Colloquiums and Special Lectures

If an unoriented graph on $n$ vertices is $d$-regular, the largest eigenvalue of its adjacency matrix is $d$. Considering a nearest neighbor random walk on this graph, an important problem is the speed of convergence to the uniform measure. This is closely related to the second largest (singular) blue of the adjacency matrix. It is known that it can’t be of order substantially less than $2\sqrt{d-1}$. Graphs achieving this rate are called Ramanujan. Ramanujan graphs are in a sense optimal expander graphs. Finding such graphs is a hard problem involving number theory, with a first series of explicit examples given by Lubotzky, Phillips and Sarnak.
Graphs being $\varepsilon$-close to this $2\sqrt{d-1}$ rate are called epsilon-Ramanujan. It was proved, among others by Friedman, that for $n$ large enough, a uniformly chosen random $d$-regular graph is epsilon-Ramanujan with high probability. There exists predictions of the same flavor for the second eigenvalue of more general models of random graphs (e.g. random coverings of graphs), in particular a very general conjecture known as Alon’s generalized second eigenvalue conjecture.
In this talk, we present a solution to this long standing open problem, and generalizations. The techniques involve tools from operator algebras (liberalization tricks and norm estimates for operator valued sums of free elements) as well as the construction of a non-commutative version of non-backtracking operator theory. Time allowing, we will also describe new applications to random matrices, random tensors and operator algebras. This is based on past and ongoing collaboration with Charles Bordenave.

The National Security Agency (NSA) in August 2015 announced plans to transition to post-quantum algorithms. Currently, Suite B cryptographic algorithms are specified by the National Institute of Standards and Technology (NIST) and are used by NSA?s Information Assurance Directorate in solutions approved for protecting classified and unclassified National Security Systems (NSS). Below, we announce preliminary plans for transitioning to quantum resistant algorithms.
Shortly after the National Institute of Standardization and Technology (NIST) announced a call to select standards for post-quantum public-key cryptosystems.
The academic and industrial communities have suggested as the quantum-resistant primitives: Lattice-based, Multivariate, Code-based, Hash-based, Isogeny-based and group-based primitives. In this talk I will focus on some ideas of (semi)group-based primitives. The one which was proposed to NIST is by SecureRF company based in Connecticut, among its founders there is a number theorist (Goldfeld) and two group theorists (Anshel and Anshel). They proposed a digital signature using a hard algorithmic problem in braid groups, namely conjugacy search problem.
I will then give a survey of some other suggested group-based cryptosystems that could be claimed as post-quantum cryptosystems, including my own recent work on this topic. I will also report on a recent joint work with Faugere, Kashefi, Kaplan, Perret, Horan on "Fast Quantum Algorithm for Solving Multivariate Quadratic Equations".

Chow motives were introduced by Alexander Grothendieck in the 60s, and they have since become a fundamental tool for investigating the structure of algebraic varieties. Moreover, the motives became one of the main languages in the algebraic geometry to formulate and to solve its problems. Computing Chow motives has also proved to be valuable for addressing questions on other topics. In my talk I will explain the concept of motives and present some applications of them to classical problems in Algebra. In particular, using motives I will answer a question of Jean-Pierre Serre, under what conditions one can embed certain finite groups in the algebraic group of type E8.

In this talk we will discuss the structure of spaces of commuting elements in a compact Lie group. Their connected components and other basic topological properties will be discussed. We will also explain how they can be assembled to produce a space which classifies certain bundles and represents an interesting cohomology theory. A number of explicit examples will be provided for orthogonal, unitary and projective unitary groups. Bio

TBA

Abstract Survey samplers have long used probability samples from multiple sources in conjunction with census and administrative data to make valid and efficient inferences on population parameters. This topic has received a lot of attention more recently in the context of data from non-probability samples such as web surveys and social media. In this talk, I will discuss some methods, based on models for the non-probability samples, which could lead to useful inferences when combined with probability samples. I will also explain how big data may be used as predictors in small area estimation.
J. N. K. Rao is a Distinguished Research Professor at Carleton University. He is also a consultant and member of the methodology advisory board at Statistics Canada. He received the first medal for outstanding contributions to small area estimation at the International Conference on Small Area Estimation, Paris 2017. He published the second edition of his highly cited Wiley book on small area estimation in 2015 jointly with Professor Isabel Molina of Carlos III University, Madrid, Spain.

Fukaya categories were introduced to algebraically study all Lagrangian submanifolds of a given symplectic manifolds. The rich algebraic structure underlying such categories allows sometime to reduce its study to a finite set of objects called generators of the category. Such finite generation allows to prove in some cases rigidity results on the topology of Lagrangian submanifolds. In this talk I will review the basic concepts of symplectic topology. Then I will give an overview of the construction the wrapped Fukaya category. Finally I will explain how in a joint work with Dimitroglou-Rizell, Ghiggini and Golovko we proved finite generation of the wrapped Fukaya category of Weinstein manifolds.

Abstract
For more information please contact J.N.K. Rao.

In this talk, we will present old and new results showing differences and analogies between symplectic manifolds and complex manifolds. First we will describe a surgery construction due to Luttinger which produces many symplectic manifolds admitting no Kaehler metrics. Then we will discuss Eliashberg's theorem showing that a large class of symplectic domains can be upgraded to complex Stein domains. Finally we will see how to combine this result with Donaldson's approximately holomorphic methods to obtain a Lefschetz-thpe decomposition for arbitrary closed integral symplectic manifolds.
Short Bio