One of the main concerns in social network science is the study of positions and roles. By "position" social scientists usually mean a collection of actors who have similar ties to other actors, while a "role" is a specific pattern of ties among actors or positions. Since the 1970s a lot of research has been done to develop these concepts in a rigorous way. An open question in the field is whether it is possible to perform role and positional analysis simultaneously. In joint work in progress with Mason Porter we explore this question by proposing a framework that relies on the principle of functoriality in category theory. In this talk I will introduce role and positional analysis, present some well-studied examples from social network science, and what new insights this framework might give us.

Eigenvalue-eigenvector pairs of combinatorial graph Laplacians are extensively used in graph theory and network analysis. It is well known that the spectrum of the Laplacian L of a given graph G encodes aspects of the geometry of G  - the multiplicity of the eigenvalue 0 counts the number of connected components while the second smallest eigenvalue (called the Fiedler eigenvalue) quantifies the well-connectedness of G . In network analysis, one uses Laplacian eigenvectors associated with small eigenvalues to perform spectral clustering. In graph signal processing, graph Fourier transforms are defined in terms of an orthonormal eigenbasis of L. Eigenvectors of L also play a central role in graph neural networks.

Motivated by this we study eigenvalue-eigenvector pairs of Laplacians of random graphs and their potential use in TDA. I will present simulation results on what persistent homology barcodes of Bernoulli random graphs G(n, p) look like when we use Laplacian eigenvectors as filter functions. Also, I will discuss the conjectures made from the simulations as well as the challenges that arise when trying to prove them. This is work in progress.
 

Abstract: The calibration problem for local stochastic volatility models leads to two-dimensional stochastic differential equations of McKean-Vlasov type. In these equations, the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. While such equations enjoy frequent application in the financial industry, their mathematical analysis poses a major challenge. I will explain how to prove the strong existence of stationary solutions for these equations, as well as the strong uniqueness in an important special case. Based on joint work with Daniel Lacker and Jiacheng Zhang.  
 

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Consider a network of identical phase oscillators with sinusoidal coupling. How likely are the oscillators to globally synchronize, starting from random initial phases? One expects that dense networks have a strong tendency to synchronize and the basin of attraction for the synchronous state to be the whole phase space. But, how dense is dense enough? In this (hopefully) entertaining Zoom talk, we use techniques from numerical linear algebra and computational Algebraic geometry to derive the densest known networks that do not synchronize and the sparsest networks that do. This is joint work with Steven Strogatz and Mike Stillman.

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The boundary integral equation method is a popular method for solving elliptic PDEs with constant coefficients, and systems of such PDEs, in bounded and unbounded domains. An attraction of the method is that it reduces solution of the PDE in the domain to solution of a boundary integral equation on the boundary of the domain, reducing the dimensionality of the problem. Second kind integral equations, featuring the double-layer potential operator, have a long history in analysis and numerical analysis. They provided, through C. Neumann, the first existence proof to the Laplace Dirichlet problem in 3D, have been an important analysis tool for PDEs through the 20^th century, and are popular computationally because of their excellent conditioning and convergence properties for large classes of domains. A standard numerical method, in particular for boundary integral equations, is the Galerkin method, and the standard convergence analysis starts with a proof that the relevant operator is coercive, or a compact perturbation of a coercive operator, in the relevant function space. A long-standing open problem is whether this property holds for classical second kind boundary integral equations on general non-smooth domains. In this talk we give an overview of the various concepts and methods involved, reformulating the problem as a question about numerical ranges. We solve this open problem through counterexamples, presenting examples of 2D Lipschitz domains and 3D Lipschitz polyhedra for which coercivity does not hold. This is joint work with Prof Euan Spence, Bath.

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