### Threshold for Steiner triple systems

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

## Abstract

We prove that with high probability $\mathbb{G}^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system for $n\equiv 1,3\pmod{6}$, establishing the exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.

This is joint work with Ashwin Sah and Michael Simkin.

### Non-branching in RCD(K,N) Spaces

## Abstract

On a smooth Riemannian manifold, the uniqueness of a geodesic given initial conditions follows from standard ODE theory. This is known to fail in the setting of RCD(K,N) spaces (metric measure spaces satisfying a synthetic notion of Ricci curvature bounded below) through an example of Cheeger-Colding. Strengthening the assumption a little, one may ask if two geodesics which agree for a definite amount of time must continue on the same trajectory. In this talk, I will show that this is true for RCD(K,N) spaces. In doing so, I will generalize a well-known result of Colding-Naber concerning the Hölder continuity of small balls along geodesics to this setting.

14:00

### Braids, Unipotent Representations, and Nonabelian Hodge Theory

## Abstract

A complex plane curve singularity gives rise to two objects: (1) a moduli space that representation theorists call an affine Springer fiber, and (2) a topological link up to isotopy. Roughly a decade ago, Oblomkov–Rasmussen–Shende conjectured a striking identity relating the homology of the affine Springer fiber to the so-called HOMFLYPT homology of the link. In unpublished writing, Shende speculated that it would follow from advances in nonabelian Hodge theory: the study of transcendental diffeomorphisms relating “Hitchin” and “Betti” moduli spaces. We make this dream precise by expressing HOMFLYPT homology in terms of the homology of a “Betti”-type space, which, we conjecture, deformation-retracts onto the affine Springer fiber. In doing so, we recast the whole story in terms of an arbitrary semisimple group. We give evidence for the nonabelian Hodge conjecture at the numerical level, using a mysterious formula that involves rational Cherednik algebras and the degrees of unipotent principal-series representations.

14:00

### Friendly bisections of random graphs

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details. Joint with the Random Matrix Theory Seminar.

## Abstract

We introduce a new method for studying stochastic processes in random graphs controlled by degree information, involving combining enumeration techniques with an abstract second moment argument. We use it to constructively resolve a conjecture of Füredi from 1988: with high probability, the random graph G(n,1/2) admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which n-o(n) vertices have at least as many neighbours in their own part as across. This work is joint with Asaf Ferber, Matthew Kwan, Bhargav Narayanan, and Mehtaab Sawhney.

### What is the role of a neuron?

## Abstract

One of the great challenges of neural networks is to understand how they work. For example: does a neuron encode a meaningful signal on its own? Or is a neuron simply an undistinguished and arbitrary component of a feature vector space? The tension between the neuron doctrine and the population coding hypothesis is one of the classical debates in neuroscience. It is a difficult debate to settle without an ability to monitor every individual neuron in the brain.

Within artificial neural networks we can examine every neuron. Beginning with the simple proposal that an individual neuron might represent one internal concept, we conduct studies relating deep network neurons to human-understandable concepts in a concrete, quantitative way: Which neurons? Which concepts? Are neurons more meaningful than an arbitrary feature basis? Do neurons play a causal role? We examine both simplified settings and state-of-the-art networks in which neurons learn how to represent meaningful objects within the data without explicit supervision.

Following this inquiry in computer vision leads us to insights about the computational structure of practical deep networks that enable several new applications, including semantic manipulation of objects in an image; understanding of the sparse logic of a classifier; and quick, selective editing of generalizable rules within a fully trained generative network. It also presents an unanswered mathematical question: why is such disentanglement so pervasive?

In the talk, we challenge the notion that the internal calculations of a neural network must be hopelessly opaque. Instead, we propose to tear back the curtain and chart a path through the detailed structure of a deep network by which we can begin to understand its logic.

14:00

### What is the role of a neuron?

## Abstract

One of the great challenges of neural networks is to understand how they work. For example: does a neuron encode a meaningful signal on its own? Or is a neuron simply an undistinguished and arbitrary component of a feature vector space? The tension between the neuron doctrine and the population coding hypothesis is one of the classical debates in neuroscience. It is a difficult debate to settle without an ability to monitor every individual neuron in the brain.

Within artificial neural networks we can examine every neuron. Beginning with the simple proposal that an individual neuron might represent one internal concept, we conduct studies relating deep network neurons to human-understandable concepts in a concrete, quantitative way: Which neurons? Which concepts? Are neurons more meaningful than an arbitrary feature basis? Do neurons play a causal role? We examine both simplified settings and state-of-the-art networks in which neurons learn how to represent meaningful objects within the data without explicit supervision.

Following this inquiry in computer vision leads us to insights about the computational structure of practical deep networks that enable several new applications, including semantic manipulation of objects in an image; understanding of the sparse logic of a classifier; and quick, selective editing of generalizable rules within a fully trained generative network. It also presents an unanswered mathematical question: why is such disentanglement so pervasive?

In the talk, we challenge the notion that the internal calculations of a neural network must be hopelessly opaque. Instead, we propose to tear back the curtain and chart a path through the detailed structure of a deep network by which we can begin to understand its logic.

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A link for this talk will be sent to our mailing list a day or two in advance. If you are not on the list and wish to be sent a link, please contact @email.

### Simple motion of stretch-limited elastic strings

A link for this talk will be sent to our mailing list a day or two in advance. If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

## Abstract

Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of tensile force is applied the stretch of the string is maximized (becomes inextensible). In this talk, we discuss a model for these stretch-limited elastic strings, in what way they model elastic behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

### 53 Matrix Factorizations, generalized Cartan, and Random Matrix Theory

This is jointly organised with Computational Mathematics and Applications Seminars.

## Abstract

An insightful exercise might be to ask what is the most important idea in linear algebra. Our first answer would not be eigenvalues or linearity, it would be “matrix factorizations.” We will discuss a blueprint to generate 53 inter-related matrix factorizations (times 2) most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978) . We will discuss the interesting history. One anecdote is that Eugene Wigner (1968) discovered factorizations such as the svd in passing in a way that was buried and only eight authors have referenced that work. Ironically Wigner referenced Sigurður Helgason (1962) but Wigner did not recognize his results in Helgason's book. This work also extends upon and completes open problems posed by Mackey²&Tisseur (2003/2005).

Classical results of Random Matrix Theory concern exact formulas from the Hermite, Laguerre, Jacobi, and Circular distributions. Following an insight from Freeman Dyson (1970), Zirnbauer (1996) and Duenez (2004/5) linked some of these classical ensembles to Cartan's theory of Symmetric Spaces. One troubling fact is that symmetric spaces alone do not cover all of the Jacobi ensembles. We present a completed theory based on the generalized Cartan distribution. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition G=K₁AK₂ plays a crucial role in sampling algorithms and the derivation of the joint probability density of A.

Joint work with Sungwoo Jeong.

### Equal Opportunity Cities (this lecture is open to everyone)

Using data from four continents, we show that diversity of consumption and of diversity of social exposure are perhaps the single most powerful predictor of life outcomes such as increasing neighborhood GDP, increasing individual wealth, and promoting intergenerational mobility, even after controlling for variables such as population density, housing price, and geographic centrality. The effects of diversity in promoting opportunity are causal, and inequality in opportunity stems more from social norms that promote segregation than from physical segregation. Policies to promote more equal opportunities within cities seem practical.

You can register here. Everyone is welcome.