# Department research seminars

**Organizers:** Arusharka Sen and Christopher Plenzich **Time: **12:00 - 2:00 p.m. **Location:** LB 921-04

Upcoming Seminars

Title: | Borel equivalence relations and classification problems in mathematics |

Speaker: | Dr. Assaf Shani |

Date: | March 22, 2024 |

Abstract: |
This talk will be an introduction to the field of Borel equivalence relations (also called invariant descriptive set theory). No background will be assumed. We will motivate the main object of study: a Borel reduction between equivalence relations on Polish spaces. This in turn allows to measure the complexity of various classification problems in mathematics, and to prove precise impossibility results regarding conjectured classifications. |

#### Past seminars

Title: | The Distribution of Gauss Sums |

Speaker: | Dr. Chantal David |

Date: | March 15, 2024 |

Abstract: |
Gauss sums are fundamental objects in number theory. Quadratic Gauss sums were studied by Gauss, and after many attempts, Gauss gave a simple formula depending only on the argument of the Gauss sums modulo 4. Higher degree Gauss sums seem to behave differently. Based on numerical evidence, it was suggested by Kummer (1846) that the angles of cubic Gauss at prime arguments are not equidistributed, and exhibit a bias towards positive values. More extensive numerical testing seemed to indicate that the bias does not persist, and that cubic Gauss sums are indeed equidistributed, which was proven by Heath-Brown and Patterson (1979). We will explain in this talk what is involved in proving equidistribution of cubic Gauss sums, in particular why it took more than 130 years after Kummer's observations. The talk will be accessible to a general mathematical audience, including graduate students. |

Title: | Log canonical threshold and transverse contact isotopy |

Speaker: | Dr. Amey Kaloti |

Date: | April 21, 2023 |

Abstract: |
A holomorphic function of n complex variables with an isolated singularity at origin gives rise to a singular hyper surface. This function and hence the hyper surface can be studied from various viewpoints such as analytical, algebraic and topological. Associated to such a function is an elementary analytical quantity known as log canonical threshold(LCT). Even though defined analytically LCT has implications for algebraic geometry, in particular Birational geometry. Another way to study these functions is through topology as initiated by Milnor. Our goal in this talk is to study LCT and its implication for a special kind of topology, known as contact topology. More importantly, we will try to understand what can topology say about LCT? If time permits, I will talk about generalizations of LCT coming from algebraic geometry, known as jumping numbers and explore their connections with topology as well. |

Title: | Continuous-State Nonlinear Branching Processes |

Speaker: | Dr. Xiaowen Zhou |

Date: | March 31, 2023 |

Abstract: |
Continuous-state branching processes are continuous-state counterparts of discrete-state Bienayme-Galton-Watson branching processes. We consider a class of continuous-state branching processes with branching rates depending on the current population sizes. They are nonnegative-valued Markov processes that can be obtained either from spectrally positive Levy processes via Lamperti type time changes or as unique nonnegative solutions to SDEs driven by Brownian motion and (or) Poisson random measure with positive jumps. The nonlinear branching mechanism allows the processes to have exotic behaviours such as coming down from innity. But at the same time it brings in new challenges to their study for lack of the additive branching property. In this talk we introduce the above continuous-state nonlinear branching processes, and present results on coming down from innity, explosion and extinguishing behaviours for such processes. It is based on joint work with Clement Foucart, Bo Li, Junping Li, Pei-Sen Li and Yingchun Tang. |

Title: | Smooth copula-based generalized extreme value model and spatial interpolation for extreme rainfall in Central Eastern Canada |

Speaker: | Dr. Mélina Mailhot |

Date: | February 24, 2023 |

Abstract: |
In this presentation, we will propose a smooth copula-based Generalized Extreme Value (GEV) model to map and predict extreme rainfall in Central Eastern Canada. The considered data contains a large portion of missing values, and one observes several non-concomitant record periods at different stations. The proposed two-steps approach combines GEV parameters' smooth functions in space through the use of spatial covariates and a flexible hierarchical copula-based model to take into account dependence between the recording stations. The hierarchical copula structure is detected via a clustering algorithm implemented with an adapted version of the copula-based dissimilarity measure. |

Title: | Solvability of conics over the integers |

Speaker: | Dr. Carlo Pagano |

Date: | November 25, 2022 |

Abstract: |
In this talk I will put in context a recent joint work with Peter Koymans, settling a conjecture of Peter Stevenhagen predicting an asymptotic formula for how often the so-called negative Pell equation is solvable over the integers . The talk is aimed for the general mathematical audience. Hence, after stating the main result, in the first half of the talk I will introduce the audience to the obstructions to solve conics (2-variable quadratic equations) over the integers and the rational numbers. In the second half I will give an overview of the proof and its possible applications to other problems. |

Title: | From Financial Losses to Solvency Capital Requirements: A Revised MCT Framework |

Speaker: | Dr. Mélina Mailhot |

Date: | November 18, 2022 |

Abstract: |
Canadian solvency capital requirement standards have been updated and reviewed. Adapted modeling techniques and dependence measures are suggested, in order to fill the gap between European and Canadian standards. In this talk, the focus will be on catastrophic risks. Spatio-temporal models and adapted Minimum Capital Test measures will be presented. |

Title: | Saddle Transport and Chaos in the Double Pendulum |

Speaker: | Dr. Jason Bramburger |

Date: | November 11, 2022 |

Abstract: | Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to illustrate a variety of phenomena in nonlinear dynamics. This work provides a direct comparison between the dynamics of the double pendulum and transport in the solar system, which exist on vastly different scales. Thus, the double pendulum may be viewed as a table-top benchmark for chaotic transport, with direct relevance to energy-efficient space mission design. This talk presents a variety of periodic orbits corresponding to acrobatic motions of the double pendulum that can be identified and controlled in a laboratory setting. |

Title: | Holditch Curves and Holditch's Envelope |

Speaker: | Dr. Ronald Stern |

Date: | October 7, 2022 |

Abstract: | Reverend Hamnet Holditch’s Theorem has been fascinating people since 1858, and is listed as one of C. Pickover's 250 Milestones in the History of Mathematics. But Holditch’s hypotheses were not made precise. In this talk we will discuss how one proves the conditions which he must have been assuming. These involve smoothness and convexity of loci of dividing points of a chord as it travels around a closed curve, and the existence of a smooth envelope of those traveling chords. This is joint work with Professors Stancu and Proppe. |

Title: | Risk-Aware Multi-Armed Bandits with Extreme Value Theory |

Speaker: | Dr. Frédéric Godin |

Date: | April 29, 2021 |

Abstract: | A multi-armed bandit’s problem is a reinforcement learning setup where an agent is faced repeatedly with the task to select from a choice of actions (arms), each providing rewards/cost from different distributions. The traditional framework where the agent attempts maximizing its aggregate reward over a fixed number of trials is introduced. A risk-aware modification of the setup where the agent attempts minimizing risk rather than simply maximizing expected rewards is then introduced. The specific case of catastrophic risk minimization is emphasized, with tools of extreme value theory being embedded in the algorithm we develop to tackle such problem. This is joint work with Dylan Troop (CIISE, Concordia) and Jia Yuan Yu (Amazon). |

Title: | The Curse of Dimensionality and the Blessings of Sparsity and Monte Carlo Sampling: From Polynomial Approximation to Deep Learning in High Dimensions |

Speaker: | Dr. Simone Brugiapaglia |

Date: | April 22, 2021 |

Abstract: | In data science and scientific computing, the approximation of high-dimensional functions from pointwise samples is a ubiquitous task, made intrinsically difficult by the so-called curse of dimensionality. In this talk, we will illustrate how to alleviate the curse thanks to the "blessings" of sparsity and Monte Carlo sampling. First, we will consider the case of sparse polynomial approximation via compressed sensing. Focusing on the case where the target function is smooth (e.g., holomorphic), but possibly highly anisotropic, we will show how to obtain sample complexity bounds only mildly affected by the curse of dimensionality and rigorous convergence rates. Then, we will illustrate how the mathematical toolkit of sparse polynomial approximation can be employed to obtain practical existence theorems for deep learning in the context of high-dimensional Hilbert-valued function approximation. These results show not only the existence of neural networks with desirable approximation properties, but also how to compute them via a suitable training procedure in order to achieve best-in-class performance guarantees. We will conclude by discussing ongoing and future research directions. |

Title: | Susceptible-Infected-Recovered Model with Stochastic Transmission |

Speaker: | Dr. Yang Lu |

Date: | April 9, 2021 |

Abstract: | The Susceptible-Infected-Recovered (SIR) model is the cornerstone of epidemiological models. However, this specification depends on two parameters only, which implies a lack of flexibility and the difficulty to replicate the volatile reproduction numbers observed in practice. We extend the classic SIR model by introducing nonlinear stochastic transmission, to get a stochastic SIR model. We derive its exact solution and discuss the condition for herd immunity. The stochastic SIR model corresponds to a population of infinite size. When the population size is finite, there is also sampling uncertainty. We propose a state-space framework under which we analyze the relative magnitudes of the observational and stochastic epidemiological uncertainties during the evolution of the epidemic. We also emphasize the lack of robustness of the notion of herd immunity when the SIR model is time discretized. |

Title: | Atoms and BMO, or: How I Learned to Stop Worrying and Love the Log |

Speaker: | Dr. Galia Dafni |

Date: | March 26, 2021 |

Abstract: | Hardy spaces and functions of bounded mean oscillation come up in various aspects of harmonic analysis and PDE. The talk will introduce these notions and their relation to other familiar spaces and objects in analysis, leading to some recent research work with former and current students. The talk should be accessible to anyone with an advanced undergraduate analysis background. |

Title: | An Analysis of Electricity Congestion Price Patterns in North America |

Speaker: | Dr. Frédéric Godin |

Date: | January 22, 2021 |

Abstract: | We will discuss the use of principal component analysis (PCA) on electricity price data to detect the most salient congestion patterns in an electricity transmission grid; discrepancies between prices at various nodes (locations) in the grid indeed indicate limitation of transmission capacity between these nodes. Outputs from the PCA along with some data visualization tools are shown to make the identification of such patterns seamless and straightforward. An empirical analysis is conducted for three North American power systems, namely NYISO, ISO New England, and PJM. |

Join the Zoom Session: | Zoom session hosted by Dr. Arusharka Sen https://concordia-ca.zoom.us/j/82238074192 |

Title: | Chebyshev's Bias and Generalizations |

Speaker: | Dr. Lucile Devin (CICMA) |

Date: | February 7, 2020 |

Abstract: | In 1853, Chebyshev observed that the prime numbers are not as well distributed as we think they are in congruence classes. We will explain that observation and give some ideas on recent results generalizing it. |

Title: | Circle Maps and Rigidity |

Speaker: | Dr. Elio Mazzeo |

Date: | November 29, 2019 |

Abstract: | We will discuss the dynamics of circle maps (orientation-preserving circle homeomorphisms). We will review the topological classification of these circle maps (the Denjoy Theory). We will introduce the dynamical partition of the circle, and talk about the connection between how well the irrational rotation number is approximated by a rational and the global smooth classification of circle maps (the Hermann Theory). We will discuss the differences in the rigidity results of three important classes of circle maps: diffeomorphisms, critical maps, and maps with break points. |

Title: | Prime Statistics |

Speaker: | Dr. Tristan Freiberg |

Date: | November 1, 2019 |

Abstract: | We discuss some major breakthroughs that have occurred in prime number theory over the past decade, and their consequences with respect to the distribution of primes in intervals. |

Title: | Robust Sparse Recovery for High-Dimensional Problems in Uncertainty Quantification |

Speaker: | Dr. Simone Brugiapaglia |

Date: | October 11, 2019 |

Abstract: | Motivated by the uncertainty quantification of PDEs with random inputs, we will consider the problem of computing sparse polynomial approximations of functions defined over high-dimensional domains from pointwise samples. In this context, recently introduced techniques based on sparse recovery and on compressed sensing are able to substantially lessen the curse of dimensionality, thus enabling the effective approximation of high-dimensional functions from small datasets. We will illustrate rigorous error estimates for these approaches by focusing on their robustness to unknown errors corrupting the data. Finally, we will demonstrate their effectiveness through numerical experiments and present some open challenges and future research directions in the field. |

Title: | Extension Theorem for Functions of Bounded Mean Oscillation and its Variations |

Speaker: | Dr. Almaz Butaev |

Date: | April 26, 2019 |

Abstract: | We make a brief overview of the class of functions called BMO and its role in some problems of analysis. The main question we discuss is, “How to extend BMO functions (and variants thereof) defined on a domain to the entire Euclidean space?” Some results of a joint work with Dr. Galia Dafni are presented. |

Title: | Ephemerals |

Speaker: | Dr. Alexander Shnirelman |

Date: | March 22, 2019, 11:45 am - 1:30 pm |

Abstract: | I willl talk about some strange objects appearing in the nonlinear problems. They first appeared in the fluid dynamics, in the study of some exotic weak solutions of the Euler equations describing the motion of the ideal incompressible fluid. Similar things are found in the stochastic analysis (where they are responsible for the "black noise"). These objects, which he proposes to call "ephemerals", maybe imagined as a sort of generalized functions; however, these "functions" are orthogonal to all smooth functions, so in fact they are not functions at all. The exact definition of these objects is given in the language of the nonstandard analysis. |

Title: | The Countable Lifting Property and its Relation to a Partial Order on Reduced Rings |

Speaker: | Dr. Bob Raphael |

Date: | February 22, 2019 |

Abstract: | Work joint with W. Burgess of the University of Ottawa. The context is commutative rings which are lattices under a natural partial order. The object of study is the countable lifting property. The work of Topping is mentioned along with advice to beginners on how to view the printed word. Joint work with Hager is mentioned. The main topic is a partial order on commutative rings which are semiprime. The lifting result is presented for the Boolean case. Some pluses and minuses for the general case are presented. They suggest study of the case when a ring is closed under taking infs in the partial order. Examples and counterexamples are given, and a discussion of topological conditions on X for the ring C(X) ensues. |

Title: | Probability in Arithmetic Conjectures |

Speaker: | Dr. Hershy Kisilevsky |

Date: | January 25, 2019 |

Abstract: | We use probabilistic models to predict the frequency vanishing of special values of elliptic L‐functions. This translates to predicting the existence of algebraic points on elliptic curves. These seem to agree well with observed computational evidence. |

Title: | Theory of Multivariate Sigma-functions: How to Construct Analytic Series. Applications and Open Problems |

Speaker: | Dr. Julia Bernatska |

Date: | December 7, 2018 |

Abstract: | This talk is devoted to the theory of multivariate sigma-functions developed by V. Buchstaber, D. Leykin, V. Enolski (see [1–4]). The sigma-function is considered as a solution of a system of linear partial differential equations. The theory gives the way how to construct the system. The unique solution arises as a series expansion, and has the advantage to be effective and easy for computation. The first part of talk describes the construction of sigma-function associated with a so called (n, s)-curve. As a by-product we obtain the basis of second kind differentials associated to the standard first kind differentials. The general scheme is illustrated by the examples of small genera. Further we discuss on some applications and open problems related to so called polylinear relations, namely, bilinear Hirota relations, which can be alternatively obtained from Klein’s bidifferential formula; and trilinear relations, which produce addition formulas. References: [1] Buchstaber V. M., Enolskii V. Z., and Leykin D. V. Rational Analogs of Abelian Functions, Functional Analysis and Its Applications, Vol. 33, No. 2, pp. 83–94, 1999. [2] Buchstaber V. M., and Leykin D. V. Polynomial Lie Algebras, Functional Analysis and Its Applications, Vol. 36, No. 4, pp. 267–280, 2002. [3] Buchstaber V. M., and Leykin D. V. Heat Equations in a Nonholonomic Frame, Functional Analysis and Its Applications, Vol. 38, No. 2, pp. 88–101, 2004. [4] Buchstaber V. M., and Leykin D. V. Solution of the Problem of Differentiation of Abelian Functions over Parameters for Families of (n, s)-Curves, Functional Analysis and Its Applications, Vol. 42, No. 4, pp. 268–278, 2008. |

Title: | Independent Component Analysis and Robust Clustering Related to Pollution and Astronomical Data |

Speaker: | Dr. Asis Kumar Chattopdhyay |

Date: | November 23, 2018 |

Abstract: | Independent Component Analysis (ICA) is closely related to Principal Component Analysis (PCA). Whereas ICA finds a set of source data that are mutually independent, PCA finds a set of data that are mutually uncorrelated. The assumption that data from different physical processes are uncorrelated does not always imply the reverse case that uncorrelated data are coming from different physical processes. This is because lack of correlation is a weaker property than independence. In the present work two data sets were considered. The first one corresponds to the study of pollution level in the Central Iowa, USA. The study was part of the Soil Moisture Experiment 2002. The second one corresponds to the investigation of evolutionary properties of the galaxy NGC 5128 on the basis of its globular clusters. The set of observable parameter includes structural parameters, spectroscopically determined Lick indices and radial velocities from the literature. For both the data sets components responsible for significant variation have been obtained through both Principal Component Analysis (PCA) and Independent Component Analysis (ICA) and clusters have been formed through k-Means clustering in order to find natural groups. It is found that ICA performs better than PCA due to non-Gaussian nature of the data sets. |

Title: | Dynamical System: An Astrophysical Application |

Speaker: | Dr. Tanuka Chattopdhyay |

Date: | October 26, 2018 |

Abstract: | In physics, a dynamical system is described as a particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives. In astrophysics we face, on regular basis, many such systems through which we can study the various activities taking place in galaxies. Galaxies are big ensembles of stars and gas, pervaded by magnetic field along with a poorly understood component, called dark matter. In search for a synthetic understanding, of the evolution of the star formation rate and the chemical abundances in galaxies, a dynamical model is proposed, combining gas in fall from galactic haloes, outflow of gas by supernova explosions. The oscillatory star formation model is a consequence of the modelling of the fractional masses changes of the hot, warm and cold components of the interstellar medium. The derived periods of oscillation vary in the range (0.1–3.0) × 10^{7} yr depending on various parameters existing from giant to dwarf galaxies. The evolution of metallicity varies in giant and dwarf galaxies and depends on the outflow process. |

Title: | The Sato-Tate Conjecture and other Equidistribution Conjectures for Elliptic Curves |

Speaker: | Dr. Chantal David |

Date: | October 5, 2018 |

Abstract: | We present in this talk several natural prime counting conjectures associated to elliptic curves over Q, concerning the reductions of a fixed elliptic curve E over the finite fields F_p, varying over all primes p. Among such conjectures is the Sato-Tate conjecture which was proven recently by Taylor in collaboration with Clozel, Harris, and Shepherd-Barron. His breakthrough work (recognized by the 2015 Breakthrough Prize in Mathematics!) proves a non-effective version of the Sato-Tate conjecture. We explain how assuming more hypotheses leads to an effective version of the Sato-Tate conjecture. As an application, we show how this effective version of the Sato-Tate conjecture gives upper bounds for other prime counting conjectures associated to elliptic curves over Q, which are still open, as the Lang-Trotter conjecture, or the extremal primes conjecture. The talk will be accessible to a general audience of mathematicians and graduate students, without number theory background. |

Title: | Perfectly Matched Layer Method for Scattering in Deformed Tubular Waveguides |

Speaker: | Dr. Victor Kalvin |

Date: | April 13, 2018 |

Abstract: | In order to obtain a good approximation of scattered waves by numerical solutions of a problem with finite computational domain, waveguides should be truncated without creating excessive reflections from the boundary of truncation. The idea is to place in front of the boundary of truncation an artificial layer strongly absorbing the scattered waves. This truncation scheme supplemented with very special construction of the layer is widely known as the Perfectly Matched Layer method. In this talk, I will show how the method of spectral deformations (originating from the theory of resonances for N-body quantum scattering) can be used to construct the layer and to analyze stability and convergence of the resulting PML method. |

Title: | A Paradoxical Game |

Speaker: | Dr. Hamid Pezeshk |

Date: | April 6, 2018 |

Abstract: | In this talk, a well-known paradox in game theory will be discussed. Some extensions to the original version of the paradox in two-player games will be introduced and some applications in biological systems will be illustrated. Motivating by the paradox the robustness of genetic switches in the noisy environment will be explained. Finally, games with three players are reviewed and some extensions of the paradox for these types of games are introduced. |

Title: | Empirical Likelihood for Complete and Censored Data |

Speaker: | Dr. Arusharka Sen |

Date/Time: | March 29, 2018 at 11:45 am. |

Abstract: | Empirical likelihood is one of the important paradigms of modern statistical inference. Introduced by Owen (1988), it provides a model-free (non-parametric, in statistical terminology) counterpart of the so-called likelihood function of parametric statistical models. In a way, it borrows ideas from the maximum likelihood and method of moments paradigms to produce high-probability upper and lower bounds (confidence intervals) for various quantitative features of the unknown distribution of the observed data. In this talk I will try to illustrate the empirical likelihood idea for complete as well as censored data which is a particular form of incomplete data encountered in survival analysis studies. |

Title: | Multivariate Geometric Risk MeasuresJoint work with Marius Hofert (University of Waterloo) and Mélina Mailhot (Concordia University) |

Speaker: | Dr. K. Herrmann |

Date: | March 16, 2018 |

Abstract: | This talk focuses on multivariate risk measures. Specifically, we consider risk measures inspired and based on the geometric quantiles introduced by Chaudhuri (1996). We first introduce a generalization of expectiles for d-dimensional multivariate distribution functions. In the univariate case, expectiles are defined in terms of asymmetric square loss minimization and have recently attracted attention as risk measures. The resulting geometric expectiles are unique solutions to a convex risk minimization problem and are given by d-dimensional vectors. We discuss their behavior under common data transformations such as translations, scaling and rotations of the underlying data. Geometric expectiles also obey symmetry properties comparable to the univariate case. We furthermore discuss elicitability in the context of geometric expectiles and multivariate risk measures in general. We show that a consistent estimator is readily available by the sample version. Aside from expectiles, we also present a possible generalization for the commonly used Tail-Value-at-Risk and Range-Value-at-Risk. The geometric Range-Value-at-Risk is obtained by integrating geometric Value-at-Risk over a suitable set of confidence levels. In the talk we will discuss properties of the geometric RVaR and highlight open research questions.
Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Association, 91(434), 862-872. .Herrmann, K., Hofert, M., Mailhot, M. (2018). Multivariate geometric expectiles. Scandinavian Actuarial Journal, DOI: 10.1080/03461238.2018.1426038. |

Title: | Enumeration of Rooted Ribbon Graphs using QFT |

Speaker: | Dr. K. Gopalakrishna |

Date: | March 2, 2018 |

Abstract: | We look at counting rooted ribbon graphs using Feynman diagrams and generalized Catalan numbers. |

Title: | PPPP |

Speaker: | Dr. Lior Bary-Soroker |

Date: | February 9, 2018 |

Abstract: | I will discuss the fascinating connection between Prime numbers, Polynomials over finite fields, and Permutations. In particular, I will give probably the most complicated proof of the derangement problem and will discuss the twin prime conjecture. |

Title: | The Hilbert's 21st Problem and Related Questions |

Speaker: | Dr. Yulia Bibilo |

Date: | January 12, 2018 |

Abstract: | The Hilbert's 21st problem was formulated as follows: show that there always exists a linear di fferential equation of a Fuchsian type (or a Fuchsian system) with given poles and a given monodromy representation. The problem is closely connected to so called isomonodromic deformations and integrability of nonlinear differential equations. In general the Hilbert's 21st problem has negative solution. There have been a number of studies to obtain necessary and sufficient conditions for a required Fuchsian system to exist. In the talk, we present an approach to get explicit solutions of this problem via middle convolution. |

Title: | Algorithmic Theory of Nilpotent Groups |

Speaker: | Dr. Jeremy Macdonald |

Date: | December 8, 2017 |

Abstract: | Group presentations give a way to define a group using some building blocks (generators) and rules according to which they interact (relations). Many important infinite groups can be described by finite presentations, which in principle provides a computer program all of the information it needs to perform computations in the group. However, even the most basic questions, such as "Is this group trivial?" can be undecidable. When such questions can be answered, we ask to determine the computational complexity of doing so. We will provide an elementary introduction to algorithmic problems in groups and survey the current state of knowledge in the class of (infinite) nilpotent groups, including new results on low-complexity problems. |

Title: | Thomas Bayes at a Legume Farm: A Bayesian Analysis of Data from Legume On-Farm Trials in Afghanistan |

Speaker: | Dr. Murari Singh |

Date: | November 24, 2017 |

Abstract: | In his Time travelling machine, Thomas Bayes visited a legume farm in 2012 in Afghanistan. Realizing that the on-farm legume trials were being conducted for a number of years and locations in the past, he also noticed the prior information that was generated from such trials, and continued with his advice on…. I will present a brief introduction to the legume on-farm trials conducted in Afghanistan during 2009-2012. In order to compare the improved technologies with farmers' practices, I will discuss the models, estimation of the productivity in terms of means and posteriors for means, standard deviation and risk associated with the technology to meet a given target. The stochastic dominance [risk] analysis of yield data from Baghlan-e-sannati showed that the improved chickpea variety, Madad, can achieve a target of 2 t/ha with 95% probability which was three times that for the local practices. Crossover in the risk curves could be used to identify and quantify the region specific risk to meet a given target. It will be argued that the Bayesian analysis is more suitable approach for analysis of on-farm trials data compared to the frequentist approach. |

Title: | Solving Equations: From Pythagoras to Modular Forms |

Speaker: | Dr. Giovanni Rosso |

Date: | November 17, 2017 |

Abstract: | A central theme of number theory, from Diophantus till nowadays, is the quest for integral solutions of polynomial equations. Starting from Pythagorean triples, we shall proceed through centuries until we arrive at the proof of Fermat's Last Theorem by Taylor and Wiles. |

Title: | Symplectic Geometry of the Moduli Space of Projective Structures on Riemann Surfaces |

Speaker: | Dr. Chaya Norton |

Date: | October 27, 2017 |

Abstract: | The moduli space of quadratic differentials on Riemann surfaces can be viewed as the total space of the cotangent bundle to the moduli space of Riemann surfaces. By choosing a base projective connection which varies homomorphically in moduli, the moduli space of projective structures is identified with the moduli space of quadratic differentials. A projective connection defines, via the monodromy map, a representation of the fundamental group of the Riemann surface into PSL (2, C), i.e. a point in the character variety. We study the symplectic geometry induced via these maps and show: The homological symplectic structure on the moduli space of quadratic differentials (defined explicitly in terms of Darboux coordinates which involve the double cover arising from a quadratic differential) is identified with the canonical symplectic structure on the cotangent bundle to the moduli space of Riemann surfaces. Choosing the base projective connection as Bergman, Schottky, and Wirtinger induces equivalent symplectic structures on the moduli space of projective connections. Finally we show that under the monodromy map with base Bergman projective connection, the homological symplectic structure induces the Goldman bracket on the character variety. Following results of Kawai, the Bers base projective connection induces from the moduli of quadratic differentials an equivalent symplectic structure on the moduli space of projective connections. This is joint work with Marco Bertola and Dmitry Korotkin. |

Title: | New Insight into the Role of Intangible Heterogeneity of Gene Effects on Survival Time of Patients with Ovarian Cancer A statistical method for variable selection in finite mixture of survival models |

Speaker: | Dr. Farhad Shokoohi |

Date: | October 13, 2017 |

Abstract: | The advent of modern technology has led to a surge of high-dimensional data in biology and health sciences such as genomics, epigenomics and medicine. The high-grade serous ovarian cancer (HGS-OvCa) data reported by The Cancer Genome Atlas (TCGA) Research Network is one example that includes information on over 9,000 genes. Our study focuses on the relationship between Disease Free Time (DFT) after surgery among ovarian cancer patients and their DNA methylation profiles of genomic features. Such studies pose additional challenges beyond the typical big data problem due to intangible population substructure and censoring. Despite the availability of several methods for analyzing time-to-event data with a large number of covariates but a small sample size, there is no method available to date that accommodates the additional feature of heterogeneity. In this talk, we propose a regularized framework based on the finite mixture of accelerated failure time model to capture intangible heterogeneity due to population substructure and to account for censoring simultaneously. Our data analysis indicates the existence of heterogeneity in the HGS-OvCa data, with one component of the mixture capturing a more aggressive form of the disease, and the second component capturing a less aggressive form. In particular, the second component portrays a significant positive relationship between methylation and DFT for BRCA1. By further unearthing the negative relationship between gene expression and methylation for this gene, a biologically reasonable explanation emerges: hyper-methylation of BRCA1 leads to downregulation of the gene, which reduces the expression of the gene and thus longer DFT, especially if the gene is mutated. |

Title: | Functional-Discrete Technique for Eigenvalue Problems |

Speaker: | Dr. Nataliia Rossokhata |

Date: | April 7, 2017 |

Abstract: | Eigenvalue problems for differential equations are models of many important physical processes. However, most of them are not solvable, and computationally efficient approximation techniques are of great applicability. Finding the eigenvalues and eigenfunctions even for linear (Sturm-Liouville) problems can be a computationally challenging task due to highly oscillatory behaviour of the eigenfunctions corresponding to high eigenvalues. I will introduce a numerical technique which uses the differential equation coefficient approximation and generates a series solution. It provides an exponential convergence rate and allows efficient approximation for high eigenvalues. This technique is applicable for both linear and nonlinear eigenvalue problems. Some qualitative results for an eigenvalue problem with interface conditions obtained with this numerical technique will be discussed. |

Title: | Mathematical and Experimental Aspects of Gravitational Waves Discovery |

Speaker: | Dr. Dmitry Korotkin |

Date: | March 9, 2017 (Thursday) |

Abstract: | Recent discovery of gravitational waves at LIGO detectors was a result of long-term efforts by mathematicians and physicists. Mathematically, it required a detailed numerical simulation of interaction and collision of two black holes. Physically, it required the construction of ultra-sensitive detectors which are able to capture gravitational waves emitted as a result of collision of two massive black holes. The detection of the gravitational waves provided another compelling confirmation of Einstein's general relativity formulated exactly 100 years before. We discuss some basics of mathematical formalism involved in the study of gravitational waves and give a brief review of the experimental setup used in LIGO detectors. |

Title: | Kernel Density Estimation: Bernstein Polynomials and Circular Data |

Speaker: | Dr. Yogendra P. Chaubey |

Date: | February 10, 2017 |

Abstract: | In this talk I will give a brief preview of the kernel density estimator along with some new developments. Specifically, the approximation lemma due to Feller (see Lemma 1, §VII.1, Feller 1965) is used to motivate the Bernstein polynomial density estimator. Further, its generalization to multivariate density estimation and adaptation to circular density estimation will be presented. The talk will conclude with a result connecting the circular kernel density estimation with orthogonal polynomials on a unit circle and some illustrations. |

Title: | Dynamic Risk Management |

Speaker: | Dr. Frédéric Godin |

Date: | December 2, 2016 |

Abstract: | The application of mathematics to a financial risk management problem is illustrated. The case of a financial institution hedging a stock option payoff with a stocks portfolio is considered. Notions of sequential decision problems and dynamic programming are discussed. The Bellman equation is used to optimize the hedge. |

Title: | Geometric characterizations of ellipsoids |

Speaker: | Dr. Alina Stancu |

Date: | October 28, 2016 |

Abstract: | Ellipsoids enjoy a very large number of characterizations that are often used toward solving diverse mathematical problems. In other words, we can describe ellipsoids as the unique shapes that have certain properties without explicitly defining their equation. In this talk, I will present some geometric characterizations of ellipsoids. |

Title: | Multivariate Risk Measures |

Speaker: | Dr. Mélina Mailhot |

Date: | September 30, 2016 |

Abstract: | The use of multivariate risk measures in Actuarial Science, Finance and Enterprise Risk Management has been investigated since the beginning of the 21st century and is gaining popularity in research and in practice. It has the advantage of approaching heterogeneous classes of homogeneous risks in a more precise way than applying a univariate risk measure to an aggregate portfolio of pooled risks. Insurance companies are constrained with regulatory capital requirements, and must carefully select risks. Financial institutions must also fulfill regulatory capital requirements and perform risk selection. Risk measures help regulators establish those required amounts in order to secure capital and protect stakeholders. Risk measures are also used to compare risks and part of the risk selection process in Actuarial Science. In this presentation, we define several multivariate risk measures, such as multivariate Value-at-Risk (VaR), Tail Value-at-Risk (TVaR) and empirical multivariate risk measures. Properties and applications will be provided. |

Title: | A two-dimensional family of transformations with very diverse behaviour |

Speaker: | Dr. Pawel Gora |

Date: | April 8, 2016 |

Abstract: | I will talk about a process called "map with memory" and a two dimensional family of transformations induces by it. The family exhibits most diverse behaviour: from absolutely continuous invariant measure to globally attracting fixed point and singular Sinaj-Ruelle-Bowen measure. I hope to show a multitude of pictures as the study is mostly based on computer experiments. |

Title: | Replication, the Monster Group and the Baby Monster Group |

Speaker: | Dr. Chris Cummins |

Date: | March 11, 2016 |

Abstract: | In their paper "Monstrous Moonshine" Conway and Norton introduced the idea of replicable" functions, whose properties form a connection between modular functions and the characters of the monster group. In this talk I will explain these properties and their use in Borcherds' proof of the moonshine conjectures. I will then discuss one way in which these ideas generalize to the case of the baby monster group, which is joint work with Rodriguo Matias. |

Title: | Determinants of Super-Size: Why and How |

Speaker: | Dr. Marco Bertola |

Date: | February 12, 2016 |

Abstract: | Determinants are familiar to every student in math (hopefully). Interesting things happens when we try to compute determinants of large (infinite) size, be it as a limit of increasing size matrices or directly of appropriate infinite dimensional operators. I will give examples of applications of such large determinants to random processes and to algebraic geometry, and mention some methods for addressing their computation. |

Title: | The Countable Lifting Property and C(X): Don’t Believe Something Just Because it is in Print. |

Speaker: | Dr. Robert Raphael (Joint work with A.W. Hager) |

Date: | December 11, 2015 |

Abstract: | We work with vector lattices and discuss the ability to lift a countable orthogonal set in W/S back to one in W, where maps preserve vector lattice structure. The talk also discusses mistakes mathematicians make in the course of their work. |

Title: | On Arithmetic of Modular Forms |

Speaker: | Dr. Adrian Iovita |

Date: | November 6, 2015 |

Abstract: | Modular forms can be seen simply as holomorphic functions on the upper half plane which satisfy a certain transformation property with respect to some finite index subgroup of SL_2(Z). Modular forms which are eigenvectors for a system of operators, called Hecke operators, have remarkable arithmetic properties: for every prime integer p, there is a p-adic representation of the absolute Galois group of the rationals attached to them. I will try to explain some of the consequences of this fact |

Title: | On Arithmetic of Modular Forms |

Speaker: | Dr. Adrian Iovita |

Date: | November 6, 2015 |

Abstract: | Modular forms can be seen simply as holomorphic functions on the upper half plane which satisfy a certain transformation property with respect to some finite index subgroup of SL_2(Z). Modular forms which are eigenvectors for a system of operators, called Hecke operators, have remarkable arithmetic properties: for every prime integer p, there is a p-adic representation of the absolute Galois group of the rationals attached to them. I will try to explain some of the consequences of this fact |

Title: | Determinants of Obesity at Different Stages of the Life |

Speaker: | Dr. Lisa Kakinami |

Date: | October 2, 2015 |

Abstract: | Obesity has doubled since 1980 and places a person at an increased risk for a number of chronic diseases including cancer, diabetes, and cardiovascular disease. Thus obesity places a large burden on the health care system and reducing obesity throughout the life span is a top public health priority. Dr. Kakinami’s research in improving our understanding of the determinants of obesity (focusing on poverty, social-familial factors and behaviours) and empirically testing the methodological tools used in identifying obesity and obesity change will be presented. |

Title: | Zeroes and Zeta Functions and Symmetry: One Level Density for Families of L-Functions Attached to Elliptic Curves |

Speaker: | Dr. Chantal David |

Date: | March 20, 2015 |

Abstract: | The one-level density is about the behavior of low-lying zeroes of L-functions in families, and it is conjectured by Katz and Sarnak that it is given by the equivalent statistics on groups of random matrices. Contrary to other statistics, as the pair correlation, which are universal (i.e. the same for all Lfunctions), it is believed that the one-level density will differ depending of the “symmetry type" of the family (unitary, symplectic, orthogonal, even orthogonal and odd orthogonal). We study the one-level density for various families of L-functions attached to elliptic curves, using the ratios conjectures as introduced by Conrey, Farmer and Zirnbauer. From the (conjectural) closed formulas that we obtain, we can determine the underlying symmetry type of the families. This cannot always be done with the classical approach to the one-level density, via the explicit formulas, as results can only be achieved for test functions with Fourier transform of limited support, and the three orthogonal symmetry types are then undistinguishable. But this can be done with the ratio conjectures, with somehow surprising results, shedding new light on “independent" and "non-independent" zeroes, and the repulsion phenomenon. |

Title: | Ephemerals |

Speaker: | Dr. Alexander Shnirelman |

Date: | February 13, 2015 |

Abstract: |
The 20-th century was a time of proliferation of function-like objects such as measures, distributions, currents, Young measures, etc. Ephemerals arrived by the end of the century from different sources and in different guises. In the probability they were introduced by Vershik and Tsirelson within their theory of "Black Noise". In the Partial Differential Equations (especially in the Fluid Mechanics) they came together with paradoxical weak solutions of the Euler equations found by Scheffer, Shnirelman, De Lellis and Schekelyhidi, and others. These solutions start "from nothing" (i.e. from zero), and vanish in a finite time. Ephemerals can be described as a sort of functions which are orthogonal to all smooth functions, but being substituted into certain nonlinear partial differential equations as a right hand side, or a boundary/initial conditions, produce a nontrivial solution. So, ephemerals have nothing to do with distributions and similar objects. For now, there exist no general theory of ephemerals. All what we have is a number of examples. Some of them will be presented in the talk. |

Title: | On the Origins of Mathematical Thinking |

Speaker: | Dr. Alexander Shnirelman |

Date: | November 18, 2014 (Tuesday) |

Abstract: |
The talk is devoted to the great mystery of the birth of mathematics. The roots of mathematical thinking are much deeper than was previously believed. The very idea of an exact mathematical truth has a distinctly religious origin as was demonstrated by Abraham Zeidenberg. Moreover, the analysis of an actual logic of mathematical thinking, both in the form of mathematical discovery and mathematical comprehension by students, shows that this logic is not the Aristotelian one, but is close to the “prelogical” logic of premodern people. This thesis is illustrated by the analysis of the “Sumerian Pythagorean Theorem” which appeared at least 1000 years before Pythagoras, and some other theorems, as well as some non-mathematical phenomena. |

Title: | MAST 217 - Introduction to Thinking Mathematically |

Speaker: | Dr. Nadia Hardy |

Date: | April 11, 2014 |

Abstract: |
In this talk, Dr. Hardy will present the approach she has taken to teach the course MAST 217. She will show the sequence of activities with which students engage, inside and outside the classroom, and discuss their rationale; how and why she came to think about these activities. By means of several examples of students work, she will illustrate and analyze their thinking, understandings, and doings. She will do so as a teacher of mathematics and as a researcher in mathematics education – and briefly address the different questions, concerns, satisfactions, and reflections that emerge from wearing these two different hats. Then, she will present some reflections on the limitations of the teaching approach. Finally, Dr. Hardy will link this approach to the one she took for teaching MATH 200 and imagine which forms such approach could take for teaching other mathematics subjects. |

Title: | On Long Sequences |

Speaker: | Dr. Alexander Shnirelman |

Date: | January 24, 2014 |

Abstract: |
(Click here to view) |

Title: | From prime ideals to complete categories of commutative rings |

Speaker: | Dr. Bob Raphael |

Date: | January 10, 2014 |

Abstract: |
We review the notion of a prime ideal, the Zariski topology and the patch topology. A category is called complete if it has products and equalizers. We are interested in the completions in the category of commutative rings of different natural categories of integral domains. These include: all domains, integrally closed domains, Noetherian domains, and Unique factorisation domains. The completions are described in details. The talk is based on joint work with M. Barr and J. Kennison. |

Title: | Curve Flows and Solitons Equations: the Vortex Filament Case |

Speaker: | Ms. Manuela Girotti |

Date: | November 29, 2013 |

Abstract: |
The Vortex Filament Equation, describing the self-induced motion of a vortex filament in an ideal fluid, is a simple but important example of integrable curve dynamics. In this talk I will give a short introduction about VFE and I will show its connection with the nonlinear Schroedinger equation through the Hasimoto map. This is just the starting point for a wide spectrum of research which is meant to study curve flows in R^3 using powerful results borrowed from soliton theory. The talk is intended to give a peek in this field and stimulate interesting discussions especially among graduate students. |

Title: | Holditch’s Theorem Via Mamikon’s Theorem and Smoothing |

Speaker: | Dr. Ron Stern |

Date: | November 15, 2013 |

Abstract: | In 1858, Reverend Hamnet Holditch, a Senior Fellow of Gonville and Caius College, Cambridge, published the following remarkable result:
Reverend Holditch’s proof was not really a proof at all, with required hypotheses omitted and several technical details overlooked. In recent years, proofs of more general versions of Holditch’s Theorem have appeared, using modern methods. In this talk, we revisit Holditch’s original approach, and show how Mamikon’s method of sweeping tangents (a result which may be as obscure as that of Holditch) in conjunction with a smoothing result from Nonsmooth Analysis, do provide a formal proof. Some interesting peripheral questions are suggested by our methods. |

Title: | Comparison of the Expected Value of Perfect Information Related to Different Stochastic Dynamic Models |

Speaker: | Dr. Saeb Hachem |

Date: | April 5 , 2013 |

Abstract: | In a deterministic dynamic programming model (DDM), all the parameters of the optimization problem are assumed to be known and the decisions to be taken progressively in time. Any optimal decisions path, along the periods of the planning horizon, results in a unique value of the objective function (costs incurred or rewards). For many real world problems, the future cannot be known with certainty and hence some parameters must be uncertain. With the scenario approach, many DDM, relative to different evolution or passes of the set of the parameters, are solved. This approach is not relevant for decision making for many reasons. It does not even make available a unique optimal decision to implement at the beginning of the planning horizon, because the multiple choices that may result from this kind of sensitivity analysis. In stochastic dynamic programming models (SDM), some of the parameters of the underlying deterministic dynamic programming model are assumed to be random. This additional uncertainty induces a distribution of costs incurred or rewards. Without exception, stochastic dynamic models try to control directly or indirectly this distribution via controlling moments and/or quantiles and related criteria such as the value at risk (VAR) and the expected value at risk (TVAR or CVAR). SDM are mainly different from each other by the way the constraints involving random parameters are handled, and by the assumption about the decisions process. Decisions can be taken prior to the revelation of all random data, assumed to be adjustable to the progressive revelation of random data or assumed to be partly adjustable and partly taken a prior to each time period of the planning horizon. The objective functions of different SDM are comparable. Their expected value of perfect information, which is the difference between their objective functions and the scenario approach, is also comparable. The inequalities that relate all these relative performance are new results. Many SDM have been suggested to solve hydro generation problems. Although a hydro generation problem is used as a thread to introduce different SDM, the talk will be sprinkled by comments related to some applications in finance, actuarial science and open sky mining project development. |

Title: | A New Kind of Learning |

Speaker: | Dr. Fred Szabo |

Date: | March 1, 2013 |

Abstract: | Now that Concordia University has a site license for
- With its totally new predicative interface,
*Mathematica*9 reduces dramatically the need for writing and memorizing programming code. - With its rapidly expanding natural-language input option,
*Mathematica*brings conceptual and computational thinking closer together. - With its unique list-based syntax and predictive options for basic commands and functions,
*Mathematica*encourages consistent structure thinking. - With its tight integration of Wolfram/Alpha and growing access to computable knowledge in vast range of disciplines,
*Mathematica*9 is fast becoming a lingua franca that encourages collaborative thinking in cognitive disciplines. - Wolfram Technologies is committing enormous resources to the development of teaching and learning tools such as lesson plans, model lectures, blog, and other tools.
- Creating slideshows from
*Mathematica*notebooks is a built-in option and facilitates the transformation of lecture notes to interactive elegant presentations, conference talks, and seminars. - The Computable Document Format is for computational knowledge what the Portable Document Format is for the rest of our documents. It is a cost-free option, even for those without access to
*Mathematica*, and works with cdf-formatted electronic textbooks, research papers, and journal articles. - The list goes on.
Examples illustrating these features are taken from EC Math 208, Math 212, Mast 232, Mast 235, and the forthcoming Math 616 graduate course in linear algebra. |

Title: | Non-Convex Conservation Laws : Models of Balanced Diffusion and Dispersion |

Speaker: | Dr. Marc Laforest (Visiting Professor from École Polytechnique de Montréal) |

Date: | January 28, 2013 |

Abstract: | Conservation laws are first order models of continuum mechanics that are degenerate in the sense that higher order stabilizing physics, for example diffusion (2nd order) and dispersion (3rd order), are neglected. For convex conservation laws, the first law of thermodynamics renders the models stable by imposing that the entropy must increase. For non-convex conservation laws, like the equations of ideal magnetohydrodynamics or of certain thin film flows, entropy growth is not a sufficient criteria to obtain stability. Typically, physicists look for solutions that are limits of a purely diffusive regularization and in that case, the well-posedness has been established through the combined efforts of several researchers. |

Title: | On Different Approaches to Teaching Prerequisite, Pre-University Mathematics Courses to Mature Students |

Speaker: | Dr. Anna Sierpinska |

Date: | January 20, 2011 |

Abstract: | I will talk about a teaching experiment on absolute value inequalities, where three approaches were tried: "procedural", "theoretical", and "visual". The results of the experiment suggest that showing students two ways of solving this type of problems and not just one improves their performance and theoretical thinking in mathematics. Time permitting, I will reflect on the possible historical roots of the tradition of procedural approaches to teaching mathematics. |

Title: | Polyakov-Alvarez Formula and Weil Reciprocity Law for Polyhedra |

Speaker: | Dr. Alexey Kokotov |

Date: | November 25, 2011 |

Abstract: | Starting with a short introduction to the spectral theory of 2d smooth compact Riemannian manifolds,I will prove the classical result due to Polyakov (1981) and Alvarez (1983) - the comparison formula for spectral determinants of Laplacians. It turns out that there exists an analog of this result for flat singular 2d manifolds (e. g. boundaries of Euclidean polyhedra or, more generally, 2d simplicial complexes). As a simple corollary of this new analog of comparison formula, one gets a reciprocity law for conformally equivalent polyhedra, which could be alternatively derived from the Weil eciprocity law for harmonic functions with logarithmic singularities. |

Title: | Prime Ideals in Commutative Rings and the Work of DeMarco and Orsatti |

Speaker: | Dr. Robert Raphael |

Date: | November 11 , 2011 |

Abstract: | We give a very elementary discussion on prime numbers, prime ideals, the Spectrum, and close with the work of DeMarco and Orsatti. |

Title: | A Curious Tale |

Speaker: | Dr. Chris Cummins |

Date: | February 18, 2011 |

Abstract: | Starting with the connections between the modular groups and finite simple groups, we discuss some torsion-free and congruence subgroups. Further curiosities lead back to our starting point. |

Title: | On Problems Related to Algebraic Connectivity of Graphs |

Speaker: | Dr. Arbind Lal |

Date: | April 1, 2011 |

Abstract: | Let $G$ be a connected graph and $L(G)$ be its Laplacian matrix. The talk will start with the basic results related with $L(G)$. Then a generalization of a result of Fiedler, commonly known as Fiedler's monotonicity theorem will be presented. Some results related with algebraic connectivity of trees and their generalizations to certain graphs will be presented. Some problems related to algebraic connectivity that are still open will also be presented in this talk. |

Title: | Mathematical Problems in Actuarial Science |

Speaker: | Dr. Jose Garrido |

Date: | February 18, 2011 |

Abstract: | Actuarial sciences are multidisciplinary in nature. Ideas from mathematics, probability, statistics, demography, computer science, finance and even health and social sciences form the basis of actuarial models. Despite this intricate relation to many mathematical fields, actuarial sciences are still somewhat of an unknown for many mathematicians. This talk will just brush a very personal survey of some mathematical problems or ideas in actuarial sciences. Depending on time, illustrations with problems in differential equations, stochastic processes, functional analysis, matrix algebra and complex analysis will be given. |

Title: | Riemann Zeta Function and Random Matrix Theory |

Speaker: | Dr. Chantal David |

Date: | February 11, 2011 |

Abstract: | We will explain in this talk the link between the distribution of the zeroes of the Riemann Zeta function and the theory of random matrices. Some applications of the random matrix model to moments and vanishing of L-functions in families will also be given. |

Title: | Number Theory as an Experimental Science |

Speaker: | Dr. Hershy Kisilevsky |

Date: | January 21, 2011 |

Abstract: | I will talk about the important influence of numerical calculation and machine computation in the discovery, formulation and verification of conjectures in number theory. There will be particular emphasis on the conjectures arising from a series of machine computations of values of L-functions done with Jack Fearnley. |

Title: | The Elephant and the Mouse |

Speaker: | Dr. John McKay |

Date: | April 7, 2010 |

Abstract: | I recount a career of nearly fifty years chasing properties of finite groups using computers. The wonders of character tables & the latest news on the conjecture that m_p(G) = m_p(N_G(P)) for all finite groups G. Properties of Coxeter-Dynkin diagrams. Exploring the monster. Ideas on a construction netting all finite simple groups. [This is intended primarily for graduate students. No knowledge of integrable systems, symplectic geometry, algebraic topology, or an evolving universe is assumed.] |

Title: | Universality: What Random Matrices and Orthogonal Polynomials have in Common with Waves |

Speaker: | Dr. Marco Bertola |

Date: | March 11, 2010 |

Abstract: | The talk will try to survey three topics that seem completely unrelated: random matrices, that is, the study of statistical properties of eigenvalues of matrices whose entries are chosen at random (with a certain distribution); Orthogonal polynomials; and Nonlinear (integrable) waves, namely, nonlinear PDEs which admit infinitely many conserved quantities. I will focus on one instance from each group; the “Hermitean'' matrix model on one side and the (focusing) Nonlinear Schro"dinger equation. It turns out that the method to investigate many spectral statistical properties of the first model when the size N of the matrices becomes increasingly large can be --almost verbatim-- exported to the analysis of the “small-dispersion limit” of certain nonlinear waves, where the “dispersion parameter'' plays the analog of the inverse of the size of the matrices. Two examples are the Korteweg-de Vries and the Nonlinear Schro"dinger equations (in one spatial dimension): the link is due to the inverse spectral method introduced decades ago by Zakharov and Shabat, and the much more recent method of the nonlinear Steepest Descent of Deift and Zhou. Thrown into this mix are certain century-old special functions (Painlev e' transcendents) which still hold mysteries and are object of open conjectures. |

Title: | Long-Time Behavior of 2-Dimensional Flows of Ideal Incompressible Fluid |

Speaker: | Dr. Alexander Shnirelman |

Date: | February 11, 2010 |

Abstract: | Consider the motion of ideal incompressible fluid in a bounded 2-d domain. It is described by the Euler equations which, in spite of their deceptive simplicity, are hard to investigate. For the initial velocity field smooth enough, the Euler equations have a unique solution for all time, and it's natural to ask what is its long-time asymptotics. The physical experiments and computer simulations show a nontrivial, counterintuitive picture of a huge attractor in the space of incompressible velocity fields, consisting of stationary, periodic, quasiperiodic and, possibly, chaotic solutions. This picture appears to contradict the conservative nature of the Euler equations; this is similar to contradiction between the microscopical reversibility of the molecular motion and macroscopical irreversibility of thermodynamical processes. I am going to demonstrate the results of computer simulation and physical experiments on the fluid motion, and discuss connections of this problem with analysis, dynamical systems and even topology. |

Title: | Reciprocal Symmetry, Unimodality and Khintchine’s Theorem |

Speaker: | Dr. Yogendra P. Chaubey |

Date: | December 11, 2009 |

Abstract: | The symmetric distributions on real line and their multivariate extensions play a central role in statistical theory and many of its applications. Furthermore, data in practice often consist of nonnegative measurements. In this respect, R-symmetric distributions defined on the positive real line may be considered analogous to symmetric distributions on the real line. Hence it is useful to investigate reciprocal symmetry in general and R-symmetry in articular. In this paper, we shall explore a number of interesting results and interplays involving reciprocal symmetry, unimodality and Khintchine's theorem with emphasis on R-symmetry. |

Title: | Determinant of Laplace Operator as Morse Function |

Speaker: | Dr. Dmitri Korotkin |

Date: | November 27, 2009 |

Abstract: | The idea to use determinant of Laplace operator to study the space of metrics on Riemann surfaces goes back to works of Osgood, Philips and Sarnak written in 1980's. In this talk we give a simple proof of the their main heorem which states that the determinant of Laplacian is maximal within given conformal class on the metric of constant curvature. Our proof makes use of Ricci flow on two-dimensional manifolds. We show also how to use the determinant of Laplacian as Morse function on the moduli space of genus two Riemann surfaces to compute the orbifold Euler characteristic of this space; this characteristic turns out to be equal to -1/120, in agreement with the classical result of Harer and Zagier. |

Title: | “Learning by Example” - The Approach to Teaching Mathematics in College Level Algebra Textbooks. What Algebra? What Mathematics? |

Speaker: | Dr. Nadia Hardy |

Date: | October 16, 2009 |

Abstract: | In this talk, I will go over the didactic and mathematical organization of nine contemporary college level Algebra textbooks (including the one we use for MATH200). These textbooks follow a teaching approach that can be named "learning by example". I assume that the striking similarities in content presentation refer to a widely adopted way of teaching in North America. I will focus on the chapters about factoring and solving quadratic equations. By analyzing the textbooks' discourses, the worked-out examples, the ad-hoc jargon, and the proposed exercises, I will argue that the resulting body of knowledge has little to do with mathematical knowledge. Heuristic activities become the knowledge to be learned and 'doing mathematics' becomes a ritual that defies mathematical rationality. |

Title: | Computers in the Teaching of Mathematics |

Speaker: | Dra. Araceli Reyes [Visiting Scholar from Instituto Technológico Autónomo de México (ITAM)] |

Date: | May 12, 2009 |

Abstract: | In this seminar, I will present the different ways that I have used computers to teach a variety of mathematical subjects. The topics will be drawn from Geometry, Calculus and Linear Algebra. I will also discuss, as deeply as possible, the theoretical educational frameworks that underlie these computer applications. Following the talk, we will have a session in the computer lab so everyone in attendance can experiment with possible applications of the computer software. We will be using Maple and GeoGebra. *This is a Joint Mathematics Education and Exceptional Pizza Seminar. |

Title: | Random Trees, Walks and Point Processes |

Speaker: | Dr. Lea Popovic |

Date: | April 3, 2009 |

Abstract: | Random Trees appear in a variety of applications, for example in recording relationships of randomly evolving populations, or keeping track of executed actions in randomized algorithms. A finite tree can be encoded by a walk around it, or by a point-process on its internal nodes. What is useful about these representations is that for certain types of random trees they turn out to be well-known objects: a random walk and an i.i.d. point-process. When a tree has a large number of nodes and edges it can be approximated by a continuum tree whose walk is related to Brownian motion and whose point-process is Poisson. We will see how these objects help us in presenting trees in an accessible manner. |

Title: | Another Introduction to Tau Functions |

Speaker: | Dr. John Harnad |

Date: | March 6, 2009 |

Abstract: | The notion of tau functions was introduced originally by Hirota and Sato in the context of completely integrable systems, but has proved to be much farther reaching in its applications than was originally conceived. Besides its original use as a generating function for classical integrable, commutative flows, allowing the dynamical equations to be expressed in bilinear form, it has found remarkable applications in a number of other distinct areas of mathematics and physics. These include: 1) Correlation functions for quantum many-body and spin systems (Ising model, Heisenberg ferromagnet, etc.), 2) Partition functions and correlators for random matrices and Coulomb gases, 3) Generating functions and partition functions for certain classes of random processes (asymmetric exclusions process, Schur processes, etc.) and certain random tilings, 4) Generating functions for topological invariants (Gromov-Witten, Donaldson-Thomas, Hurwitz numbers, etc.). It is a central ingredient, in particular, in most of the recent work of Fields medalist Andrey Okounkov and his collaborators. Some of the key tools and building blocks for tau functions are borrowed from group representation theory, geometry and combinatorics (partitions, group characters (Schur functions), sorting algorithms) and some from a very simple version of quantum field theory (free fermionic operators, vertex operators, vacuum expectation values, Wick's theorem). This talk will present a small sample of these applications and will try include a very elementary introduction to the methods involved. It is also related to the topics covered in the Aisenstadt lecture series given by Craig Tracy at the CRM during the same week. |

Title: | The Div-Curl Lemma |

Speaker: | Dr. Galia Dafni |

Date: | February 6, 2009 |

Abstract: | The Div-Curl Lemma, due to F. Murat, is a tool in "compensated compactness", a way of obtaining convergence results for nonlinear quantities in partial differential equations. This talk will explain the lemma and its more recent versions, involving the use of Hardy spaces. |

Title: | The Twin Prime Conjecture for Elliptic Curves |

Speaker: | Dr. Chantal David |

Date: | November 28, 2008 |

Abstract: | A well-known open problem in number theory is that of showing that there exists infinitely many primes p such that p+2 is also a prime. The problem is known as the twin prime conjecture, and was made precise by Hardy and Littlewood in 1933, who predicted an asymptotic for the number of twin primes up to x. One can generalise the twin prime conjecture to distribution of primes represented by general polynomials (the polynomials being n and n+2 for the case of the twin prime conjecture). For example, are there infinitely many primes of the form n^2+1? In 1988, Neil Koblitz formulated another analogue conjecture, for elliptic curves. For each prime p, let N_p(E) be the order of the group of points of E modulo p. Are there infinitely many primes p such that N_p(E) is prime? This has application to cryptography. This conjecture is still an open question, and there are no example of elliptic curves with infinitely many such primes. This talk will explain the twin prime conjecture and the Koblitz conjecture, without assuming any background from the audience. |

Title: | On Commutative Clean Rings |

Speaker: | Dr. Bob Raphael |

Date: |
October 31, 2008 |

Abstract: | Clean rings are defined and elementary examples are given. An embedding theorem is proved, and extensions by idempotents are discussed. Applications to rings of the form C(X) are given. Some of the work is joint with W. Burgess of the University of Ottawa. The level of the talk will be elementary. |

Title: | Why Should We Care About Philosophy of Mathematics? |

Speaker: | Dr. Bill Byers |

Date: | September 26, 2008 |

Abstract: | People who study, teach, and do mathematics don't often take the time to think about what they are doing. In this talk, I suggest that a little more reflection would be a good thing. I'll consider questions like: What does it mean for a result to be deep or trivial? Is there a difference between following an argument and understanding what is really going on? What are the roles of logic and ambiguity in mathematics? |

Title: | Combustion and Asymptotics |

Speaker: | Dr. Iana Anguelova |

Date: | March 30, 2007 |

Abstract: | I will introduce some concepts and problems in the mathematical theory of combustion, for instance what is the difference between flame propagation and detonation. I will then show how different asymptotic methods are (invented and) applied to solving combustion problems. |

Title: | Stochastic Filtering and its Applications |

Speaker: | Dr. Wei Sun |

Date: | March 2, 2007 |

Abstract: | Filtering is concerned with estimating the conditional probability distribution of a signal through a partial and noisy sequence of observations of the signal. Recently, there is an increasing interest in applying filtering theory to many real-world problems. In this talk, I will first demonstrate some applications of filtering. Then I will introduce the three fundamental problems of nonlinear filtering: filtering equations, particle filters and stability of filters. |

Title: | Maps of Interval, Invariant Measures and Some Related Problems |

Date: | January 6, 2007 |

Speaker: | Dr. Pawel Gora |

Abstract: | We will mainly discuss absolutely continuous invariant measures for maps of an interval. Some related problems will also be considered: representations of numbers and their properties. If time permits we will describe connections with the Ising model. |

Title: | Sources of Frustration Among Students in Prerequisite Mathematics Courses |

Speaker: | Dr. Anna Sierpinska |

Date: | December 8, 2006 |

Abstract: | I will be talking about my research on students' frustration in mathematics courses that are required for admission into academic programs such as Psychology, Computer Engineering, Business School, etc. The research was based on a questionnaire sent to students enrolled in MATH 200, 201, 206 and 209 courses in the years 2003 and 2004; 96 students responded. The questionnaire and frequencies of responses are available on the web through a link at http://www.asjdomain.ca/. In the talk, I'll present the theoretical framework underlying the analysis of the data (a concept of frustration and a theory of institutions) and results related to the main sources of students' frustration identified through the study. |

Title: | An Excursion into Affine Geometry |

Speaker: | Dr. Alina Stancu |

Date: | November 17, 2006 |

Abstract: | Affine geometry is devoted to the study of invariant quantities of curves, surfaces or hypersurfaces under the group of volume preserving affine transformations of Rn. Think of them as transformations of Rn which map spheres centered at the origin into ellipsoids of equal volume placed arbitrarily in the space. If we take an n-dimensional convex body, a ball for example, with the metric inherited from the Euclidean space, its volume is an affine invariant, but its surface area is not. However, there exists a notion of affine surface area which is invariant under affine transformations and there exists a famous affine isoperimetric inequality which relates it to the volume. In this talk we will give two simple geometric interpretations of the affine surface area of convex bodies. Along the way, we will touch upon some new characterizations of ellipsoids. |

Title: | Equity-Indexed Annuities and Dynamic Hedging Errors |

Speaker: | Dr. Patrice Gaillardetz |

Date: | October 27, 2006 |

Abstract: | In this talk, I present different features of Equity-Indexed Annuities (EIAs). I explain among other things types of design, law constraints, and advantages of these equity-linked products. Because of their sophisticated designs, pricing EIAs in an incomplete market is complex. Therefore, I also propose a new pricing principle that combines the actuarial as well as the financial approaches. The financial approach underlies a dynamic hedging strategy that is not self-financing. This non self-financing strategy is leading to two different types of errors that are due to the mortality risk. A loaded premium that protects the insurance company against this mortality risk is presented. Numerical examples on EIAs are provided to illustrate the implementation of this approach. |

Title: | Estimation Problems for Censored Data |

Speaker: | Dr. Arush Sen |

Date: | April 28, 2006 |

Abstract: | Censored data poses a major constraint in survival analysis, an area of statistics that deals with 'life-time' data, e.g., time until death for patients with a certain disease, time until infection after exposure etc. Censoring means being able to observe data only partially. Another issue is the possibility of 'cure', i.e., patients not dying or not catching the disease. We shall discuss these issues for two important models of censoring, viz., random censoring (for uni-variate as well as bi-variate data) and interval censoring (Case-1), and methods of dealing with them. *Part of the talk is based on joint works with W.Stute and F.Tan. |

Title: | Random Matrices, Random processes, Integrable Systems |

Speaker: | Dr. John Harnad |

Date: | March 17, 2006 |

Abstract: | The spectral theory of random matrices has appeared and re-appeard in various applications over the past few decades. Aside from well-known applications of multivariate-statistics, it has been of importance in such diverse and interesting physical problems as the statistical theoeyr of hiigh-lying energy levels of large atomic nuclei (Wigner, Byson, 1960's) and attempts at discretixation of the Feyman path integralunderlying 2D-quantum gravity and conformal models (1990's). More recently, connections have also been made with supersymmetric Yang-Mills theory, and also some quite different problems amenable to similar analysis, such as growth problems in random media, random words, random tilings and random permutations, as well as the seemingly unrelated domain of classical and quantum integrable systems. A key step in understanding these relations is to note, first, that there is an immediate connection with the theory of orthogonal polynomials, which dates back to the work of Stieltjes in the 19th century, and second, that an effective way to study the relevant statistics of the eigenvalues is by varying the parameters governing the measure and support of the spectrum. The latter leads directly to the types of deformation equations studied in the theory of integrable systems. |

Title: | Arithmetic of Elliptic Curves and Modular Forms |

Speaker: | Dr. Adrian Iovita |

Date: | January 27, 2006 |

Abstract: | I will discuss a famous conjecture of Mazur-Tate-Teirelbaum relating special values of the p-acidic and complex L-functions of an elliptic curve (respectively modular form) in the presence of a trivial zero. |

Title: | Determinants of Laplace Operators on Riemann Surfaces and Tau-Functions of Riemann-Hilbert Problems |

Speaker: | Dr. Dmitri Korotkin |

Date: | December 2, 2005 |

Abstract: | Determinants of Laplacian on a compact Riemann surface is an important spectral characteristic of both the conformal class of the Riemann surface and the metric. These determinants play an important role in many applications of Riemann surfaces - from string theory to geometry of moduli space of Riemann surfaces. The tau-functions of Riemann-Hilbert problems arise in a completely different context: they correspond to equations of isomonodromy deformations (the classical Schlesinger equations) of a given Riemann-Hilbert problem, and play the central role in solvability of these problems. In our talk, we discuss these objects, and show that the are very closely related to each other. In particular, we find new expressions for determinants of Laplacians on Riemann surfaces in two classes of metrics: the metrics of constant curvatures and flat metrics with conical singularities. |

Title: | Exit Problems for Reflected Levy Processes |

Speaker: | Dr. Xiaowen Zhou |

Date: | October 28, 2005 |

Abstract: | Levy processes are stochastic stationary and independent increments. Some of the most important examples are Brownian motion, the compound Poisson process and the stable process. In this talk we will first give a brief introduction of (spectrally negative) Levy processes and their exit problems. We will then present Bertoin's solutions to the exit problems. The rest of the talk will focus on some recent results on the exit problems for the reflected Levy processes. Connections with risk models will be mentioned. |

Title: | A Mystery of the 2-Dimensional Fluid |

Speaker: | Dr. Alexander Shnirelman |

Date: | October 7, 2005 |

Abstract: | N/A |

Title: | Visualization of Hypperelliptic Solutions to Integrable Equations |

Speaker: | Dr. Christian Klein |

Date: | April 22, 2005 |

Abstract: | Almost periodic solutions to certain integrable equations as to Korteweg-de Vries and the Kadomtsev-Petviashvili equation describing waves on shallow water are given in terms of theta functions associated to certain Riemann surfaces. Corresponding solutions to the Ernst equation describe solutions to the Einstein equations in a stationary axisymmetric vacuum, i.e., the graviational field of stars and galaxies. In the latter case, the underlying Riemann surface depends explicitly on the physical coordinates. The numerical evaluation and the visualization of these solutions thus requires an efficient code of high precision which is achieved by using so-called spectral methods. |

Title: | Ghostly Curve and Partition Functions |

Speaker: | Dr. Marco Bertola |

Date: | April 08, 2005 |

Abstract: | We define the notion of partition function for a certain statistical model of random matrices and show how it is related to a ghostly curve (a.k.a. spectral curve). |