Past

The construction of the measure of the $\Phi^4_3$ model in the 1970s has been one of the major achievements of constructive quantum field theory. In the 1980s Parisi and Wu suggested an alternative way of constructing quantum field theory measures by viewing them as invariant measures of certain stochastic PDEs. However, the highly singular nature of these equations prevented their application in rigorous constructions until the breakthroughs in the area of singular stochastic PDEs in the past decade. After explaining the basic idea behind stochastic quantization proposed by Parisi and Wu I will show how to apply this technique to construct the measure of a certain quantum field theory model generalizing the $\Phi^4_3$ model called the fractional $\Phi^4$ model. The measure of this model is obtained as a perturbation of the Gaussian measure with covariance given by the inverse of a fractional Laplacian. Since the Gaussian measure is supported in the space of Schwartz distributions and the quartic interaction potential of the model involves pointwise products, to construct the measure it is necessary to solve the so-called renormalization problem. Based on joint work with M. Gubinelli and P. Rinaldi.

A well-known problem in algebraic geometry is to construct smooth projective Calabi--Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi--Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov--Tian--Todorov theorem as well as its variant for pairs of a log Calabi--Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.
 

We consider minimizing the sum of a large number of smooth and possibly non-convex functions, which is the typical problem encountered in the training of deep neural networks on large-size datasets. 

Improving the Controlled Minibatch Algorithm (CMA) scheme proposed by Liuzzi et al. (2022), we propose CMALight, an ease-controlled incremental gradient (IG)-like method. The control of the IG iteration is performed by means of a costless watchdog rule and a derivative-free line search that activates only sporadically to guarantee convergence. The schemes also allow controlling the updating of the learning rate used in the main IG iteration, avoiding the use of preset rules, thus overcoming another tricky aspect in implementing online methods.

Convergence to a stationary point holds under the lonely assumption of Lipschitz continuity of the gradients of the component functions without knowing the Lipschitz constant or imposing any growth assumptions on the norm of the gradients.

We present two sets of computational tests. First, we compare CMALight against state-of-the-art mini-batch algorithms for training standard deep networks on large-size datasets, and deep convolutional neural networks and residual networks on standard image classification tasks on CIFAR10 and CIFAR100. 

Results shows that CMALight easily scales up to problem with order of millions  variables and has an advantage over its state-of-the-art competitors.

Finally, we present computational results on generative tasks, testing CMALight scaling capabilities on image generation with diffusion models (U-Net architecture). CMA Light achieves better test performances and is more efficient than standard SGD with weight decay, thus reducing the computational burden (and the carbon footprint of the training process).

Laura Palagi, @email

Department of Computer, Control and Management Engineering,

Sapienza University of Rome, Italy

Joint work with 

Corrado Coppola, @email

Giampaolo Liuzzi, @email

Lorenzo Ciarpaglini, @email

Since Hawking first discovered that black holes radiate, the evaporation of black holes has been a subject of great interest. In this talk, based on [2411.03447], we review some recent results about the evaporation of charged (Reissner-Nordström) black holes. We consider in particular the difference between neutral and charged particle emission, and explain how this drives the black hole near extremality, as well as how evaporation is then changed in that limit.

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.

A non-nilpotent graph Γ(G) of a finite group G has elements of G as vertices, with x and y joined by an edge iff a subgroup generated by these two elements is non-nilpotent. During the talk we will prove several (often unrelated) properties of this construction; for instance, any simple graph can be found as an induced subgraph of Γ(G) for some (solvable) group G. The talk is based on my article "A few remarks on the theory of non-nilpotent graphs" (May 2023).

Physics beyond relativistic invariance and without Lorentz (or Poincaré) symmetry and the geometry underlying these non-Lorentzian structures have become very fashionable of late. This is primarily due to the discovery of uses of non-Lorentzian structures in various branches of physics, including condensed matter physics, classical and quantum gravity, fluid dynamics, cosmology, etc. In this talk, I will be talking about one such theory - Carrollian theory, where the Carroll group replaces the Poincare group as the symmetry group of interest. Interestingly, any null hypersurface is a Carroll manifold and the Killing vectors on the null manifold generate Carroll algebra. Historically, Carroll group was first obtained from the Poincaré group via a contraction by taking the speed of light going to zero limit as a “degenerate cousin of the Poincaré group”.  I will shed some light on Carrollian fermions, i.e. fermions defined on generic null surfaces. Due to the degenerate nature of the Carroll manifold, there exist two distinct Carroll Clifford algebras and, correspondingly, two different Carroll fermionic theories. I will discuss them in detail. Then, I will show some examples; when the dispersion relation becomes trivial, i.e. energy bands flatten out, there can be a possibility of the emergence of Carroll symmetry. 

Understanding the mechanisms behind the spatial distribution, self-organisation and aggregation of organisms is a central issue in both ecology and cell biology. Since self-organisation at the population level is the cumulative effect of behaviours at the individual level, it requires a mathematical approach to be elucidated.
In nature, every individual, be it a cell or an animal, inspects its territory before moving. The process of acquiring information from the environment is typically non-local, i.e. individuals have the ability to inspect a portion of their territory. In recent years, a growing body of empirical research has shown that non-locality is a key aspect of movement processes, while mathematical models incorporating non-local interactions have received increasing attention for their ability to accurately describe how interactions between individuals and their environment can affect their movement, reproduction rate and well-being. In this talk, I will present a study of a class of advection-diffusion equations that model population movements generated by non-local species interactions. Using a combination of analytical and numerical tools, I will show that these models support a wide variety of spatio-temporal patterns that are able to reproduce segregation, aggregation and time-periodic behaviours commonly observed in real systems. I will also show the existence of parameter regions where multiple stable solutions coexist and hysteresis phenomena.
Overall, I will describe various methods for analysing bifurcations and pattern formation properties of these models, which represent an essential mathematical tool for addressing fundamental questions about the many aggregation phenomena observed in nature.

Understanding the mechanisms behind the spatial distribution, self-organisation and aggregation of organisms is a central issue in both ecology and cell biology. Since self-organisation at the population level is the cumulative effect of behaviours at the individual level, it requires a mathematical approach to be elucidated.
In nature, every individual, be it a cell or an animal, inspects its territory before moving. The process of acquiring information from the environment is typically non-local, i.e. individuals have the ability to inspect a portion of their territory. In recent years, a growing body of empirical research has shown that non-locality is a key aspect of movement processes, while mathematical models incorporating non-local interactions have received increasing attention for their ability to accurately describe how interactions between individuals and their environment can affect their movement, reproduction rate and well-being. In this talk, I will present a study of a class of advection-diffusion equations that model population movements generated by non-local species interactions. Using a combination of analytical and numerical tools, I will show that these models support a wide variety of spatio-temporal patterns that are able to reproduce segregation, aggregation and time-periodic behaviours commonly observed in real systems. I will also show the existence of parameter regions where multiple stable solutions coexist and hysteresis phenomena.
Overall, I will describe various methods for analysing bifurcations and pattern formation properties of these models, which represent an essential mathematical tool for addressing fundamental questions about the many aggregation phenomena observed in nature.
 

It is well known that the theory of differential-difference fields in characteristic zero has a model companion. Here by a differential-difference field I mean a field with a differential and a difference structure where the operators commute (in other words the difference structure is a differential-endomorphism). The theory DCFA_0 was studied in a series of papers by Bustamante. In this talk I will address the case of positive characteristic.

I will report on work with Andrew Graham in which we prove new results towards the Bloch--Kato conjecture for automorphic forms on $\mathrm{GSp}_4 \times \mathrm{GL}_2$.

Kinetic and mean-field equations are central to understanding complex systems across fields such as classical physics, engineering, and the socio-economic sciences. Efficiently solving these equations remains a significant challenge due to their high dimensionality and the need to preserve key structural properties of the models. 

In this talk, we will focus on recent advancements in deterministic numerical methods, which provide an alternative to particle-based approaches (such as Monte Carlo or particle-in-cell methods) by avoiding stochastic fluctuations and offering higher accuracy. We will discuss strategies for designing these methods to reduce computational complexity while preserving fundamental physical properties and maintaining efficiency in stiff regimes. 
Special attention will be given to the role of these methods in addressing multi-scale problems in rarefied gas dynamics and plasma physics. Time permitting, we will also touch on emerging techniques for uncertainty quantification in these systems.

Anomalies characterize the breaking of a classical symmetry at the quantum level. They play an important role in quantum field theories, and constitute robust observables which appear in various contexts from phenomenological particle physics to black hole microstates, or to classify phases of matter. The anomalies of a d-dimensional QFT are naturally encoded via descent equations into the so-called anomaly polynomial in (d+2)-dimensions. The aim of this seminar is to review the descent procedure, anomaly polynomial, anomaly inflow, and in particular their realisation in M-theory. While this is quite an old story, there has been some more recent developments involving holography that I'll describe if time permits. 

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

The Martinet flat structure is one of the simplest sub-Riemannian manifolds that has many non-Riemannian features: it is not equiregular, it has abnormal geodesics, and the Carnot-Carathéodory sphere is not sub-analytic. I will review how the geometry of the Martinet flat structure is tied to the equations of the pendulum. Surprisingly, the Measure Contraction Property (a weak synthetic formulation of Ricci curvature bounds in non-smooth spaces) fails, and we will try to understand why. If time permits, I will also discuss how this can be generalised to some Carnot groups that have abnormal extremals. This is a joint work in progress with Luca Rizzi.

Tensor methods are methods for unconstrained continuous optimization that can incorporate derivative information of up to order p > 2 by computing a step based on the pth-order Taylor expansion at each iteration. The most important among them are regularization-based tensor methods which have been shown to have optimal worst-case iteration complexity of finding an approximate minimizer. Moreover, as one might expect, this worst-case complexity improves as p increases, highlighting the potential advantage of tensor methods. Still, the global complexity results only guarantee pessimistic sublinear rates, so it is natural to ask how local rates depend on the order of the Taylor expansion p. In the case of strongly convex functions and a fixed regularization parameter, the answer is given in a paper by Doikov and Nesterov from 2022: we get pth-order local convergence of function values and gradient norms. 
The value of the regularization parameter in their analysis depends on the Lipschitz constant of the pth derivative. Since this constant is not usually known in advance, adaptive regularization methods are more practical. We extend the local convergence results to locally strongly convex functions and fully adaptive methods. 
We discuss how for p > 2 it becomes crucial to select the "right" minimizer of the regularized local model in each iteration to ensure all iterations are eventually successful. Counterexamples show that in particular the global minimizer of the subproblem is not suitable in general. If the right minimizer is used, the pth-order local convergence is preserved, otherwise the rate stays superlinear but with an exponent arbitrarily close to one depending on the algorithm parameters.

Tension-induced giant actuation in elastic sheets

Dr. Marc Suñé

Buckling is normally associated with a compressive load applied to a slender structure; from railway tracks in extreme heat to microtubules in cytoplasm, axial compression is relieved by out-of-plane buckling. However, recent studies have demonstrated that tension applied to structured thin sheets leads to transverse motion that may be harnessed for novel applications, such as kirigami grippers, multi-stable `groovy-sheets', and elastic ribbed sheets that close into tubes. Qualitatively similar behaviour has also been observed in simulations of thermalized graphene sheets, where clamping along one edge leads to tilting in the transverse direction. I will discuss how this counter-intuitive behaviour is, in fact, generic for thin sheets that have a relatively low stretching modulus compared to the bending modulus, which allows `giant actuation' with moderate strain.

Embedding the group algebra into a division ring has proven to be a powerful tool for detecting structural properties of the group, especially in relation to its homology. In this talk, we will show how division rings can be used to identify residual properties of groups, one-ended groups, and coherent groups. We will place special emphasis on the class of free-by-cyclic groups to provide a clear, explicit exposition.

A C*-algebra is said to be residually finite-dimensional (RFD) when it has `sufficiently many' finite-dimensional representations. The RFD property is an important, and still somewhat mysterious notion, with subtle connections to residual finiteness properties of groups. In this talk I will present certain characterisations of the RFD property for C*-algebras of amenable étale groupoids and for C*-algebraic crossed products by amenable actions of discrete groups, extending (and inspired by) earlier results of Bekka, Exel, and Loring. I will also explain the role of the amenability assumption and describe several consequences of our main theorems. Finally, I will discuss some examples, notably these related to semidirect products of groups.

We study networks of firms in which inputs for production are not easily substitutable, as in several real-world supply chains. Building on Robert May's original argument for large ecosystems, we argue that such networks generically become dysfunctional when their size increases, when the heterogeneity between firms becomes too strong, or when substitutability of their production inputs is reduced. At marginal stability and for large heterogeneities, crises can be triggered by small idiosyncratic shocks, which lead to “avalanches” of defaults. This scenario would naturally explain the well-known “small shocks, large business cycles” puzzle, as anticipated long ago by Bak, Chen, Scheinkman, and Woodford. However, an out-of-equilibrium version of the model suggests that other scenarios are possible, in particular that of `turbulent economies’.

I’ll discuss joint work of mine with with Ascencio-Martin, Britnell, Duncan, Francoeurs and Koutny to set up and study algebraic models of concurrent computation. 

Trace monoids were introduced by Mazurkiewicz as algebraic models of Petri nets, where some pairs of actions can be applied in either of two orders and have the same effect. Abstractly, a trace monoid is simply a right-angled Artin monoid. More recently Koutny et al. introduced the concept of a step trace monoid, which allows the additional possibility that a pair of actions may have the same effect performed simultaneously as sequentially. 

I shall introduce these monoids, discuss some problems we’d like to be able to solve, and the methods with which we are trying to solve them. In particular I’ll discuss normal forms for traces, comtraces and step traces, and generalisations of Stallings folding techniques for finitely presented groups and monoids.