Past

The Zilber-Pink conjecture predicts that if V is a proper subvariety of an arithmetic variety S (e.g. abelian variety, Shimura variety, others) not contained in a proper special subvariety of V, then the “unlikely intersections” of V with the proper special subvarieties of S are not Zariski dense in V. In this talk I will present a strong counterpart to the Zilber-Pink conjecture, namely that under some natural conditions, likely intersections are in fact Euclidean dense in V.  This is joint work with Tom Scanlon.

The main object of study of the talk is the balanced triple product p-adic L-function; this is a p-adic L-function associated with a triple of families of (quaternionic) modular forms. The first instances of these functions appear in the works of Darmon-Lauder-Rotger, Hsieh, and Greenberg-Seveso. They have proved to be effective tools in studying cases of the p-adic equivariant Birch & Swinnerton-Dyer conjecture. With this aim in mind, we discuss the construction of a new p-adic L-function, extending Hsieh's construction, and allowing classical weight one modular forms in the chosen families. Such improvement does not come for free, as it coincides with the increased dimension of certain Hecke-eigenspaces of quaternionic modular forms with non-Eichler level structure; we discuss how to deal with the problems arising in this more general setting. One of the key ingredients of the construction is a p-adic extension of the Jacquet-Langlands correspondence addressing these more general quaternionic modular forms. This is joint work in progress with Aleksander Horawa.

On supercomputers that exist today, achieving even close to the peak performance is incredibly difficult if not impossible for many applications. Techniques designed to improve the performance of matrix computations - making computations less expensive by reorganizing an algorithm, making intentional approximations, and using lower precision - all introduce what we can generally call ``inexactness''. The questions to ask are then:

1. With all these various sources of inexactness involved, does a given algorithm still get close enough to the right answer?
2. Given a user constraint on required accuracy, how can we best exploit and balance different types of inexactness to improve performance?

Studying the combination of different sources of inexactness can thus reveal not only limitations, but also new opportunities for developing algorithms for matrix computations that are both fast and provably accurate. We present few recent results toward this goal, icluding mixed precision randomized decompositions and mixed precision sparse approximate inverse preconditioners.

We propose a unified theory of brain function called ‘Thermodynamics of Mind’ which provides a natural, parsimonious way to explain the underlying computational mechanisms. The theory uses tools from non-equilibrium thermodynamics to describe the hierarchical dynamics of brain states over time. Crucially, the theory combines correlative (model-free) measures with causal generative models to provide solid causal inference for the underlying brain mechanisms. The model-based framework is a powerful way to use regional neural dynamics within the hierarchical anatomical brain connectivity to understand the underlying mechanisms for shaping the temporal unfolding of whole-brain dynamics in brain states. As such this model-based framework fitted to empirical data can be exhaustively investigated to provide objectively strong causal evidence of the underlying brain mechanisms orchestrating brain states. 

This week we will be discussing the theme of resilience. Free lunch provided!

The speakers are Maya Stein (University of Chile), Mathias Schacht (Hamburg), János Pach (Rényi Institute, Hungary and IST Austria), Marthe Bonamy (Bordeaux), Mehtaab Sawhney (Cambridge/MIT), and Julian Sahasrabudhe (Cambridge). Please see the event website for further details including titles, abstracts, and timings. Anyone interested is welcome to attend, and no registration is required.

In this talk, I will firstly give asymptotic formulas for the moments of the n-th derivative of the characteristic polynomials from the CUE. Secondly, I will talk about the connections between them and a solution of certain Painleve differential equation. This is joint work with Jonathan P. Keating.
 

COURSE_PROPOSAL (12)_1.pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

After a brief introduction to subject of spherical representations of hyperbolic groups, I will present a new construction motivated by a spectral formulation of the so-called Shalom conjecture.This a joint work with Dr Jan Spakula.

Let G and H be real reductive groups. To any L-homomorphism e: H^L \to G^L one can associate a map e_* from virtual representations of H to virtual representations of G. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which e_* exhibits especially nice behavior is the case when H is an endoscopic group. In this talk, I will introduce a more general class of L-homomorphisms that exhibit similar behavior to the endoscopic case. I will explain how this more general notion of endoscopic lifting relates to the theory of cohomological induction. I will also explain how this generalized notion of endoscopic lifting can be used to prove the unitarity of many Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.

Recent years have seen many new ideas appearing in the solution theories of singular stochastic partial differential equations. An exciting application of SPDEs that is beginning to emerge is to the construction and analysis of quantum field theories. In this talk, I will describe how stochastic quantisation of Parisi–Wu can be used to study QFTs, especially those arising from gauge theories, the rigorous construction of which, even in low dimensions, is largely open.

I will begin by speaking about Ito SDEs on manifolds, how their meaning depends on the choice of a connection, and an example in which the Ito formulation is preferable to the more common Stratonovich one. SDEs are naturally generalised to the case of more irregular driving signals by rough differential equations (RDEs), i.e. equations driven by rough paths. I will explain how it is possible to give a coordinate-invariant definition of rough integral and rough differential equation on a manifold, even in the case of arbitrarily low regularity and when the rough path is not geometric, i.e. it does not satisfy a classical integration by parts rule. If time permits, I will end on a more recent algebraic result that makes it possible to canonically convert non-geometric RDEs to geometric ones.

Optimal transport (OT) has recently gained a lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this talk, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book "Computational Optimal Transport".

Optimal mass transport is a versatile tool that can be used to prove various geometric and functional inequalities. In this talk we focus on the class of Sobolev inequalities.

In the first part of the talk I present the main idea of this method, based on the work of Cordero-Erausquin, Nazaret and Villani (2004).

The second part of the talk is devoted to the joint work with Ch. Gutierrez and A. Kristály about Sobolev inequalities with weights. 

Symplectic duality predicts that affine symplectic singularities come in pairs that are in a sense dual to each other. The Hikita conjecture relates the cohomology of the symplectic resolution on one side to the functions on the fixed points on the dual side.  

In a recent work with Ivan Losev and Lucas Mason-Brown, we suggested an important example of symplectic dual pairs. Namely, a Slodowy slice to a nilpotent orbit should be dual to an affinization of a certain cover of a special orbit for the Langlands dual group. In that paper, we explain that the appearance of the special unipotent central character can be seen as a manifestation of a slight generalization of the Hikita conjecture for this pair.

However, a further study shows that several things can (and do!) go wrong with the conjecture. In this talk, I will explain a modified version of the statement, recent progress towards the proof, and how special unipotent characters appear in the picture. It is based on a work in progress with Do Kien Hoang and Vasily Krylov.

We will first introduce the modular method for solving Diophantine Equations, famously used to
prove the Fermat Last Theorem. Then, we will see how to generalize it for a totally real number field $K$ and
a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the 
signature of the equation. We will see various results concerning the solutions to the Fermat equation with
signatures $(r,r,p)$ (fixed $r$). This will involve image of inertia comparison and the study of certain
$S$-unit equations over $K$. If time permits, we will discuss briefly how to attack the very similar family
of signatures $(p,p,2)$ and $(p,p,3)$. 

The Anderson operator is a perburbation of the Laplace-Beltrami operator by a space white noise potential. I will explain how to get a short self-contained functional analysis construction of the operator and how a sharp description of its heat kernel leads to useful quantitative estimates on its eigenvalues and eigenfunctions. One can associate to Anderson operator a (doubly) random field called the Anderson Gaussian free field. The law of its (random) partition function turns out to characterize the law of the spectrum of the operator. The square of the Anderson Gaussian free field turns out to be related to a probability measure on paths built from the operator, called the polymer measure.

In 1983 Gromov proved the systolic inequality: if M is a closed, essential n-dimensional Riemannian manifold where every loop of length 2 is null-homotopic, then the volume of M is at least a constant depending only on n.  He also proved a version that depends on the simplicial volume of M, a topological invariant generalizing the hyperbolic volume of a closed hyperbolic manifold.  If the simplicial volume is large, then the lower bound on volume becomes proportional to the simplicial volume divided by the n-th power of its logarithm.  Nabutovsky showed in 2019 that Papasoglu's method of area-minimizing separating sets recovers the systolic inequality and improves its dependence on n.  We introduce simplicial volume to the proof, recovering the statement that the volume is at least proportional to the square root of the simplicial volume.

The SYZ conjecture is a geometric way of understanding mirror symmetry via the existence of dual special Lagrangian fibrations on mirror Calabi-Yau manifolds. Motivated by this conjecture, it is expected that $G_2$ and $Spin(7)$-manifolds admit calibrated fibrations as well. I will explain how to construct examples of a weaker type of fibration on compact $Spin(7)$-manifolds obtained via gluing, and give a hint as to why the stronger fibrations are still elusive. The key ingredient is the stability of the weak fibration property under deformation of the ambient $Spin(7)$-structure.

I will discuss various possible ways a global symmetry can act on operators in a quantum field theory. The possible actions on q-dimensional operators are referred to as q-charges of the symmetry. Crucially, there exist generalized higher-charges already for an ordinary global symmetry described by a group G. The usual charges are 0-charges, describing the action of the symmetry group G on point-like local operators, which are well-known to correspond to representations of G. We find that there is a neat generalization of this fact to higher-charges: i.e. q-charges are (q+1)-representations of G. I will also discuss q-charges for generalized global symmetries, including not only invertible higher-form and higher-group symmetries, but also non-invertible categorical symmetries. This talk is based on a recent (arXiv: 2304.02660) and upcoming works with Sakura Schafer-Nameki.

For week 4's @email session we welcome the SIAM-IMA student chapter, running their annual Three Minute Thesis competition.

The Three Minute Thesis competition challenges graduate students to present their research in a clear and engaging manner within a strict time limit of three minutes. Each presenter will be allowed to use only one static slide to support their presentation, and the panel of esteemed judges (details TBC) will evaluate the presentations based on criteria such as clarity, pacing, engagement, enthusiasm, and impact. Each presenter will receive a free mug and there is £250 in cash prizes for the winners. If you're a graduate student, sign up here (https://oxfordsiam.com/3mt) by Friday of week 3 to take part! And if not, come along to support your DPhil friends and colleagues, and to learn about the exciting maths being done by our research students.