Speaker: Fouad Naderi
Title: Non-commutative Lebesgue decomposition of non-commutative measures
Abstract: A positive non-commutative (NC) measure is a positive linear functional on the free disk operator system generated by a d-tuple of non-commuting isometries. By using NC repreducing kernel Hilbert theory (RKHS), we construct a natural Lebesgue decomposition for any positive NC measure against any other such measure. This approach is new even in its classical form since when reduced to the case d=1, it retrieves the classical Lebesgue decomposition. In fact, we translate everything in 1 dimension into the language of operator theory, operator algebra, and RKHS so that they are free from dimension, and then we generalize the decomposition into the higher dimensions by the language of non-commutative function theory (NCF). I have previously given a seminar on NCF.
Speaker: Tommy Cai
Title: Orderable groups and generalized torsions
Abstract: A strict total order on a set S is a binary relation < on S which is transitive (a<b, b<c implies that a<c) and trichotomous( for all a,b in S, exactly one of a<b, a=b, and b<a holds) .
We say a group is (left) orderable if there is a strict total order < on it, which is invariant under (left) multiplication. For example, the group of real numbers is orderable, with the natural order. As a nonexample, if a group has a torsion -a nontrivial element whose order is finite- it is not (left) orderable. (Hint of proof: if a<e, then a^2<a, then a^3<a^2 and so on.)
A generalized torsion of a group is a nontrivial element $a$ such that a product of conjugates of $a$ is trivial, i.e., there exists g_1, g_2,\dotsc,g_n in G, such that $g_1^{-1}ag_1g_2^{-1}ag_2\dotsm g_n^{-1}ag_n=e$.
Very little things are known about generalized torsions. For instance, if $a$ is a generalized torsion, does it mean that $a^2$ is a generalized torsion? If a group doesn't have a generalized torsion, does it mean that it is left-orderable? We will talk about some known results and some progress about generalized torsion and orderability.
Speaker: Jared Gobin
Title: Sums of Boolean Complementary Pairs
Abstract: Boolean complementary pairs (BCPs) are pairs of polynomials $f(x), g(x) \in \Z_2[x,x^{-1}]$ that satisfy $f(x)f(x^{-1})+g(x)g(x^{-1})=\lambda \in \Z_2$. BCPs were introduced by Craigen to help unveil the mystery of a related object called ternary complementary pairs. Craigen solved the structure of even-weight BCPs and found an algebraic factorization of odd-weight BCPs with his student Woodford. Upon beginning research for my thesis, I realized that this factorization had beautiful algebraic structure and much more application than originally perceived! Despite this, all questions about what these factorizations could tell us about BCPs led to dead ends...except for one. "Which polynomials are of the form f(x)+g(x), where (f(x),g(x)) is a BCP?" turns out to be a completely answerable question with a surprising amount of structure! Join me as I present this result for the first time (it was sadly cut from my thesis due to length problems) and talk about its consequences.
Speaker: Mehadi Hasan
Title: Reducing Health Systems Financial Pressure and Optimizing ICU Bed Availability during an Epidemic: A Binary and Fractional Control Approach
Abstract: The COVID-19 pandemic has revealed the significant financial pressure on health systems and the need for effective strategies to optimize ICU bed availability during an epidemic. We propose a binary and fractional control approach based on an $SICC_HC_IRS$ model to manage ICU bed availability based on first come first served service and predict the need for additional beds during an epidemic. The model accounts for susceptible, infected, critically ill, hospitalized, isolated, and removed individuals, considering resource constraints and shifting patients between hospitals and isolation camps. We apply the proposed approach to the COVID-19 epidemic in Ontario, Canada, in May 2021, a challenging time with high case numbers, hospitalizations, and strict public health measures. Our analysis investigates the impact of various parameters on the critically ill population, identifying those with the most positive and negative effects. We find that $\beta$, the contact rate between susceptible and infected individuals, has the most significant positive impact on the critically ill population, while $\gamma$, the recovery rate, has the most substantial negative impact. Sensitivity analysis and response curve analysis reveal that a 10\% increase in $\beta$ results in an 18\% increase in the critical population, with $\beta \in [0.9,20]$ contributing to its growth. Using this information, we create an uncontrolled situation by increasing $\beta$ to 10, which leads to a surge in critically ill patients exceeding the hospital's carrying capacity. The model predicts the need for 11,481 additional beds outside of the hospital to accommodate severe cases. Our binary and fractional control approach offers valuable insights for managing healthcare resources and informing decision-making during an epidemic, contributing to reduced financial pressure on health systems and optimized ICU bed availability.
Speaker: Dinamo Djounvuna
Title: Lie 2-algebras arising from Quasi-Hamiltonian G-spaces
Abstract: The study of Hamiltonian group actions has traditionally been bound to the framework where moment maps lie in the dual of a Lie algebra. This paradigm was expanded with the advent of Quasi-Hamiltonian G-spaces, as introduced by Alekseev, Malkin, and Meinrenken, which consider actions where the moment map targets the group itself. This insightful development opened new avenues in exploring symmetries within differential geometry. Concurrently, 2-plectic manifolds, a generalization of symplectic manifolds characterized by nondegenerate 3-forms, began to draw attention for their rich structural properties and potential for higher-dimensional generalization of classical mechanics.
At first glance, Quasi-Hamiltonian G-spaces and 2-plectic manifolds may seem to reside in disparate areas of mathematical study. However, a deeper investigation reveals a shared feature: both are equipped with a closed 3-form that facilitates the construction of Lie 2-algebras. Specifically, the G-equivariant map and the 2-form accompanying a Quasi-Hamiltonian G-space can be used to define a 3-form in the relative sense, leading to a relative 2-plectic manifold.
In this talk, we will follow the path laid out by Christopher Rogers in his construction of Lie 2-algebras and adapt it to the setting of Quasi-Hamiltonian G-spaces. We aim to elucidate how the quasi-Hamiltonian structure, through its inherent 2-form and G-equivariant map, naturally leads to a Lie 2-algebra.