York University

In this talk I will review how the BV formalism is used in quantizing theories with local gauge symmetries within the framework of perturbative algebraic quantum field theory. The latter is a mathematically rigorous approach to QFT that combines the locality idea going back to Haag and Kastler with Epstein-Glaser renormalization. In my talk I will also show how these methods can also lead to the construction of a factorization algebra.

Morse theory has a long history with many spectacular applications in different areas of mathematics. In this talk I will explain an extension of the main theorem of Morse theory that works for a large class of functions on singular spaces. The main example to keep in mind is that of moment maps on varieties, and I will present some applications to the topology of symplectic quotients of singular spaces.
 

Graph sampling with noise is a fundamental problem in graph signal processing (GSP). A popular biased scheme using graph Laplacian regularization (GLR) solves a system of linear equations for its reconstruction. Assuming this GLR-based reconstruction scheme, we propose a fast sampling strategy to maximize the numerical stability of the linear system--i.e., minimize the condition number of the coefficient matrix. Specifically, we maximize the eigenvalue lower bounds of the matrix that are left-ends of Gershgorin discs of the coefficient matrix, without eigen-decomposition. We propose an iterative algorithm to traverse the graph nodes via Breadth First Search (BFS) and align the left-ends of all corresponding Gershgorin discs at lower-bound threshold T using two basic operations: disc shifting and scaling. We then perform binary search to maximize T given a sample budget K. Experiments on real graph data show that the proposed algorithm can effectively promote large eigenvalue lower bounds, and the reconstruction MSE is the same or smaller than existing sampling methods for different budget K at much lower complexity.

In this work we study a stochastic three-dimensional Landau-Lifschitz-Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show existence of weak martingale solutions taking values in a two-dimensional sphere $\mathbb{S}^3$ and discuss certain regularity results. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. This is a joint work with Utpal Manna (Triva

Heckman introduced N operators on the space of polynomials in N variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these N operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. We compute the limits of our operators at N → ∞ . These limits yield a Lax operator for Macdonald symmetric functions. This is a joint work with Evgeny Sklyanin.

We study infinite (random) systems of interacting particles living in a Euclidean space X and possessing internal parameter (spin) in R¹. Such systems are described by Gibbs measures on the space Γ(X,R¹) of marked configurations in X (with marks in R¹). For a class of pair interactions, we show the occurrence of phase transition, i.e. non-uniqueness of the corresponding Gibbs measure, in both 'quenched' and 'annealed' counterparts of the model.