Mon, 03 Feb 2025
14:15
L5

ALC G2-manifolds

Lorenzo Foscolo
(La Sapienza, Rome)
Abstract

ALF gravitational instantons, of which the Taub-NUT and Atiyah-Hitchin metrics are prototypes, are the complete non-compact hyperkähler 4-manifolds with cubic volume growth. Examples have been known since the 1970's, but a complete classification was only given around 10 years ago. In this talk, I will present joint work with Haskins and Nordström where we extend some of these results to complete non-compact 7-manifolds with holonomy G2 and an asymptotic geometry, called ALC (asymptotically locally conical), that generalises to higher dimension the asymptotic geometry of ALF spaces.

Sun, 11 Feb 2024
14:00
L5

TBA

Itay Glazer
(Technion - Israel Institute of Technology)
Fri, 06 Dec 2024
15:00
L5

From single neurons to complex human networks using algebraic topology

Lida Kanari
(École Polytechnique Fédérale de Lausanne (EPFL))

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Abstract

Topological data analysis, and in particular persistent homology, has provided robust results for numerous applications, such as protein structure, cancer detection, and material science. In the field of neuroscience, the applications of TDA are abundant, ranging from the analysis of single cells to the analysis of neuronal networks. The topological representation of branching trees has been successfully used for a variety of classification and clustering problems of neurons and microglia, demonstrating a successful path of applications that go from the space of trees to the space of barcodes. In this talk, I will present some recent results on topological representation of brain cells, with a focus on neurons. I will also describe our solution for solving the inverse TDA problem on neurons: how can we efficiently go from persistence barcodes back to the space of neuronal trees and what can we learn in the process about these spaces. Finally, I will demonstrate how algebraic topology can be used to understand the links between single neurons and networks and start understanding the brain differences between species. The organizational principles that distinguish the human brain from other species have been a long-standing enigma in neuroscience. Human pyramidal cells form highly complex networks, demonstrated by the increased number and simplex dimension compared to mice. This is unexpected because human pyramidal cells are much sparser in the cortex. The number and size of neurons fail to account for this increased network complexity, suggesting that another morphological property is a key determinant of network connectivity. By comparing the topology of dendrites, I will show that human pyramidal cells have much higher perisomatic (basal and oblique) branching density. Therefore greater dendritic complexity, a defining attribute of human L2 and 3 neurons, may provide the human cortex with enhanced computational capacity and cognitive flexibility.

Tue, 03 Dec 2024
14:00
L5

Gecia Bravo-Hermsdorff: What is the variance (and skew, kurtosis, etc) of a network? Graph cumulants for network analysis

Gecia Bravo-Hermsdorff
(University College London)
Abstract

Topically, my goal is to provide a fun and instructive introduction to graph cumulants: a hierarchical set of subgraph statistics that extend the classical cumulants (mean, (co)variance, skew, kurtosis, etc) to relational data.  

Intuitively, graph cumulants quantify the propensity (if positive) or aversion (if negative) for the appearance of any particular subgraph in a larger network.  

Concretely, they are derived from the “bare” subgraph densities via a Möbius inversion over the poset of edge partitions.  

Practically, they offer a systematic way to measure similarity between graph distributions, with a notable increase in statistical power compared to subgraph densities.  

Algebraically, they share the defining properties of cumulants, providing clever shortcuts for certain computations.  

Generally, their definition extends naturally to networks with additional features, such as edge weights, directed edges, and node attributes.  

Finally, I will discuss how this entire procedure of “cumulantification” suggests a promising framework for a motif-centric statistical analysis of general structured data, including temporal and higher-order networks, leaving ample room for exploration. 

Mon, 03 Mar 2025
14:15
L5

Seiberg-Witten equations in all dimensions

Joel Fine
(Université libre de Bruxelles (ULB))
Abstract

I will describe a generalisation of the Seiberg-Witten equations to a Spin-c manifold of any dimension. The equations are for a U(1) connection A and spinor \phi and also an odd-degree differential form b (of inhomogeneous degree). Clifford action of the form is used to perturb the Dirac operator D_A. The first equation says that (D_A+b)(\phi)=0. The second equation involves the Weitzenböck remainder for D_A+b, setting it equal to q(\phi), where q(\phi) is the same quadratic term which appears in the usual Seiberg-Witten equations. This system is elliptic modulo gauge in dimensions congruent to 0,1 or 3 mod 4. In dimensions congruent to 2 mod 4 one needs to take two copies of the system, coupled via b. I will also describe a variant of these equations which make sense on manifolds with a Spin(7) structure. The most important difference with the familiar 3 and 4 dimensional stories is that compactness of the space of solutions is, for now at least, unclear. This is joint work with Partha Ghosh and, in the Spin(7) setting, Ragini Singhal.

Mon, 20 Jan 2025
14:15
L5

Yang-Mills on an ALF-fibration

Jakob Stein
(UNICAMP)
Abstract

In this talk, we will make an explicit link between self-dual Yang-Mills instantons on the Taub-NUT space, and G2-instantons on the BGGG space, by displaying the latter space as a fibration by the former. In doing so, we will discuss analysis on non-compact manifolds, circle symmetries, and a new method of constructing solutions to quadratically singular ODE systems. This talk is based on joint work with Matt Turner: https://arxiv.org/pdf/2409.03886

Mon, 10 Feb 2025
14:15
L5

The Schubert variety of a hyperplane arrangement

Nick Proudfoot
(University of Oregon)
Abstract

I’ll tell you about some of my favorite algebraic varieties, which are beautiful in their own right, and also have some dramatic applications to algebraic combinatorics.  These include the top-heavy conjecture (one of the results for which June Huh was awarded the Fields Medal), as well as non-negativity of Kazhdan—Lusztig polynomials of matroids.

Tue, 05 Nov 2024
14:00
L5

María Reboredo Prado: Webs in the Wind: A Network Exploration of the Polar Vortex

María Reboredo Prado
(Mathematical Institute)
Abstract

All atmospheric phenomena, from daily weather patterns to the global climate system, are invariably influenced by atmospheric flow. Despite its importance, its complex behaviour makes extracting informative features from its dynamics challenging. In this talk, I will present a network-based approach to explore relationships between different flow structures. Using three phenomenon- and model-independent methods, we will investigate coherence patterns, vortical interactions, and Lagrangian coherent structures in an idealised model of the Northern Hemisphere stratospheric polar vortex. I will argue that networks built from fluid data retain essential information about the system's dynamics, allowing us to reveal the underlying interaction patterns straightforwardly and offering a fresh perspective on atmospheric behaviour.

Tue, 19 Nov 2024
14:00
L5

Brennan Klein: Network Comparison and Graph Distances: A Primer and Open Questions

Brennan Klein
(Northeastern University Network Science Institute)
Further Information

Brennan Klein is an associate research scientist at the Network Science Institute at Northeastern University, where he studies complex systems across nature and society using tools from network science and statistics. His research sits in two broad areas: First, he develops methods and theory for constructing, reconstructing, and comparing complex networks based on concepts from information theory and random graphs. Second, he uses an array of interdisciplinary approaches to document—and combat—emergent or systemic disparities across society, especially as they relate to public health and public safety. In addition to his role at Northeastern University, Brennan is the inaugural Data for Justice Fellow at the Institute on Policing, Incarceration, and Public Safety in the Hutchins Center for African and African American Studies at Harvard University. Brennan received a PhD in Network Science from Northeastern University in 2020 and a B.A. in Cognitive Science from Swarthmore College in 2014. Website: brennanklein.com. Contact: @email; @jkbren.bsky.social.

Abstract
Quantifying dissimilarities between pairs of networks is a challenging and, at times, ill-posed problem. Nevertheless, we often need to compare the structural or functional differences between complex systems across a range of disciplines, from biology to sociology. These techniques form a family of graph distance measures, and over the last few decades, the number and sophistication of these techniques have increased drastically. In this talk, I offer a framework for categorizing and benchmarking graph distance measures in general: the within-ensemble graph distance (WEGD), a measure that leverages known properties of random graphs to evaluate the effectiveness of a given distance measure. In doing so, I hope to offer an invitation for the development of new graph distances, which have the potential to be more informative and more efficient than the tools we have today. I close by offering a roadmap for identifying and addressing open problems in the world of graph distance measures, with applications in neuroscience, material design, and infrastructure networks.
Mon, 10 Mar 2025
14:15
L5

A functorial approach to quantization of symplectic singularities

Lewis Topley
(University of Bath)
Abstract

Namikawa has shown that the functor of flat graded Poisson deformations of a conic symplectic singularity is unobstructed and pro-representable. In a subsequent work, Losev showed that the universal Poisson deformation admits, a quantization which enjoys a rather remarkable universal property. In a recent work, we have repackaged the latter theorem as an expression of the representability of a new functor: the functor of quantizations. I will describe how this theorem leads to an easy proof of the existence of a universal equivariant quantizations, and outline a work in progress in which we describe a presentation of a rather complicated quantum Hamiltonian reduction: the finite W-algebra associated to a nilpotent element in a classical Lie algebra. The latter result hinges on new presentations of twisted Yangians.

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