Past Fridays@4

Are you thinking of launching your own start-up or considering joining an early-stage company? Navigating the entrepreneurial landscape can be both exciting and challenging. Join Pete for an interactive exploration of the unwritten rules and hidden insights that can make or break a start-up journey.

Drawing from personal experience, Pete's talk will offer practical wisdom for aspiring founders and team members, revealing the challenges and opportunities of building a new business from the ground up.

Whether you're an aspiring entrepreneur, a potential start-up team member, or simply curious about innovative businesses, you'll gain valuable perspectives on the realities of creating something from scratch.

This isn't a traditional lecture – it will be a lively conversation that invites participants to learn, share, and reflect on the world of start-ups. Come prepared to challenge your assumptions and discover practical insights that aren't found in standard business guides.
 

Speaker: Professor Pete Grindrod

In the fast-paced world of trading, where exabytes of data and advanced mathematical models offer powerful insights, how do you harness these to anticipate market shifts and evolving prices? Numbers alone only tell part of the story. Beneath the surface lies the unpredictable force of human behaviour – the delicate balance of buyers and sellers shaping the market’s course. 

In this talk, we’ll uncover how these forces intertwine, revealing insights that not only harness data but challenge conventional thinking about the future of trading.

Speaker: Chris Horrobin (Head of European and US people development for Optiver)

Speaker bio

Chris Horrobin is Head of European and US people development for Optiver. Chris started his career trading US and German bond options, adding currency and European index options into the mix before moving to focus primarily on index options. Chris spent his first three years in Amsterdam before transferring to Sydney. 

During these years, Chris traded some of the biggest events of his career including Brexit and Trump (first time around) and before moving back to Europe led the positional team in his last year. Chris then moved out of trading and into our training team running our trading education space for four years, owning both the design and execution of our renowned internship and grad programs. 

The Education Team at Optiver is central to the Optiver culture and focus on growth – both of employees and the company. Chris has now extended his remit to cover the professional development of hires throughout the business.

North meets South is a tradition founded by and for early-career researchers. One speaker from the North of the Andrew Wiles Building and one speaker from the South each present an idea from their work in an accessible yet intriguing way. 

North Wing

Speaker: Paul-Hermann Balduf
Title: Statistics of Feynman integral
Abstract: In quantum field theory, one way to compute predictions for physical observables is perturbation theory, which means that the sought-after quantity is expressed as a formal power series in some coupling parameter. The coefficients of the power series are Feynman integrals, which are, in general, very complicated functions of the masses and momenta involved in the physical process. However, there is also a complementary difficulty: A higher orders, millions of distinct Feynman integrals contribute to the same series coefficient.

My talk concerns the statistical properties of Feynman integrals, specifically for phi^4 theory in 4 dimensions. I will demonstrate that the Feynman integrals under consideration follow a fairly regular distribution which is almost unchanged for higher orders in perturbation theory. The value of a given Feynman integral is correlated with many properties of the underlying Feynman graph, which can be used for efficient importance sampling of Feynman integrals. Based on 2305.13506 and 2403.16217.

South Wing

Speaker: Marc Suñé 
Title: Extreme mechanics of thin elastic objects
Abstract: Exceptionally hard --- or soft -- materials, materials that are active and response to different stimuli, elastic objects that undergo large deformations; the advances in the recent decades in robotics, 3D printing and, more broadly, in materials engineering, have created a new world of opportunities to test the (extreme) mechanics of solids. 

In this colloquium I will focus on the elastic instabilities of slender objects. In particular, I will discuss the transverse actuation of a stretched elastic sheet. This problem is a peculiar example of buckling under tension and it has a vast potential scope of applications, from understanding the mechanics of graphene and cell tissues, to the engineering of meta-materials.

The goal of Post-Quantum Cryptography (PQC) is to design cryptosystems which are secure against classical and quantum adversaries. A topic of fundamental research for decades, the status of PQC drastically changed with the NIST PQC standardization process. Recently there have been AI attacks on some of the proposed systems to PQC. In this talk, we will give an overview of the progress of quantum computing and how it will affect the security landscape. 

Group-based cryptography is a relatively new family in post-quantum cryptography, with high potential. I will give a general survey of the status of post-quantum group-based cryptography and present some recent results.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.

CANCELLED DUE TO ILLNESS

The response of the Greenland and Antarctic ice sheets to a changing climate is one of the largest sources of uncertainty in future sea level predictions.  The behaviour of the subglacial environment, where ice meets hard rock or soft sediment, is a key determinant in the flux of ice towards the ocean, and hence the loss of ice over time.  Predicting how ice sheets respond on a range of timescales brings together mathematical models of the elastic and viscous response of the ice, subglacial sediment and water and is a rich playground where the simplified models of the contact between ice, rock and ocean can shed light on very large scale questions.  In this talk we’ll see how these simplified models can make sense of a variety of field and laboratory data in order to understand the dynamical phenomena controlling the transient response of large ice sheets.

There will be free pizza provided for all attendees directly after the event just outside L1, so please do come along!

North Wing
Speaker: Alexandru Pascadi 
Title: Points on modular hyperbolas and sums of Kloosterman sums
Abstract: Given a positive integer c, how many integer points (x, y) with xy = 1 (mod c) can we find in a small box? The dual of this problem concerns bounding certain exponential sums, which show up in methods from the spectral theory of automorphic forms. We'll explore how a simple combinatorial trick of Cilleruelo-Garaev leads to good bounds for these sums; following recent work of the speaker, this ultimately has consequences about multiple problems in analytic number theory (such as counting primes in arithmetic progressions to large moduli, and studying the greatest prime factors of quadratic polynomials).

South Wing
Speaker: Tim LaRock
Title: Encapsulation Structure and Dynamics in Hypergraphs
Abstract: Within the field of Network Science, hypergraphs are a powerful modelling framework used to represent systems where interactions may involve an arbitrary number of agents, rather than exactly two agents at a time as in traditional network models. As part of a recent push to understand the structure of these group interactions, in this talk we will explore the extent to which smaller hyperedges are subsets of larger hyperedges in real-world and synthetic hypergraphs, a property that we call encapsulation. Building on the concept of line graphs, we develop measures to quantify the relations existing between hyperedges of different sizes and, as a byproduct, the compatibility of the data with a simplicial complex representation–whose encapsulation would be maximum. Finally, we will turn to the impact of the observed structural patterns on diffusive dynamics, focusing on a variant of threshold models, called encapsulation dynamics, and demonstrate that non-random patterns can accelerate spreading through the system.

What makes a good talk? This year, graduate students and postdocs will give a series talks on how to give talks! There may even be a small prize for the audience’s favourite.

If you’d like to have a go at informing, entertaining, or just have an axe to grind about a particularly bad talk you had to sit through, we’d love to hear from you (you can email Ric Wade or ask any of the organizers).
 

Speaker: Mattia Magnabosco (Newton Fellow, Maths)
Title: Synthetic Ricci curvature bounds in sub-Riemannian manifolds
Abstract: In Riemannian manifolds, a uniform bound on the Ricci curvature tensor allows to control the volume growth along the geodesic flow. Building upon this observation, Lott, Sturm and Villani introduced a synthetic notion of curvature-dimension bounds in the non-smooth setting of metric measure spaces. This condition, called CD(K,N), is formulated in terms of the optimal transport interpolation of measures and consists in a convexity property of the Rényi entropy functionals along Wasserstein geodesics. The CD(K,N) condition represents a lower Ricci curvature bound by K and an upper bound on the dimension by N, and it is coherent with the smooth setting, as in a Riemannian manifold it is equivalent to a lower bound on the Ricci curvature tensor. However, the same relation between curvature and CD(K,N) condition does not hold for sub-Riemannian (and sub-Finsler) manifolds. 

Speaker: Rebecca Lewis (Florence Nightingale Bicentenary Fellow, Stats)
Title: High-dimensional statistics
Abstract: Due to the increasing ease with which we collect and store information, modern data sets have grown in size. Whilst these datasets have the potential to yield new insights in a variety of areas, extracting useful information from them can be difficult. In this talk, we will discuss these challenges.

From genetic studies and astrophysics simulations to statistical modelling and AI, scientific computing has enabled amazing discoveries and there is no doubt it will continue to do so. However, the corresponding environmental impact is a growing concern in light of the urgency of the climate crisis, so what can we all do about it? Tackling this issue and making it easier for scientists to engage with sustainable computing is what motivated the Green Algorithms project. Through the prism of the GREENER principles for environmentally sustainable science, we will discuss what we learned along the way, how to estimate the impact of our work and what levers scientists and institutions have to make their research more sustainable. We will also debate what hurdles exist and what is still needed moving forward.

Speaker: James Taylor
Title: D-Modules and p-adic Representations

Abstract: The representation theory of finite groups is a beautiful and well-understood subject. However, when one considers more complicated groups things become more interesting, and to classify their representations is often a much harder problem. In this talk, I will introduce the classical theory, the particular groups I am interested in, and explain how one might hope to understand their representations through the use of D-modules - the algebraic incarnation of differential equations.

Speaker: Anthony Webster
Title: An Introduction to Epidemiology and Causal Inference

Abstract: This talk will introduce epidemiology and causal inference from the perspective of a statistician and former theoretical physicist. Despite their studies being underpinned by deep and often complex mathematics, epidemiologists are generally more concerned by seemingly mundane information about the relationships between potential risk factors and disease. Because of this, I will argue that a good epidemiologist with minimal statistical knowledge, will often do better than a highly trained statistician. I will also argue that causal assumptions are a necessary part of epidemiology, should be made more explicitly, and allow a much wider range of causal inferences to be explored. In the process, I will introduce ideas from epidemiology and causal inference such as Mendelian Randomisation and the "do calculus", methodological approaches that will increasingly underpin data-driven population research.  

Neural network architectures play a key role in determining which functions are fit to training data and the resulting generalization properties of learned predictors. For instance, imagine training an overparameterized neural network to interpolate a set of training samples using weight decay; the network architecture will influence which interpolating function is learned. 

In this talk, I will describe new insights into the role of network depth in machine learning using the notion of representation costs – i.e., how much it “costs” for a neural network to represent some function f. Understanding representation costs helps reveal the role of network depth in machine learning. First, we will see that there is a family of functions that can be learned with depth-3 networks when the number of samples is polynomial in the input dimension d, but which cannot be learned with depth-2 networks unless the number of samples is exponential in d. Furthermore, no functions can easily be learned with depth-2 networks while being difficult to learn with depth-3 networks. 

Together, these results mean deeper networks have an unambiguous advantage over shallower networks in terms of sample complexity. Second, I will show that adding linear layers to a ReLU network yields a representation cost that favors functions with latent low-dimension structure, such as single- and multi-index models. Together, these results highlight the role of network depth from a function space perspective and yield new tools for understanding neural network generalization. 

Sarah Livermore has worked in the Civil Service for over 10 years, using the maths skills gained in her physics degrees (MPhys, DPhil) whilst studying at Oxford. In this session she’ll discuss some of the roles available to people with a STEM background in the Civil Service, a ‘day in the life’ of a civil servant, typical career paths and how to apply.

Conferences and networking are important parts of academic life, particularly early in your academic career.  But how do you make the most out of conferences?  And what are the does and don'ts of networking?  Learn about the answers to these questions and more in this panel discussion by postdocs from across the Mathematical Institute.

Oxford University Innovation, the University’s commercialisation team, will explain the support they can give to Maths researchers who want to generate commercial impact from their work and expertise. In addition to an overview of consulting, this talk will explain how mathematical techniques and software can be protected and commercialised.

Join us for a session with Keith Macksoud, Executive Director at global recruitment consultant Options Group in London and who previously has > 20 years’ experience in Prime Brokerage Sales at Morgan Stanley, Citi, and Deutsche Bank.  Keith will discuss the recruitment process for financial institutions, and how to increase your chances of a successful application. 

Keith will detail his finance background in Prime Brokerage and provide students with an exclusive look behind the scenes of executive search and strategic consulting firm Options Group. We will look at what Options Group does, how executive search firms work and the Firm’s 30-year track record of placing individuals at many of the industries’ largest and most successful global investment banks, investment managers and other financial services-related organisations. 

About Options Group

Options Group is a leading global executive search and strategic consulting firm specializing in financial services including capital markets, global markets, alternative investments, hedge funds, and private banking/wealth management.

https://www.optionsgroup.com/

Speaker: Cedric Pilatte 
Title: Convolution of integer sets: a galaxy of (mostly) open problems

Title: Manifold-Free Riemannian Optimization

Abstract: Optimization problems constrained to a smooth manifold can be solved via the framework of Riemannian optimization. To that end, a geometrical description of the constraining manifold, e.g., tangent spaces, retractions, and cost function gradients, is required. In this talk, we present a novel approach that allows performing approximate Riemannian optimization based on a manifold learning technique, in cases where only a noiseless sample set of the cost function and the manifold’s intrinsic dimension are available.

Title: Elliptic curves and modularity

Abstract: The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

Speaker: Ximena Laura Fernandez
Title: Let it Be(tti): Topological Fingerprints for Audio Identification

Abstract: Ever wondered how music recognition apps like Shazam work or why they sometimes fail? Can Algebraic Topology improve current audio identification algorithms? In this talk, I will discuss recent collaborative work with Spotify, where we extract low-dimensional homological features from audio signals for efficient song identification despite continuous obfuscations. Our approach significantly improves accuracy and reliability in matching audio content under topological distortions, including pitch and tempo shifts, compared to Shazam.

Talk based on the work: https://arxiv.org/pdf/2309.03516.pdf
 

Speaker: Brett Kolesnik
Title: Coxeter Tournaments

Abstract: We will present ongoing joint work with three Oxford PhD students: Matthew Buckland (Stats), Rivka Mitchell (Math/Stats) and Tomasz Przybyłowski (Math). We met last year as part of the course SC9 Probability on Graphs and Lattices. Connections with geometry (the permutahedron and generalizations), combinatorics (tournaments and signed graphs), statistics (paired comparisons and sampling) and probability (coupling and rapid mixing) will be discussed.

What opportunities are available outside of academia? What skills beyond strong academic background are companies looking for to be successful in transitioning to industry? Come along and hear from video technology company V-Nova and Dr Anne Wolfes from the Careers Service to get some invaluable advice on careers outside academia.

Speaker: Lasse Grimmelt (North Wing)
Title: Modular forms and the twin prime conjecture

Abstract: Modular forms are one of the most fruitful areas in modern number theory. They play a central part in Wiles proof of Fermat's last theorem and in Langland's far reaching vision. Curiously, some of our best approximations to the twin-prime conjecture are also powered by them. In the existing literature this connection is highly technical and difficult to approach. In work in progress on this types of questions, my coauthor and I found a different perspective based on a quite simple idea. In this way we get an easy explanation and good intuition why such a connection should exists. I will explain this in this talk.

Speaker: Yang Liu (South Wing)
Title: Obtaining Pseudo-inverse Solutions With MINRES

Abstract: The celebrated minimum residual method (MINRES) has seen great success and wide-spread use in solving linear least-squared problems involving Hermitian matrices, with further extensions to complex symmetric settings. Unless the system is consistent whereby the right-hand side vector lies in the range of the matrix, MINRES is not guaranteed to obtain the pseudo-inverse solution. We propose a novel and remarkably simple lifting strategy that seamlessly integrates with the final MINRES iteration, enabling us to obtain the minimum norm solution with negligible additional computational costs. We also study our lifting strategy in a diverse range of settings encompassing Hermitian and complex symmetric systems as well as those with semi-definite preconditioners.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

Job applications involve a lot of work and can be overwhelming. Join us for a workshop and Q+A session focused on breaking down academic applications: we’ll talk about approaching reference letter writers, writing research statements, and discussing what makes a great CV and covering letter.