Past

Euclidean Ramsey Theory is a natural multidimensional version of Ramsey Theory. A subset of Euclidean space is called Ramsey if, for any $k$, whenever we partition Euclidean space of sufficiently high dimension into $k$ classes, one class much contain a congruent copy of our subset. It is still unknown which sets are Ramsey. We will discuss background on this and then proceed to some recent results.

There are several functional encodings of random trees which are commonly used to prove (among other things) scaling limit results.  We consider two of these, the height process and Lukasiewicz path, in the classical setting of a branching process tree with critical offspring distribution of finite variance, conditioned to have n vertices.  These processes converge jointly in distribution after rescaling by n^{-1/2} to constant multiples of the same standard Brownian excursion, as n goes to infinity.  Their difference (taken with the appropriate constants), however, is a nice example of a discrete snake whose displacements are deterministic given the vertex degrees; to quote Marckert, it may be thought of as a “measure of internal complexity of the tree”.  We prove that this discrete snake converges on rescaling by n^{-1/4} to the Brownian snake driven by a Brownian excursion.  We believe that our methods should also extend to prove convergence of a broad family of other “globally centred” discrete snakes which seem not to be susceptible to the methods of proof employed in earlier works of Marckert and Janson.

This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.

The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether given two morphisms g and h between two free semigroups $A$ and $B$, there is any nontrivial $x$ in $A$ such that $g(x)=h(x)$? This question can be phrased in terms of equalisers, asked in the context of free groups, and expanded: if the `equaliser' of $g$ and $h$ is defined to be the subgroup consisting of all $x$ where $g(x)=h(x)$, it is natural to wonder not only whether the equaliser is trivial, but what its rank or basis might be. 

While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalisations. In this talk I will give an overview of what is known about the PCP in hyperbolic groups, nilpotent groups and beyond (joint work with Alex Levine and Alan Logan).

I will talk about several results on Hecke algebras attached to Bernstein blocks of (arbitrary) reductive p-adic groups, where we construct a local Langlands correspondence for these Bernstein blocks. Our techniques draw inspirations from the foundational works of Deligne, Kazhdan and Lusztig. 

As an application, we prove the Local Langlands Conjecture for G_2, which is the first known case in literature of LLC for exceptional groups. Our correspondence satisfies an expected property on cuspidal support, which is compatible with the generalized Springer correspondence, along with a list of characterizing properties including the stabilization of character sums, formal degree property etc. In particular, we obtain (not necessarily unipotent) "mixed" L-packets containing "F-singular" supercuspidals and non-supercuspidals. Such "mixed" L-packets had been elusive up until this point and very little was known prior to our work. I will give explicit examples of such mixed L-packets in terms of Deligne-Lusztig theory and Kazhdan-Lusztig parametrization. 

If time permits, I will explain how to pin down certain choices in the construction of the correspondence using stability of L-packets; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover, I will mention how to adapt our general strategy to construct explicit LLC for other reductive groups, such as GSp(4), Sp(4), etc. Such explicit description of the L-packets has been useful in number-theoretic applications, e.g. modularity lifting questions as in the recent work of Whitmore. 

Some parts of this talk are based on my joint work with Aubert, and some other parts are based on my joint work with Suzuki. 
 

 I will explore subtle aspects of rank-one 4d N=2 supersymmetric QFTs through their low-energy Coulomb-branch physics. This low-energy Lagrangian is famously encoded in the Seiberg-Witten (SW) curve, which is a one-parameter family of elliptic curves. Less widely appreciated is the fact that various properties of the QFTs, including properties that cannot be read off from the Lagrangian, are nonetheless encoded into the SW curve, in particular in its Mordell-Weil group. This includes the global form of the flavour group, the one-form symmetries under which defect lines are charged, and the "global form" of the theory. In particular, I will discuss in detail the difference between the pure SU(2) and the pure SO(3) N=2 SYM theories from this perspective. I will also comment on 5d SCFTs compactified on a circle in this context.

A representation on the space of paths is a map which is compatible with the concatenation operation of paths, such as the path signature and Cartan development (or equivalently, parallel transport), and has been used to define characteristic functions for the law of stochastic processes. In this talk, we consider representations of surfaces which are compatible with the two distinct algebraic operations on surfaces: horizontal and vertical concatenation. To build these representations, we use the notion of higher parallel transport, which was first introduced to develop higher gauge theories. We will not assume any background in geometry or category theory. Based on a preprint (https://arxiv.org/abs/2311.08366) with Harald Oberhauser.

R. Schoen has conjectured that an asymptotically flat Riemannian n-manifold (M,g) with non-negative scalar curvature is isometric to Euclidean space if it admits a non-compact area-minimizing hypersurface. This has been confirmed by O. Chodosh and M. Eichmair in the case where n=3. In this talk, I will present recent work with M. Eichmair where we confirm this conjecture in the case where 3<n<8 and the area-minimizing hypersurface arises as the limit of large isoperimetric hypersurfaces. By contrast, we show that a large part of spatial Schwarzschild of dimension 3<n<8 is foliated by non-compact area-minimizing hypersurfaces.

In 1989, Mazur defined the deformation functor associated to a residual Galois representation, which played an important role in the proof by Wiles of the modularity theorem. This was used as a basis over which many mathematicians constructed variations both to further specify it or to expand the contexts where it can be applied. These variations proved to be powerful tools to obtain many strong theorems, in particular of modular nature. In this talk I will give an overview of the deformation theory of Galois representations and describe two variants of Mazur's functor that allow one to properly deform reducible residual representations (which is one of the shortcomings of Mazur's original functor). Namely, I will present the theory of determinant-laws initiated by Bellaïche-Chenevier on the one hand, and an idea developed by Calegari-Emerton on the other.
If time permits, I will also describe results that seem to indicate a possible comparison between the two seemingly unrelated constructions.

One can think of the stabilisation of an ∞-category as the ∞-category of objects that admit infinite deloopings. For example, the ∞-category of spectra is the stabilisation of the ∞-category of homotopy types. Costabilisation is the opposite notion of stabilisation, where we are interested in objects that allow infinite desuspensions. It is easy to see that the costabilisation of the ∞-category of homotopy types is trivial. Fix a prime number p. In this talk I will show that the costablisation of the ∞-category of T(h)-local spectral Lie algebras is equivalent to the ∞-category of T(h)-local spectra, where T(h) denotes a p-local telescope spectrum of height h. A key ingredient of the proof is to relate spectral Lie algebras to (spectral) Eₙ algebras via Koszul duality.
 

We consider stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter less than 1/2. The drift is a measurable function of time and space which belongs to a certain Lebesgue space. Under subcritical regime, we show that a strong solution exists and is unique in path-by-path sense. When the noise is formally replaced by a Brownian motion, our results correspond to the strong uniqueness result of Krylov and Roeckner (2005). Our methods forgo standard approaches in Markovian settings and utilize Lyons' rough path theory in conjunction with recently developed tools. Joint work with Toyomu Matsuda and Oleg Butkovsky.

After a brief introduction to the semiregularity maps of Severi, Kodaira and Spencer, and Bloch, I will focus on the Buchweitz-Flenner semiregularity map and on its importance for the deformation theory of coherent sheaves.
The subject of this talk is the construction of a lifting of each component of the Buchweitz-Flenner semiregularity map to an L-infinity morphism between DG-Lie algebras, which allows to interpret components of the semiregularity map as obstruction maps of morphisms of deformation functors.

As a consequence, we obtain that the semiregularity map annihilates all obstructions to deformations of a coherent sheaf on a complex projective manifold. Based on a joint work with R. Bandiera and M. Manetti.

Deep learning has revolutionised many scientific fields and so it is no surprise that state-of-the-art solutions to several inverse problems also include this technology. However, for many inverse problems (e.g. in medical imaging) stability and reliability are particularly important.

Furthermore, unlike other image analysis tasks, usually only a fairly small amount of training data is available to train image reconstruction algorithms.

Thus, we require tailored solutions which maximise the potential of all ingredients: data, domain knowledge and mathematical analysis. In this talk we discuss a range of such hybrid approaches and will encounter along the way connections to various topics like generative models, convex optimization, differential equations and equivariance.

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.

I will discuss various aspects of multi-parameter persistence related to representation theory and decompositions into indecomposable summands, based on joint work with Magnus Botnan, Steffen Oppermann, Johan Steen, Luis Scoccola, and Benedikt Fluhr.

A classification of indecomposables is infeasible; the category of two-parameter persistence modules has wild representation type. We show [1] that this is still the case if the structure maps in one parameter direction are epimorphisms, a property that is commonly satisfied by degree 0 persistent homology and related to filtered hierarchical clustering. Furthermore, we show [2] that indecomposable persistence modules are dense in the interleaving distance, and that being nearly-indecomposable is a generic property of persistence modules. On the other hand, the two-parameter persistence modules arising from interleaved sets (relative interleaved set cohomology) have a very well-behaved structure [3] that is encoded as a complete invariant in the extended persistence diagram. This perspective reveals some important but largely overlooked insights about persistent homology; in particular, it highlights a strong reason for working at the level of chain complexes, in a derived category [4].

[1] Ulrich Bauer, Magnus B. Botnan, Steffen Oppermann, and Johan Steen, Cotorsion torsion triples and the representation theory of filtered hierarchical clustering, Adv. Math. 369 (2020), 107171, 51. MR4091895

[2] Ulrich Bauer and Luis Scoccola, Generic multi-parameter persistence modules are nearly indecomposable, 2022.

[3] Ulrich Bauer, Magnus Bakke Botnan, and Benedikt Fluhr, Structure and interleavings of relative interlevel set cohomology, 2022.

[4] Ulrich Bauer and Benedikt Fluhr, Relative interlevel set cohomology categorifies extended persistence diagrams, 2022.

Mathematical and computational techniques can improve our understanding of diseases. In this talk, I’ll present ways in which data from cancer patients can be combined with mathematical modelling and used to improve cancer treatments.

Given the variability in individual responses to cancer treatments, agent-based modelling has been a useful technique for accurately capturing cellular behaviours that may lead to stochasticity in patient outcomes. Using a hybrid agent-based model and partial differential equation system, we developed a model for brain cancer (glioblastoma) growth informed by ex-vivo patient samples. Extending the model to capture patient treatment with an oncolytic virus rQNestin, we used our model to propose reasons for treatment failure, which was later confirmed with further patient samples. More recently, we extended this model to investigate the effectiveness of combination treatments (chemotherapy, virotherapy and immunotherapy) informed by individual patient imaging mass cytometry.

This talk hopes to provide examples of ways mathematical and computational modelling can be used to run “virtual” clinical trials with the goal of obtaining more effective treatments for diseases.  

In the context of categorical Langlands, there are many ways in which one could define the notion of n-finitely presented smooth representation. We will explore and compare two different definitions, relating them with the notion of n-coherence for the corresponding Iwasawa ring.

In this talk I will present joint work with Ehud Hrushovski on imaginaries in the ring of adeles and more generally in products and restricted products of structures (including the generalised products of Feferman-Vaught).

We prove a general theorem on weak elimination of imaginaries in products with respect to additional sorts which we deduce from an elimination of imaginaries for atomic and atomless Booleanizations of a theory. This combined with uniform elimination of imaginaries for p-adic numbers in a language with extra sorts as p-adic lattices proved first by Hrushovski-Martin-Rideau and more recently by Hils-Rideau-Kikuchi in a slightly different language, yields weak elimination of imaginaries for the ring of adeles in a language with extra sorts as adelic versions of the p-adic lattices. 

The proofs of the general results on products use Boolean valued model theory, stability theory, analysis of definable groups and liaison groups, and descriptive set theory of smooth Borel equivalence relations including Harrington-Kechris-Louveau and Glimm-Efros dichotomy. 

I will report on current work in progress on the construction of anticyclotomic p-adic L-functions for Rankin--Selberg products. I will explain how by p-adically interpolating the branching law for the spherical pair (U(n)xU(n+1), U(n)) we can construct a p-adic L-function attached to cohomological automorphic representations of U(n) x U(n+1), including anticyclotomic variation. Due to the recent proof of the unitary Gan--Gross--Prasad conjecture, this p-adic L-function interpolates the square root of the central L-value. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation.

We propose a novel framework for exploring weak and $L_2$ generalization errors of algorithms through the lens of differential calculus on the space of probability measures. Specifically, we consider the KL-regularized empirical risk minimization problem and establish generic conditions under which the generalization error convergence rate, when training on a sample of size $n$ , is $\matcal{O}(1/n)$. In the context of supervised learning with a one-hidden layer neural network in the mean-field regime, these conditions are reflected in suitable integrability and regularity assumptions on the loss and activation functions.

We develop an algorithm for parameter-free stochastic convex optimization (SCO) whose rate of convergence is only a double-logarithmic factor larger than the optimal rate for the corresponding known-parameter setting. In contrast, the best previously known rates for parameter-free SCO are based on online parameter-free regret bounds, which contain unavoidable excess logarithmic terms compared to their known-parameter counterparts. Our algorithm is conceptually simple, has high-probability guarantees, and is also partially adaptive to unknown gradient norms, smoothness, and strong convexity. At the heart of our results is a novel parameter-free certificate for the step size of stochastic gradient descent (SGD), and a time-uniform concentration result that assumes no a-priori bounds on SGD iterates.

Additionally, we present theoretical and numerical results for a dynamic step size schedule for SGD based on a variant of this idea. On a broad range of vision and language transfer learning tasks our methods performance is close to that of SGD with tuned learning rate. Also, a per-layer variant of our algorithm approaches the performance of tuned ADAM.

This talk is based on papers with Yair Carmon and Maor Ivgi.

We consider degenerate fully nonlinear equations, whose degeneracy rate depends on the gradient of solutions. We work under a Dini-continuity condition on the degeneracy term and prove that solutions are continuously differentiable. Then we frame this class of equations in the context of a free transmission problem. Here, we discuss the existence of solutions and establish a result on interior regularity. We conclude the talk by discussing a boundary regularity estimate; of particular interest is the case of point-wise regularity at the intersection of the fixed and the free boundaries. This is based on joint work with David Stolnicki.