Date
Wednesday October 26, 2022Location
Jeffery Hall, Room 222Wednesday, October 26th, 2022
Time: 1:00 p.m. Place: Jeffery Hall, Room 222
Speaker: Sonja Ruzic
Title: Cones and affine toric varieties.
Abstract: We will continue our discussion of cones and how they relate to affine toric varieties. Time permitting, we will go over some properties of affine toric varieties.
Date
Friday November 4, 2022Location
Jeffery Hall, Room 234Friday, November 4th, 2022
Time: 2:30 p.m. Place: Jeffery Hall, Room 234
Speaker: Huizhen (Janey) Yu (University of Alberta)
Title: Average-Cost Markov Decision Processes with Borel Spaces and Universally Measurable Policies
Abstract: In this talk, I will present results for discrete-time Markov decision processes (MDPs) with infinite state and action spaces, specifically Borel-space MDPs, under the long-run average-cost criterion. While these MDPs have been extensively studied in the cases of discounted and total cost, the average-cost case is harder to analyze and still not fully understood. In formulating my results, I have adopted a general mathematical framework that, unlike most prior work on average cost, does not require continuity of the state transition and one-stage cost functions nor compactness of the admissible action sets for each state. I will begin by introducing Borel-space MDPs and reviewing past results on average-cost optimality, which depend on these assumptions, and then devote the remainder of the talk to two recent optimality results in the more general setting. The first result establishes the average-cost optimality inequality (ACOI) for two classes of MDPs, one with nonnegative one-stage costs and the other with a Lyapunov-type stability property and unbounded one-stage costs. The ACOI is the inequality counterpart of the standard average-cost optimality equation (ACOE) and implies that the optimal average-cost function is constant and that there exist stationary, universally measurable, $\epsilon$-optimal policies. In deriving this result, I use the vanishing discount factor approach and a set of new conditions to handle discontinuous system dynamics and one-stage cost functions, which are motivated by Egoroff's theorem and a relationship between the epi-limits and pointwise limits of sequences of functions. The second result concerns the more general case where the ACOI may not hold, the optimal average cost may depend on the initial state, and stationary $\epsilon$-optimal policies may not exist. Using submartingale arguments, I show that, for a given set of reachability and boundedness conditions, the optimal average-cost function is constant almost everywhere with respect to certain $\sigma$-finite measures. This result provides a characterization of the structure of the optimal average-cost functions for the class of Borel-space multichain MDPs satisfying these conditions.
Huizhen (Janey) Yu is a research associate at the Reinforcement Learning and Artificial Intelligence Group (RLAI) in the Department of Computing Science, University of Alberta. She received the Ph.D. degree from the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology. Her research interests include reinforcement learning and stochastic approximation based computational methods for solving MDPs, as well as theoretical properties of MDPs with general state and action spaces. She has served as an associate editor for several journals in the past and is a recipient of this year's Top Reviewer Award from the journal Operations Research Letters.
Date
Friday October 28, 2022Location
Jeffery Hall, Room 234
Friday, October 28th, 2022
Time: 2:30 p.m. Place: Jeffery Hall, Room 234
Speaker: Michael Albanese (University of Waterloo)
Title: The Yamabe Invariant of Complex Surfaces
Abstract: To any suitable geometric space (closed smooth manifold), one can associate a real number called the Yamabe invariant which arises from considerations in Riemannian geometry. For surfaces, this number is a familiar quantity, but in higher dimensions, it is less understood. However, as we will see, more can be said if one restricts to those spaces which admit a complex structure, e.g. orientable surfaces. This talk is partly based on joint work with Claude LeBrun.
Michael Albanese earned his PhD from Stony Brook University under the supervision of Claude LeBrun. He recently started a postdoc at the University of Waterloo after completing a postdoc at UQAM. His research interests are in complex and Riemannian geometry, as well as some areas of algebraic topology. In particular, non-Kahler surfaces, almost complex structures, and spin geometry.
Date
Tuesday October 25, 2022Location
Jeffery Hall, Room 222Tuesday, October 25th, 2022
Time: 2:00 p.m. Place: Jeffery Hall, Room 222
Speaker: Matt Litman (UC Davis)
Title: Error Approximation for Backwards and Simple Continued Fractions.
Abstract: In this talk, we provide a new framework for studying continued fractions utilizing the backwards continued fraction (BCF). We show an approximation theory for BCFs, the correspondence between continued fractions and their backwards continued fractions counterpart, and illustrate a rich approximation theory for continued fractions (CFs) utilizing the methods of the approximation theory for the backwards continued fractions. In particular, we construct explicit functions that bound the BCF or CF error over any BCF or CF cylinder set, and along the way work out the details to pass seamlessly between the BCF and CF expansion of any real number. This is joint work with Cameron Bjorklund.
Date
Friday October 28, 2022Location
Jeffery Hall, Room 422Friday, October 28th, 2022
Time: 10:00 a.m. Place: Jeffery Hall, Room 422
Speaker: Matt Litman (UC Davis)
Title: Markoff-type K3 Surfaces: Local and Global Finite Orbits
Abstract: Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran.
Date
Monday November 7, 2022Location
TBAMonday, November 7th, 2022
Time: 4:30 p.m. Place: TBA
Speaker: Taylor Brysiewicz (University of Western Ontario)
Title: Trace Tests in Numerical Algebraic Geometry
Abstract:
Website details here: https://mast.queensu.ca/~georep/Fall%20'22.html
Date
Monday October 31, 2022Location
TBAMonday, October 31st, 2022
Time: 4:30 p.m. Place: TBA
Speaker: Nasrin Altafi (Queen's University)
Title: Lefschetz properties of families of graded Artinian Gorenstein algebras
Abstract:
Website details here: https://mast.queensu.ca/~georep/Fall%20'22.html
Date
Wednesday October 19, 2022Location
Jeffery Hall, Room 222Wednesday, October 19th, 2022
Time: 1:00 p.m. Place: Jeffery Hall, Room 222
Speaker: Alexandre (Sasha) Zotine
Title: Polyhedral Cones II
Abstract: We will continue our discussion of polyhedral cones and connect them to affine toric varieties.
Date
Tuesday October 18, 2022Location
Jeffery Hall, Room 422Tuesday, October 18th, 2022
Time: 2:00 p.m. Place: Jeffery Hall, Room 422
Speaker: Seoyoung Kim (Georg-August Universität Göttingen, Germany)
Title: Diophantine m-tuples and the clique number of Paley graphs.
Abstract: The clique number of Paley graphs has been studied actively in graph theory. In this talk, we are going to discuss a graph-theoretic approach to understand Diophantine m-tuples, in particular, by using bounds on the clique number of Paley graphs.
Date
Friday October 21, 2022Location
Jeffery Hall, Room 234
Friday, October 21st, 2022
Time: 2:30 p.m. Place: Jeffery Hall, Room 234
Speaker: Kasia Jankiewicz (University of California Santa Cruz)
Title: Questions about Artin groups
Abstract: Artin groups are a family of infinite discrete groups that includes free groups, free abelian groups, and braid groups. They are closely related to Coxeter groups. They are defined by simple-looking presentations, but their geometry is mysterious, and many basic questions about them remain open. I will discuss some of my contributions to understanding Artin groups.
Kasia is an Assistant Professor at the University of California Santa Cruz. Previously she was an L.E. Dickson Instructor at the University of Chicago, and a Research Member at the MSRI program Random and Arithmetic Structures in Topology. She completed her PhD at McGill University in 2018, under the supervision of Piotr Przytycki and Daniel Wise. She holds an NSF grant, has held AWM-NSF and AMS-Simons Travel grants, and will be visiting the CRM in the winter as a CRM-Simons Scholar in Residence. Broadly speaking, Kasia's research is in geometric group theory. She is particularly interested in non-positive curvature, cube complexes, Artin groups, Coxeter groups, random groups, small cancellation theories, separability properties, algebraic bering and coherence.
