Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Probability Seminar - Daniel Perales Anaya (Waterloo)

Tuesday, March 3rd, 2020

Time: 1:30-2:30 p.m.  Place: Jeffery Hall 422

Speaker: Daniel Perales Anaya (Waterloo)

Title: Relations between infinitesimal non-commutative cumulants.

Abstract:  Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory (e.g, using Boolean cumulants to study free infinite divisibility via the Boolean Bercovici-Pata bijection). On the other hand, we have the concept of infinitesimally free cumulants (and the analogue concept for Boolean and monotone under the name of cumulants for differential independence).

In this talk we first are going to discuss the basic properties of non-commutative infinitesimal cumulants. Then we will use the Grassmann algebra to show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Later we will review how the relations between the various types of cumulants are captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. Finally, we will observe how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space, and we will present some interesting formulas in this setting. This is a joint work with Adrian Celestino and Kurusch Ebrahimi-Fard.

Free Probability and Random Matrices Seminar Webpage:

Topological Data Analysis - Catherine Pfaff (Queen's University)

Monday, March 2nd, 2020

Time: 2:30-4:00 p.m. Place: Goodes Hall 120

Speaker: Catherine Pfaff (Queen's University)

Topics:  Past Presentations & Directions Forward.
We've had a series of recent presentations on applications that have led to really great discussions (often cut short by time). This week, I'll lead a recollection of our thoughts on these applications (& directions forward), record them, & continue discussion on further directions forward (possibly to be explored by undergrad & grad students over the summer & next year).

All are welcome!

Number Theory Seminar - Didi Zhang

Monday, March 2nd, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Didi Zhang

Title: Induced subgraphs of hypercubes and a proof of the sensitivity conjecture.

Abstract: In this talk, we follow Hao Huang’s 2019 paper and show that every $(2^{n−1} + 1)$-vertex induced subgraph of the $n$-dimensional cube graph has maximum degree at least $\sqrt{n}$. This is the best possible result, and it improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.

Department Colloquium - Anup Dixit (Queen's University)

Anup Dixit (Queen's University)

Friday, February 28th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Anup Dixit (Queen's University)

Title: The generalized Brauer-Siegel conjecture.

Abstract: One of the principal objects of study in number theory are number fields $K$, which are finite field extensions of $\mathbb{Q}$. The ring of integers of $K$ is analogous to integers $\mathbb{Z}$ in $\mathbb{Q}$. A natural question to investigate is if the ring of integers of $K$ is a unique factorization domain. The answer lies in the study of the invariant class number of $K$, which captures how far the ring of integers of $K$ is from having unique factorization. The origins of this problem can be traced back to Gauss, who conjectured that there are finitely many imaginary quadratic fields with this property. This was proved in the mid-twentieth century, independently by Baker, Heegner and Stark. A more intricate question is to understand how class number varies on varying number fields. In this context, the generalized Brauer-Siegel conjecture, formulated by M. Tsfasman and S. Vl\u{a}du\c{t} in 2002, predicts the behavior of class number times the regulator over certain families of number fields. In this talk, we will discuss recent progress towards this conjecture, in particular, establishing it in special cases.

Anup Dixit is a Coleman Postdoctoral Fellow at Queen's University under the supervision of M. Ram Murty. He obtained his Ph.D. in Mathematics from the University of Toronto in 2018. He is interested in analytic as well as algebraic number theory. He has worked on families of L-functions, behaviour of the class number on varying number fields, infinite extensions of number fields, universality of functions and Euler-Kronecker constants.

Dynamics, Geometry, & Groups - Abdullah Zafar (U of T)

Friday, February 28th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Abdullah Zafar (University of Toronto)

Title: Flows, Braids and Fractal Dynamics: Understanding Patterns of Play in Professional Football.

Abstract: Team movement analysis in football is critical in the preparation and evaluation of performance. Previous studies have analyzed team performance using reductionist approaches such as team centroid analysis with limited consideration of individual player movement and physical performance. In this talk we will look at how we can model the movement of a team as a flow field, derive metrics to quantify the team tempo, and then demonstrate the utility and importance of tempo to the physical training of players as well as team performance as a whole. We will then explore the tempo time series in more depth to better understand the dynamics of a football match: discovering the underlying Fractional Brownian Motion, the changing multi-fractal spectrum, and the difference between attractors around goal-scoring moments. We will then supplement the modelling of team movement with the use of algebraic braids by considering the space-time trajectories of the players. The prominence of different braid groups within the team and during different moments of the match demonstrate the varying movement strategies used by teams, and the computation of braid entropy is explored in order to determine the intra-team structure and coherence in team movement.

Math Club - Jamie Mingo (Queen's University)

Thursday, February 27th, 2020

Time:  5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker:  Jamie Mingo (Queen's University)

Title:  Random Walks and a Truncation of Pascal's Triangle.

Abstract: Pascal's triangle has long been used for counting combinatorial objects like random walks. A random walk means that we move on some kind of grid and at each tick of the clock we take a step in a random direction.

We will see how we can use a truncation of Pascal's triangle to count walks that return their starting point on some high dimensional trees. We shall also see how this can be done in fractional dimensions.

Statistics & Biostatistics - Xiang Li (Queen's University)

Wednesday, February 26th, 2020

Time: 11:30-12:20 Place: Jeffery Hall 225

Speaker: Prof. Xiang Li (Queen's University, Dept. of Chemical Engineering)

Title: Decomposition Based Global Optimization

Abstract: Large-scale nonconvex optimization arises from a variety of scientific and engineering problems. Often such optimization problem is simplified into an easier convex or mixed-integer convex optimization problem, but the solution of the simplified problem is unlikely to be optimal or feasible for the original problem. Recent advances in decomposition based global optimization provides a promising way to solve large-scale nonconvex optimization problems within reason time. In this presentation, we will first discuss the principle of generalized Benders decomposition (GBD), including the reformulation into a master problem using strong Lagrangian duality, the construction of upper and lower bounding problems, and the finite convergence property. We also show how GBD can be applied to decompose multi-scenario problems. Then we introduce two variants of GBD. The first variant, called nonconvex generalized Benders decomposition (NGBD), is able to solve a class of nonconvex problems that GBD cannot solve. The second variant, called joint decomposition (JD), enhances GBD/NGBD via the integration of Lagrangian decomposition. Finally, we demonstrate the computational advantages of GBD, NGBD and JD via some engineering problems.

Probability Seminar - Jamie Mingo (Queen's University)

Tuesday, February 25th, 2020

Time: 1:30-2:30 p.m.  Place: Jeffery Hall 422

Speaker: Jamie Mingo (Queen's University)

Title: Free Compression of Bernoulli Random Variables.

Abstract:  If we represent a Bernoulli random variable by a diagonal matrix with ± 1 entries (half 1, half -1) and then randomly rotate it with an orthogonal matrix, we can then cut out the matrix in the upper left hand corner, of arbitrary size. This is the free compression of an Bernoulli random variable (approximately). This compression has the distribution of the sum of several free Bernoulli random variables, including fractionally many. We will relate this to the Kesten-McKay law for random regular graphs.

Free Probability and Random Matrices Seminar Webpage:

Topological Data Analysis - David Riegert & Troy Zeier

Monday, February 24th, 2020

Time: 2:30-4:00 p.m. Place: Goodes Hall 120

Speaker: David Riegert & Troy Zeier (Queen's University)

Title: A View from the Pitcher's Mound: The Statistics of Persistence Landscapes

Topics:  We (re)introduce a topological summary for data called the "persistence landscape" and discuss how this summary can be viewed as a random variable. Next, we provide a brief review of hypothesis testing and demonstrate how to apply standard statistical tests to persistence landscapes from the safety of simulations. Finally, we take these methods for a spin; applying them to Major League Baseball pitchers using a data set with measurements for ~3 million pitches across 2015-18.

All are welcome!