Title: Minimizing Kemeny's constant for partial stochastic matrices with a single specified column
Abstract: Given a finite, discrete-time, time-homogeneous Markov chain on n states with an irreducible transition matrix T, we may compute Kemeny's constant K(T) in terms of the eigenvalues of T. Kemeny's constant on such matrices may be interpreted in terms of the expected number of steps to get from a random initial state to a random destination state, and hence, may be viewed as average travel time on a network when the states of the Markov chain are viewed as vertices on a graph. We similarly define K(T) in terms of eigenvalues of a stochastic matrix T having a single essential class, an extension of irreducible stochastic matrices. Suppose we have a partial stochastic matrix where some entries are specified and the rest are unspecified, how do we choose values for the unspecified entries so that the resulting stochastic matrix has a single essential class and K(T) is minimized? Steve Kirkland solved this question for partial stochastic matrices where the only specified entries are the diagonal entries and where the only specified entries are in a single row. In this talk, we present our results for the case where the only specified entries are in a single column. This is a work in progress with Prof. Kirkland.
Title: Enhancing Mathematics Education through Context-Based, Learner-Centered Approaches
Abstract: In this talk, the context-based, learner-centered approach we introduced in the classroom to teach mathematics will be discussed. By incorporating real-world examples and activities that connected mathematical concepts to practical applications, we aimed to bridge the gap between theoretical knowledge and its real-world use. To measure the success of these methods, we used pre-and post-course student surveys to assess changes in students' understanding, engagement, and attitudes toward mathematics. The pre-course survey gathered insights into students' prior experiences and challenges, while the post-course survey evaluated their perceptions after experiencing the context-based approach. Our findings provide valuable insights into how these methods can enhance student learning, offering a foundation for refining and improving context-based teaching strategies in mathematics education.
Title: Career Talk
Abstract: The talk will focus on Jane sharing her academic career journey from being a PhD student at U of M to becoming an Associate Professor at Ontario Tech University in Oshawa, Ontario.
Title: Website-making & Proposal-writing
Abstract: We invited Prof. Adam Clay to give a presentation about research proposal writing and answer your questions regarding UMGF. Please note that the UMGF application deadline is Friday, February 27.
Our agenda for a website-making workshop will be: Introduction to Google sites, Introduction to GitHub pages, Examples of some websites you can build, Custom domains, search engine optimisation and more advanced tools (if time permits), Discussion of any technical questions you might have
Title: Matrix Analysis of the Ultraspherical Spectral Method
Abstract: Spectral methods are prized in scientific computing for their exponential convergence rates, yet classical collocation approaches suffer from severe algebraic bottle necks: they generate dense differentiation matrices with condition numbers scaling as O(N4). This research project investigates the Ultraspherical Spectral Method, a structured approach that overcomes these limitations by utilizing a hierarchy of orthogonal polynomial bases to represent differentiation as a sparse, banded operation. We perform a rigorous matrix analysis of the infinite-dimensional operators arising from this discretization, specifically constructing the banded differentiation (D(2)), multiplication (X), and basis conversion (R(2)) matrices. A key focus is the solution of ”almost-banded” linear systems structures arising from the imposition of boundary conditions on banded operators. We implement a stable QR factorization algorithm utilizing Givens rotations to solve these systems while minimizing fill-in. Numerical experiments performed in Julia validate the theoretical claims on two canonical problems: the Airy boundary value problem and the Harmonic Oscillator eigenvalue problem. Our benchmarks confirm that the ultraspherical solver achieves O(N) computational complexity, a significant improvement over the O(N3) scaling of dense spectral methods, while maintaining bounded or slowly growing condition numbers. This work demonstrates that exploiting structured linear algebra is essential for unlocking the full potential of spectral discretizations in modern differential equation solvers.
Title: Fast algorithms for Sobolev orthogonal polynomials
Abstract: Sobolev orthogonal polynomials are those polynomials orthogonal with respect to an inner product including derivatives. They have a beautiful and rich theoretical history. We propose fast algorithms for Sobolev orthogonal polynomials by careful consideration of the properties of the Sobolev–Gram matrix. We begin with a description of a matrix equation for the Sobolev–Gram matrix with a number of terms proportional to the order of the Sobolev inner product. We describe conditions on the vectorial measure that cause the Sobolev–Gram matrix to be banded, leading to linear complexity Cholesky factorization. Next, we convert the problem of including Dirac measures into that of a finite-rank perturbation of a known Cholesky factorization, which can also be performed in linear complexity. Finally, in the case of conversion to Chebyshev polynomials, we harness the power of randomized numerical linear algebra to solve the connection problem in O(n log^{O(1)}n) flops.