Past

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.

This talk is rescheduled and will take place on 21 January 2025

A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some 1-entries with 0-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted $\operatorname{ex}(n,A)$, is the maximum number of 1-entries that an $n\times n$ zero-one matrix can have without containing $A$. The systematic study of this function for various patterns $A$ goes back to the work of Furedi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (that is, the case where $A$ is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if $A$ has at most $t$ 1-entries in every row, then $\operatorname{ex}(n,A)\leq n^{2-1/t+o(1)}$. This verifies a conjecture of Methuku and Tomon.

Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number two, generalizing a celebrated result of Furedi, and Alon, Krivelevich and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree $t$ in one of the vertex classes.

Joint work with Barnabas Janzer, Van Magnan and Abhishek Methuku.

A modern perspective on symmetry in quantum theories identifies the topological invariance of a symmetry operator within correlation functions as its defining property. Within this paradigm, a framework has emerged enabling a calculus of topological defects in terms of a higher-dimensional topological quantum field theory. In this seminar, I will discuss aspects of this construction for Euclidean lattice field theories. Exploiting this framework, I will present generalisations of the celebrated Kramers-Wannier duality of the Ising model, as combinations of gauging procedures and generalised Fourier transforms of the local weights encoding the dynamics. If time permits, I will discuss implications of this framework for the real-space renormalisation group flow of these theories.

The "index of hypocoercivity" is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it quantifies `how degenerate' a hypocoercive evolution equation is, both for ODEs and for evolutions equations in a Hilbert space. We show that this index characterizes the polynomial decay of the propagator norm for short time and illustrate these concepts for the Lorentz kinetic equation on a torus. Discrete time analogues of the above systems (obtained via the mid-point rule) are contractive, but typically not strictly contractive. For this setting we introduce "hypocontractivity" and an "index of hypocontractivity" and discuss their close connection to the continuous time evolution equations.

This talk is based on joint work with F. Achleitner, E. Carlen, E. Nigsch, and V. Mehrmann.

References:
1) F. Achleitner, A. Arnold, E. Carlen, The Hypocoercivity Index for the short time behavior of linear time-invariant ODE systems, J. of Differential Equations (2023).
2) A. Arnold, B. Signorello, Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium, Kinetic and Related Models (2022).
3) F. Achleitner, A. Arnold, V. Mehrmann, E. Nigsch, Hypocoercivity in Hilbert spaces, J. of Functional Analysis (2025).
 

Heegner points are a powerful tool for understanding the structure of the group of rational points on elliptic curves. In this talk, I will describe these points and the ideas surrounding their generalisation to other situations.

June Huh proved in 2012 that the Betti numbers of the complement of a complex hyperplane arrangement form a log concave sequence.  But what if the arrangement has symmetries, and we regard the cohomology as a representation of the symmetry group?  The motivating example is the braid arrangement, where the complement is the configuration space of n points in the plane, and the symmetric group acts by permuting the points.  I will present an equivariant log concavity conjecture, and show that one can use representation stability to prove infinitely many cases of this conjecture for configuration spaces.
 

The talk will focus on recent developments regarding the (near-)critical behaviour of certain statistical physics models with long-range dependence in dimensions larger than 2, but smaller than 6, above which mean-field behaviour is known to set in. This “intermediate” regime remains a great challenge for mathematicians. The models revolve around a certain percolation phase transition that brings into play very natural probabilistic objects, such as random walk traces and the Gaussian free field. 

I will discuss recent and ongoing work with Davesh Maulik that explains how Gromov-Witten invariants behave under simple normal crossings degenerations. The main outcome of the study is that if a projective manifold $X$ undergoes a simple normal crossings degeneration, the Gromov-Witten theory of $X$ is determined, via universal formulas, by the Gromov-Witten theory of the strata of the degeneration. Although the proof proceeds via logarithmic geometry, the statement involves only traditional Gromov-Witten cycles. Indeed, one consequence is a folklore conjecture of Abramovich-Wise, that logarithmic Gromov-Witten theory “does not contain new invariants”. I will also discuss applications of this to a conjecture of Levine and Pandharipande, concerning the relationship between Gromov-Witten theory and the cohomology of the moduli space of curves.

In recent years, a new class of deep neural networks has emerged, which finds its roots at model-based iterative algorithms solving inverse problems. We call these model-based neural networks deep unfolding networks (DUNs). The term is coined due to their formulation: the iterations of optimization algorithms are “unfolded” as layers of neural networks, which solve the inverse problem at hand. Ever since their advent, DUNs have been employed for tackling assorted problems, e.g., compressed sensing (CS), denoising, super-resolution, pansharpening. 

In this talk, we will revisit the application of DUNs on the CS problem, which pertains to reconstructing data from incomplete observations. We will present recent trends regarding the broader family of DUNs for CS and dive into their theory, which mainly revolves around their generalization performance; the latter is important, because it informs us about the behaviour of a neural network on examples it has never been trained on before. 
Particularly, we will focus our interest on overparameterized DUNs, which exhibit remarkable performance in terms of reconstruction and generalization error. As supported by our theoretical and empirical findings, the generalization performance of overparameterized DUNs depends on their structural properties. Our analysis sets a solid mathematical ground for developing more stable, robust, and efficient DUNs, boosting their real-world performance.

I will discuss the preprint arXiv:2409.18130 describing an interesting connection between 4d N=2 SCFTs and 2d chiral CFTs (alias: Vertex Operator Algebras) by way of 3d EFTs.

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the workload, presenting mathematics, structuring the dissertation and creating a narrative, and avoiding plagiarism.

After recalling how Hecke algebras occur in the representation theory of reductive groups, we will introduce affine Hecke algebras through a combinatorial object called a root datum. Through a worked example we will construct a filtration on the affine Hecke algebra from which we obtain the graded Hecke algebra. This has a role analogous to the Lie algebra of an algebraic group.

We will discuss star operations on these rings, with a view towards the classical problem of studying unitary representations of reductive groups.

We consider a system of interacting particles as a model for a foraging ant colony, where each ant is represented as an active Brownian particle. The interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of a pheromone field. Unlike conventional models, our study introduces a parameter that enables the reproduction of two distinctive behaviours: the conventional Keller-Segel aggregation and the formation of travelling clusters without relying on external constraints such as food sources or nests. We consider the associated mean-field limit of this system and establish the analytical and numerical foundations for understanding these particle behaviours.

The monadic second-order theory S1S of (ℕ,<) is decidable (it essentially describes ω-automata). Undecidability of the monadic theory of (ℝ,<) was proven by Shelah. Previously, Rabin proved decidability if the monadic quantifier is restricted to Fσ-sets.
We discuss decidability for Borel sets, or even σ-combinations of analytic sets. Moreover, the Boolean combinations of Fσ-sets form an elementary substructure. Under determinacy hypotheses, the proof extends to larger classes of sets.

A non-local game involves two non-communicating players who cooperatively play to give winning pairs of answers to questions posed by an external referee. Non-local games provide a convenient framework for exhibiting quantum supremacy in accomplishing certain tasks and have become increasingly useful in quantum information theory, mathematics, computer science, and physics in recent years. Within mathematics, non-local games have deep connections with the field of operator algebras, group theory, graph theory, and combinatorics. In this talk, I will provide an introduction to the theory of non-local games and quantum correlation classes and show their connections to different branches of mathematics. We will discuss how entanglement-assisted strategies for non-local games may be interpreted and studied using tools from operator algebras, group theory, and combinatorics. I will then present a general framework of non-local games involving quantum questions and answers.

Given an elliptic curve over the rationals, a natural problem is to find an explicit point of infinite order over a given number field when there is expected to be one. Geometric constructions are known in only two different settings. That of Heegner points, developed since the 1950s, which yields points over abelian extensions of imaginary quadratic fields. And that of Stark-Heegner points, from the late 1990s: here the points constructed are conjectured to be defined over abelian extensions of real quadratic fields. I will describe a new analytic formula which encompasses both of these, and conjecturally yields points in many other settings. This is joint work with Henri Darmon and Victor Rotger.

In this talk we study the numerical approximation of the jump-diffusion McKean--Vlasov SDEs with super-linearly growing drift, diffusion and jump-coefficient. In the first step, we derive the corresponding interacting particle system and define a Milstein-type approximation for this. Making use of the propagation of chaos result and investigating the error of the Milstein-type scheme we provide convergence results for the scheme. In a second step, we discuss potential simplifications of the numerical approximation scheme for the direct approximation of the jump-diffusion McKean--Vlasov SDE. Lastly, we present the results of our numerical simulations.

AI tools like ChatGPT, Microsoft Copilot, GitHub Copilot, Claude and even older AI-enabled tools like Grammarly and MS Word, are becoming an everyday part of our research environment.  This last-minute opening up of a seminar slot due to the unfortunate illness of our intended speaker (who will hopefully re-schedule for next term) gives us an opportunity to discuss what this means for us as researchers; what are good helpful uses of AI, and are there uses of AI which we might view as inappropriate?  Please come ready to participate with examples of things which you have done yourselves with AI tools.

The Parker conjecture, which explores whether magnetic fields in perfectly conducting plasmas can develop tangential discontinuities during magnetic relaxation, remains an open question in astrophysics. Helicity conservation provides a topological barrier against topologically nontrivial initial data relaxing to a trivial solution. Preserving this mechanism is therefore crucial for numerical simulation.  

This paper presents an energy- and helicity-preserving finite element discretization for the magneto-frictional system for investigating the Parker conjecture. The algorithm enjoys a discrete version of the topological mechanism and a discrete Arnold inequality. 
We will also discuss extensions to domains with nontrivial topology.

This is joint work with Prof Patrick Farrell, Dr Kaibo Hu, and Boris Andrews

The McCullough-Miller space is a contractible simplicial complex that admits an action of the pure symmetric (outer) automorphisms of the free group, with stabilizers that are free abelian. This space has been used to derive several cohomological properties of these groups, such as computing their cohomology ring and proving that they are duality groups. In this talk, we will generalize the construction to right-angled Artin groups (RAAGs), and use it to obtain some interesting cohomological results about the pure symmetric (outer) automorphisms of RAAGs.

I will present the basic idea of the flow equation approach developed by Paweł Duch to study singular stochastic partial differential equations. In particular, I will show how it can be used to prove the existence of a solution of the stochastic Burgers equation on the one-dimensional torus.

The Aldous-Lyons conjecture from probability theory states that every (unimodular) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long-standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.

In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.

This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.

No special background in probability theory or complexity theory will be assumed.

A C*-correspondence between two C*-algebras is a generalization of a *-homomorphism. Laca and Neshveyev showed that, like a *-homomorphism, there is an induced map between traces of the algebras. Given sufficient regularity conditions, the map defines a bounded operator between the spaces of (bounded) tracial linear functionals. 

This operator can be of independent interest - a special case of correspondence gives Ruelle's operator associated to a non-invertible discrete-time dynamical system, and the study of Ruelle's operator is the basis of his thermodynamic formalism. Moreover, by the work of Laca and Neshveyev, the operator's positive eigenvectors determine the KMS states of the gauge action on the Cuntz-Pimsner algebra of the correspondence.

Given a C*-correspondence from a C*-algebra to itself, we will present a sufficient condition on the C*-correspondence that implies the operator on traces has a unique positive eigenvector, and moreover a spectral gap. This result recovers the Perron-Frobenius theorem, aspects of Ruelle's thermodynamic formalism, and unique KMS state results for a variety of constructions of Cuntz-Pimsner algebras, including the C*-algebras associated to self-similar groupoids. The talk is based on work in progress.

The celebrated Szemeredi-Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $\mathcal{O}(n^{4/3})$, which is asymptotically tight.

Solymosi conjectured that this bound drops to $o(n^{4/3})$ if we exclude subconfigurations isomorphic to any fixed point-line configuration. We describe a construction disproving this conjecture. On the other hand, we prove new upper bounds on the number of incidences for configurations that avoid certain subconfigurations. Joint work with Martin Balko.

Let G be the points of a reductive group over a p-adic field. According to Bernstein, the category of smooth complex representations of G decomposes as a product of indecomposable subcategories (blocks), each determined by inertial supercuspidal support. Moreover, each of these blocks is equivalent to the category of modules over a Hecke algebra, which is understood in many (most) cases. However, when the coefficients of the representations are now allowed to be in a more general ring (in which p is invertible), much of this fails in general. I will survey some of what is known, and not known.

Come along to hear from several PhD students and PostDocs about their research. There will also be a Q&A about doing a Master's/PhD and a chance to mingle with postgraduate students. 

Speakers include:

• Shaked Bader, DPhil Student in Geometric Group Theory, 
• Eoin Hurley, PostDoc in Combinatorics, 
• Patricia Lamirande, DPhil Student in Mathematical Biology

Several PhD students from the department will give short 5 minute talks on their research. This is also targeted at undergraduates interested in doing PhDs .

Quantum computers are becoming a reality and current generations of machines are already well beyond the 50-qubit frontier. However, hardware imperfections still overwhelm these devices and it is generally believed the fault-tolerant, error-corrected systems will not be within reach in the near term: a single logical qubit needs to be encoded into potentially thousands of physical qubits which is prohibitive.
 
Due to limited resources, in the near term, hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage but these need to resort to quantum error mitigation techniques. I will explain the basic concepts and introduce hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will detail a range of application areas of randomised quantum circuits, such as quantum algorithms, classical shadows, and quantum error mitigation introducing recent results that help lower the barrier for practical quantum advantage.

 

: With motivation from string compactifications, I will present work on the use of machine learning methods for the computation of geometric and topological properties of Calabi-Yau manifolds.

The Seventeenth Brooke Benjamin Lecture 2024

The Elusive Singularity

I will describe the open problems of singularity formation in incompressible fluids. I will discuss a list of related models, some results, and some more open problems.

Date: Monday, 11 November 2024 

Time: 5pm GMT

Location: Lecture Theatre 1, Mathematical Institute 

Speaker: Professor Peter Constantin        

More information about The Brooke Benjamin Lecture.

The Elusive Singularity

I will describe the open problems of singularity formation in incompressible fluids. I will discuss a list of related models, some results, and some more open problems.

Date: Monday, 11 November 2024 

Time: 5pm GMT

Location: Lecture Theatre 1, Mathematical Institute 

Speaker: Professor Peter Constantin            

Peter Constantin is the John von Neumann Professor of Mathematics and Applied and Computational Mathematics at Princeton University. Peter Constantin received his B.A and M.A. summa cum laude from the University of Bucharest, Faculty of Mathematics and Mechanics. He obtained his Ph.D. from The Hebrew University of Jerusalem under the direction of Shmuel Agmon.

Constantin’s work is focused on the analysis of PDE and nonlocal models arising in statistical and nonlinear physics. Constantin worked on scattering for Schr¨odinger operators, on finite dimensional aspects of the dynamics of Navier-Stokes equations, on blow up for models of Euler equations. He introduced active scalars, and, with Jean-Claude Saut, local smoothing for general dispersive PDE. Constantin worked on singularity formation in fluid interfaces, on turbulence shell models, on upper bounds for turbulent transport, on the inviscid limit, on stochastic representation of Navier-Stokes equations, on the Onsager conjecture. He worked on critical nonlocal dissipative equations, on complex fluids, and on ionic diffusion in fluids.

Constantin has advised thirteen graduate students in mathematics, and served in the committee of seven graduate students in physics. He mentored twenty-five postdoctoral associates. 

Constantin served as Chair of the Mathematics Department of the University of Chicago and as the Director of the Program in Applied and Computational Mathematics at Princeton University.

Constantin is a Fellow of the Institute of Physics, a SIAM Fellow, and Inaugural Fellow of the American Mathematical Society, a Fellow of the American Academy of Arts and Sciences and a member of the National Academy of Sciences

In general, the classification of finitely generated subgroups of a given group is intractable. Restricting to two-generator subgroups in a geometric setting is an exception. For example, a two-generator subgroup of a right-angled Artin group is either free or free abelian. Jaco and Shalen proved that a two-generator subgroup of the fundamental group of an orientable atoroidal irreducible 3-manifold is either free, free-abelian, or finite-index. In this talk I will present recent work proving a similar classification theorem for two generator mapping-torus groups of free group endomorphisms: every two generator subgroup is either free or conjugate to a sub-mapping-torus group. As an application we obtain an analog of the Jaco-Shalen result for free-by-cyclic groups with fully irreducible atoroidal monodromy. While the statement is algebraic, the proof technique uses the topology of finite graphs, a la Stallings. This is joint work with Naomi Andrew, Ilya Kapovich, and Stefano Vidussi.
 

I will present a notion of spin structure on a perfect complex in characteristic zero, generalizing the classical notion for an (algebraic) vector bundle. For a complex $E$ on $X$ with an oriented quadratic structure one obtains an associated ${\mathbb Z}/2{\mathbb Z}$-gerbe over X which obstructs the existence of a spin structure on $E$. This situation arises naturally on moduli spaces of coherent sheaves on Calabi-Yau fourfolds. Using spin structures as orientation data, we construct a categorical refinement of a K-theory class constructed by Oh-Thomas on such moduli spaces.

Self-predictive learning (aka non-contrastive learning) has become an increasingly important paradigm for representation learning. Self-predictive learning is simple yet effective: it learns without contrastive examples yet extracts useful representations through a self-predicitve objective. A common myth with self-predictive learning is that the optimization objective itself yields trivial representations as globally optimal solutions, yet practical implementations can produce meaningful solutions. 

We reconcile the theory-practice gap by studying the learning dynamics of self-predictive learning. Our analysis is based on analyzing a non-linear ODE system that sheds light on why despite a seemingly problematic optimization objective, self-predictive learning does not collapse, which echoes with important implementation "tricks" in practice. Our results also show that in a linear setup, self-predictive learning can be understood as gradient based PCA or SVD on the data matrix, hinting at meaningful representations to be captured through the learning process.

This talk is based on our ICML 2023 paper "Understanding self-predictive learning for reinforcement learning".

Celestial holography posits that the 4D S-matrix may be calculated holographically by a 2D conformal field theory. However, bulk translation invariance forces low-point massless celestial amplitudes to be distributional, which is an unusual property for a 2D CFT. In this talk, I show that translation-invariant MHV gluon amplitudes can be extracted from smooth 'leaf' amplitudes, where a bulk interaction vertex is integrated only over a hyperbolic slice of spacetime. After describing gluon leaf amplitudes' soft and collinear limits, I will show that MHV leaf amplitudes can be generated by a simple 2D system of free fermions and the semiclassical limit of Liouville theory, showing that translation-invariant, distributional amplitudes can be obtained from smooth correlation functions. An important step is showing that, in the semiclassical limit of Liouville theory, correlation functions of light operators are given by contact AdS Witten diagrams. This talk is based on a series of papers with Atul Sharma, Andrew Strominger, and Tianli Wang [2312.07820, 2402.04150,2403.18896]. 

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.

In this talk, I will give an overview of recent joint work on Topological Data Analysis (TDA). The first one is an application of TDA to quantify porosity in pathological bone tissue. The second is an extension of persistent homology to directed simplicial complexes. Lastly, we present an evaluation of the persistent Laplacian in machine learning tasks. This is joint work with Ysanne Pritchard, Aikta Sharma, Claire Clarkin, Helen Ogden, and Sumeet Mahajan; David Mendez; and Tom Davies and Zhengchao Wang, respectively.
 

We will explore the connection between Celestial and Euclidean Anti-de Sitter (EAdS) holography in the massive scalar case. Specifically, exploiting the so-called hyperbolic foliation of Minkowski space-time, we will show that each contribution to massive Celestial correlators can be reformulated as a linear combination of contributions to corresponding massive Witten correlators in EAdS. This result will be demonstrated explicitly both for contact diagrams and for the four-point particle exchange diagram, and it extends to all orders in perturbation theory by leveraging the bootstrapping properties of the Celestial CFT (CCFT).  Within this framework, the Kantorovic-Lebedev transform plays a central role. This transform will allow us to make broader considerations regarding non-perturbative properties of a CCFT.

This week's Fridays@2 will be a panel discussion focusing on what it is like to pursue a research degree. The panel will share their thoughts and experiences in a question-and-answer session, discussing some of the practicalities of being a postgraduate student, and where a research degree might lead afterwards.

Introduction to flat space holography in three dimensions and Carrollian CFT2, with selected results on correlation functions, thermal entropy, entanglement entropy and an outlook to Bondi news in 3d.

Ramification theory serves the dual purpose of a diagnostic tool and treatment by helping us locate, measure, and treat the anomalous behavior of mathematical objects. In the classical setup, the degree of a finite Galois extension of "nice" fields splits up neatly into the product of two well-understood numbers (ramification index and inertia degree) that encode how the base field changes. In the general case, however, a third factor called the defect (or ramification deficiency) can pop up. The defect is a mysterious phenomenon and the main obstruction to several long-standing open problems, such as obtaining resolution of singularities. The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many adjustments" fails when the defect is non-trivial. I will discuss my previous and ongoing work in ramification theory that allows us to understand and treat the defect.

We frame dynamic persuasion in a partial observation stochastic control game with an ergodic criterion. The receiver controls the dynamics of a multidimensional unobserved state process. Information is provided to the receiver through a device designed by the sender that generates the observation process. 

The commitment of the sender is enforced and an exogenous information process outside the control of the sender is allowed. We develop this approach in the case where all dynamics are linear and the preferences of the receiver are linear-quadratic.

We prove a verification theorem for the existence and uniqueness of the solution of the HJB equation satisfied by the receiver’s value function. An extension to the case of persuasion of a mean field of interacting receivers is also provided. We illustrate this approach in two applications: the provision of information to electricity consumers with a smart meter designed by an electricity producer; the information provided by carbon footprint accounting rules to companies engaged in a best-in-class emissions reduction effort. In the first application, we link the benefits of information provision to the mispricing of electricity production. In the latter, we show that when firms declare a high level of best-in-class target, the information provided by stringent accounting rules offsets the Nash equilibrium effect that leads firms to increase pollution to make their target easier to achieve.

This is a joint work with Prof. René Aïd, Prof. Giorgia Callegaro and Prof. Luciano Campi.