes such problems as colouring maps, distinguishing knots, classifying surfaces and their higher dimensional analogs. The influence of topology is also important in other mathematical disciplines such as dynamical systems, algebraic geometry (the study of polynomial equations in many variables) and certain aspects of analysis and combinatorics.

My research in recent years has focused on the study of the topology of manifolds and cell complexes by means of algebraic K-theory, pseudoisotopy theory, gauge theory, and representation theory.

Research Area: probability-statistics

Research Profile: Statistics, biological and medical applications, mixture distributions, nonlinear estimation, computing
New applications demand new statistical methods. Statisticians must ensure that new methodology is available to those who need it in the form of reliable, well-documented computer software, especially when the methods involve intensive computation.

While my interests in statistical applications are very general, much of my research has been in collaboration with fisheries biologists: enumerating migrating populations of juvenile Pacific salmon and inferring the age composition of populations from length-frequency data. For the latter problem I use computer graphics and non-linear estimation methods to resolve age groups from overlapping peaks in the length-frequency distribution. This is called mixture distribution analysis and my software developed at McMaster is now used world-wide for diverse applications. With the assistance of graduate students, I have implemented this methodology as the mixdist package in the open source R environment. Future work will develop the interactive graphical interface and bootstrap calculation of standard errors.

I served on Scientific Advisory Panels for the United States Environmental Protection Agency from 2000 to 2014, evaluating models to estimate human uptake of pesticide residues.

I am now an associate member of the GERAS Centre for Aging Research and collaborate with the GERAS team in study design and analysis and in medical education.

Research Area: Combinatorics

Research Area: probability-statistics

Research Profile: Statistical Inference, reliability, Industrial Statistics, Quality Control

An area of industrial research that has motivated many practical and theoretical statistical procedures is the study of the reliability of products and processes. One of my interests is the development of statistical methods to model and analyze reliability data. Another aspect focuses on the modeling and analysis of discrete data. Specifically, I have studied models and methods for the estimation of population size with application to animal abundance, the use of adaptive rules in clinical trials, and the study of correlated Bernoulli trials.

Currently, I am working on quality control methods, specifically of the multivariate type. Here I study methods to monitor processes where the number of quality indicators varies from small to quite large. In the latter case, dimension-reduction techniques such as principal components are first used, followed by control charts on leading components. The methods have been applied to the monitoring of product profiles.

Research Area: Analysis

Research Profile: Harmonic analysis, partial differential equations and function theory

Harmonic analysis, partial differential equations and function theory My research interests in harmonic analysis center on weighted norm inequalities for fractional and singular integrals, with applications to problems in partial differential equations such as regularity of solutions to the oblique derivative problem, sharp estimates for the eigenvalues of Schrodinger operators, regularity for the Monge-Ampere equation and distortion of Hausdorff measure under quasiconformal maps. My interests in function theory include interpolation and corona problems on the unit ball in several complex variables.

Research Area: probability-statistics

Research Profile: Stochastic processes, interacting particle systems
My research interests are in nonlinear stochastic model, large deviation, hydrodynamic limit and stochastic models in genetics. Some concrete problems are the long-time behaviour, the phase structure and metastability of the corresponding system. The method used comes from probability theory, stochastic processes and stochastic analysis.

Research Area: Mathematical Logic

Research Profile: Model Theory and Real Analytic Geometry

Research Area: Financial Mathematics

Research Profile: My main research lie in the area of financial mathematics with a special emphasis on optimal investment and pricing in incomplete markets. My projects are on equilibrium pricing of non tradable risks, optimal portfolio selection with regulatory constraints and time consistent portfolio management.

Research Area: Mathematical Logic

Research Profile: Mathematical logic,universal algebra and computational complexity

Mathematical logic,universal algebra and computational complexity I am involved in the study and classification of general algebraic systems. This area of mathematics is often called Universal Algebra and got its start in the 1930s. In order to compare and classify algebras they are often grouped together according to the equations that they satisfy.

Borrowing and expanding on techniques and ideas from mathematical logic, classical abstract algebra, and also from newer branches of mathematics such as lattice theory and category theory, powerful tools have been developed to help organize and understand the structure of varieties (classes of algebras defined by equations) and the algebras they contain. Recent advances in the field have opened up a new area of study dealing with the local structure of finite algebras. This new local theory of finite algebras has not only been useful in solving several longstanding problems but it has also suggested a number of new and challenging research problems.

My current research program involves studying the computational complexity of subclasses of the Constraint Satisfaction Problem (CSP). Many well known complexity problems, such as graph coloring or Boolean satisfiability, can be naturally presented within the vast CSP framework. Recent work of Bulatov, Jeavons, Krokhin and others has established a strong connection between the CSP and universal algebra and some of the important open problems in the field can be expressed in purely algebraic terms.

Research Area: Combinatorics

Research Area:  Algebra

Research Interests: My research area is commutative algebra and its connection to other areas like combinatorics and algebraic geometry. I am interested in homological invariants, edge ideals, simplicial complexes, sets of points in (multi)-projective spaces, toric ideals, symbolic powers of ideals, and combinatorial matrix theory

Research Area: geometry-topology

Research Profile: Differential geometry, geometric analysis

My research is about prescribing the curvature of manifolds, which could mean requiring curvature to be positive, zero, or negative in some suitable sense, or requiring some natural differential equation involving curvature to hold on the manifold.

One class of manifolds which I study is motivated from physics. According to Einstein’s theory of relativity, gravity exhibits itself mathematically in the bending of space-time. The famous Einstein field equation relates the Ricci curvature, which is the relevant notion of curvature or “bending” in relativity, to the matter fields present. Can Einstein’s equation hold on any manifold, or is the topology of space-time restricted, or determined by it ? Is the geometry of relativity also compatible with the elementary particles we observe ? Are there continuous families of solutions of Einstein’s equation ? What kind (range) of geometric properties are exhibited by Einstein manifolds? The analogs of these questions in all dimensions and for manifolds with positive definite inner products provide the driving force behind some of my research projects.

Research Area: applied-mathematics, mathematical-biology

Research Profile: Dynamical systems, bifurcation theory, population dynamics, mathematical ecology and epidemiology

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. For example, one goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of micro-organisms most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, the prevention of species’ extinction or the control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures, and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, integro- and functional differential equations, and bifurcation theory.

Research Area: Computing 

Research area: low-dimensional topology

Research interest: I study low-dimensional topology: mainly knots in 3-manifolds and surfaces embedded in 4-manifolds. I’m particularly interested in understanding how different notions of equivalence for 2-dimensional knots inform 4-dimensional exotica.

Research interests: ecology, economics, dynamical systems

Research Area: Mathematical Biology

Department of Mathematics
University of Pittsburgh

Title

Car Traffic on Knot Diagrams and Some Cool Knot Invariants

Abstract

We will study some strange traffic rules for cars driving through an interchange: When they enter via an underpass, they just go through. But when they enter via an overpass, they fall down to the underpass with some small probability p, and then keep going unharmed, down under. 

We will learn that to analyze this traffic game we need matrices and that to play it better we need probabilities that are not numbers between 0 and 1. We will also learn how this game can be used to define some knot invariants (a notion we will explain) which may be the best we presently have. 

Website

Title

Fun with Conditional Probability

Abstract

Conditional probability explains how the odds of an outcome change in the presence of partial information. Even very simple examples using coins and dice aren’t always obvious and can be surprising! They also have connections to classic questions such as Disease Testing, the Monty Hall Problem, the Sleeping Beauty Problem, Monte Carlo Simulations, and more. Time permitting, the extension to sub-sigma-algebras in continuous models will also be described. Best of all, this talk will use PollEverywhere software to let the audience guess the answers — can you get them right?

Title

On Gelfand spectra of Beurling algebras

Abstract

For an infinite abelian group, a Beurling algebra over it is a type of Banach algebra containing its complex group ring as a dense subalgebra. My recent student Z. Zhang and I became interested in determining exactly what sets comprise the Gelfand spectrum of some Beurling algebra. The answer uses very classical functional analysis, including the bipolar theorem in real vector spaces.

Title

Computational Enumerative Geometry

Abstract

Classically, an enumerative problem takes the form of a deceptively simple question: “How many geometric objects satisfy a given collection of geometric constraints, given by some input data?”. For example, there is exactly 1 circle through three points, 27 lines on a smooth cubic surface, and 3264 conics tangent to five conics in the plane.

But counting is only the beginning. Once we know there are finitely many solutions, we can ask richer questions:

(Monodromy) How do the solutions permute as the input data varies

(Reality) If the input is real, how many solutions can be real?

(Formula) Is there a formula for the solutions, analogous to the quadratic formula?

(Computation) How can we compute all of the solutions?

These questions connect classical enumerative geometry with other areas including group theory, field theory, computation, and topology.

In this talk, I will introduce ‘numerical algebraic geometry’ as a computational approach to these problems. Through many examples from both ‘pure’ and ‘applied’ mathematics, I will show how tools from numerical algebraic geometry allow us to explore enumerative problems through computation and experimentation. I will conclude by highlighting some of the projects currently being explored by our research group at Western University.

Title

An Introduction to Multivariate Cryptography

Abstract

Multivariate cryptography is a promising candidate for post-quantum digital signatures. In practice, a multivariate cryptographic instance corresponds to a system of multivariate polynomial equations, so the security of such protocols depends on the complexity of solving these systems. In this talk, I will discuss two key invariants that play a central role in this context. Finally, I will examine the notion of generic systems as it appears in the cryptographic literature.

Title

You’re good at math. Help others be too.

Abstract

As future graduate students, teaching assistants, and professors, participants will have much influence over how hundreds- or thousands – of others experience math. The talk would be about the gap between how people think students learn math and what the evidence actually shows, drawing on topics from cognitive science, such as desirable difficulties (learning techniques that feel hard in the moment, but improve long-term learning), and designing student practice in an effective way. I would discuss what kind of math instruction leads to long-term success, and why mathematicians have a responsibility to get it right. 

Title

Mathematical Modelling in Fluid Mechanics

Abstract

In this talk, we explore two examples of how mathematical modelling may be used to gain insight into physical phenomena observed in fluid mechanics. First, we consider pilot-wave hydrodynamics, a system in which a millimetric droplet bouncing on the surface of a vibrating liquid bath may self-propel through an interaction with its own wavefield. We explore how integro-differential equations may be used to predict the stability and dynamics of bound droplet states, including rings and lattices. Second, we turn to stratified turbulence, which plays a central role in governing ocean circulation. We focus on a statistical technique, involving dimensionality reduction and unsupervised clustering, to classify distinct turbulent regimes in experimental data. 

Title

What is mathematical modelling? And how to be great at it.

Abstract

During the COVID-19 pandemic, decision-makers and the public expected mathematical modelling to answer questions such as “How many new cases do we expect next week?” and “What is the value of social distancing measures?” The pandemic made mathematical modelling a conspicuous part of our lives, but what is mathematical modelling and how is it related to other disciplines such as statistics, data science, and artificial intelligence?  

In this talk, I will use the COVID-19 pandemic in Atlantic Canada and the territories as an example to discuss these issues. Prior to 2022, these jurisdictions experienced long periods with no COVID-19 infections, so the questions, modelling, and data needs of these jurisdictions were different. To date, there have been few discussions of these issues. The motivation for this talk is that by discussing the strengths and weaknesses of mathematical modeling and adjacent disciplines, we are better able to design methods to answer impactful research questions. 

Title

An introduction to algebraic topology via vector calculus.

Abstract

I will give an introduction to homology theory and the fundamental group, motivated by familiar properties of integration in vector calculus. I will present these ideas informally by appealing to geometric intuition and offer some generalizations and applications.

Title

Explicit Results in Number Theory and How to Find Them

Abstract

In this talk we will give a brief introduction to the study of certain weighted prime counting functions related to the Riemann zeta function. From here we will discuss what is known about these functions, how this connects to the famous Riemann Hypothesis and what information we are able to prove. There will be special attention paid to making the results explicit, the unique challenges this comes with and we aim to end on a discussion of some possible directions in the field.