Virtual

https://arxiv.org/abs/2003.02860

In this talk I will present results from an ongoing joint research  program with Konrad Waldorf. Its main goal is to understand the  relation between gerbes on a manifold M and open-closed smooth field  theories on M. Gerbes can be viewed as categorified line bundles, and  we will see how gerbes with connections on M and their sections give  rise to smooth open-closed field theories on M. If time permits, we  will see that the field theories arising in this way have several characteristic properties, such as invariance under thin homotopies,  and that they carry positive reflection structures. From a physical  perspective, ourconstruction formalises the WZW amplitude as part of  a smooth bordism-type field theory.

Contagion maps are a family of maps that map nodes of a network to points in a high-dimensional space, based on the activations times in a threshold contagion on the network. A point cloud that is the image of such a map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on several different data sets, and compare its performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, when Isomap is prone to noise-induced error. This consolidates contagion maps as a technique for manifold learning. 

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.
 

We present a recipe for constructing filter functions on graphs with parameters that can optimised by gradient descent. This recipe, based on graph Laplacians and spectral wavelet signatures, do not require additional data to be defined on vertices. This allows any graph to be assigned a customised filter function for persistent homology computations and data science applications, such as graph classification. We show experimental evidence that this recipe has desirable properties for optimisation and machine learning pipelines that factors through persistent homology. 

The last fifty years of dynamical systems theory have established that dynamical systems can exhibit extremely complex behavior with respect to both the system variables (chaos theory) and parameters (bifurcation theory). Such complex behavior found in theoretical work must be reconciled with the capabilities of the current technologies available for applications. For example, in the case of modelling biological phenomena, measurements may be of limited precision, parameters are rarely known exactly and nonlinearities often cannot be derived from first principles. 

The contrast between the richness of dynamical systems and the imprecise nature of available modeling tools suggests that we should not take models too seriously. Stating this a bit more formally, it suggests that extracting features which are robust over a range of parameter values is more important than an understanding of the fine structure at some particular parameter.

The goal of this talk is to present a high-level introduction/overview of computational Conley-Morse theory, a rigorous computational approach for understanding the global dynamics of complex systems.  This introduction will wander through dynamical systems theory, algebraic topology, combinatorics and end in game theory.

Vafa-Witten theory is a topologically twisted version of 4d N=4 super Yang-Mills theory. In my talk I will tell how to derive a holomorphic anomaly equation for its partition function on a Kaehler 4-manifold with b_2^+=1 and b_1=0 from the path integral of the effective theory on the Coulomb branch. I will also briefly mention an alternative and somewhat similar computation of the same holomorphic anomaly in the effective 2d theory obtained by compactification of the corresponding 6d (2,0) theory on the 4-manifold.
 

Abstract: Gradient descent is known to converge quickly for convex objective functions, but it can be trapped at local minimums. On the other hand, Langevin dynamic can explore the state space and find global minimums, but in order to give accurate estimates, it needs to run with small discretization step size and weak stochastic force, which in general slows down its convergence. This work shows that these two algorithms can “collaborate” through a simple exchange mechanism, in which they swap their current positions if Langevin dynamic yields a lower objective function. This idea can be seen as the singular limit of the replica-exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighbourhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. This is joint work with Xin Tong at National University of Singapore.