Past

I’ll tell you about some of my favorite algebraic varieties, which are beautiful in their own right, and also have some dramatic applications to algebraic combinatorics.  These include the top-heavy conjecture (one of the results for which June Huh was awarded the Fields Medal), as well as non-negativity of Kazhdan—Lusztig polynomials of matroids.

Cost-sensitive loss functions are crucial in many real-world prediction problems, where different types of errors are penalized differently; for example, in medical diagnosis, a false negative prediction can lead to worse consequences than a false positive prediction. However, traditional learning theory has mostly focused on the symmetric zero-one loss, letting cost-sensitive losses largely unaddressed. In this work, we extend the celebrated theory of boosting to incorporate both cost-sensitive and multi-objective losses. Cost-sensitive losses assign costs to the entries of a confusion matrix, and are used to control the sum of prediction errors accounting for the cost of each error type. Multi-objective losses, on the other hand, simultaneously track multiple cost-sensitive losses, and are useful when the goal is to satisfy several criteria at once (e.g., minimizing false positives while keeping false negatives below a critical threshold). We develop a comprehensive theory of cost-sensitive and multi-objective boosting, providing a taxonomy of weak learning guarantees that distinguishes which guarantees are trivial (i.e., can always be achieved), which ones are boostable (i.e., imply strong learning), and which ones are intermediate, implying non-trivial yet not arbitrarily accurate learning. For binary classification, we establish a dichotomy: a weak learning guarantee is either trivial or boostable. In the multiclass setting, we describe a more intricate landscape of intermediate weak learning guarantees. Our characterization relies on a geometric interpretation of boosting, revealing a surprising equivalence between cost-sensitive and multi-objective losses.

It was recently argued that topological operators (at least those associated with continuous symmetries) need regularization. However, such regularization seems to be ill-defined when the underlying QFT is coupled to gravity. If both of these claims are correct, it means that charges cannot be meaningfully measured in the presence of gravity. I will review the evidence supporting these claims as discussed in [arXiv:2411.08858]. Given the audience's high level of expertise, I hope this will spark discussion about whether this is a promising approach to understanding the fate of global symmetries in quantum gravity.

Dey and Xin (J. Appl.Comput.Top. 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice.

Our algorithm is FPT with respect to the maximal number of relations with the same degree and with further optimisation we obtain an O(n³) algorithm for interval-decomposable modules. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida which is the first to enable the decomposition of large inputs.

This is joint work with Tamal Dey and Michael Kerber.

Bernstein–Gelfand–Gelfand (BGG) resolutions and the Grothendieck–Cousin complex both play central roles in modern algebraic geometry and representation theory. The BGG approach provides elegant, combinatorial resolutions for important classes of modules especially those arising in Lie theory; while Grothendieck–Cousin complexes furnish a powerful framework for computing local cohomology via filtrations by support. In this talk, we will give an overview of these two constructions and illustrate how they arise from the same categorical consideration.

Understanding how living systems dynamically self-organise across spatial and temporal scales is a fundamental problem in biology; from the study of embryo development to regulation of cellular physiology. In this talk, I will discuss how we can use mathematical modelling to uncover the role of microscale physical interactions in cellular self-organisation. I will illustrate this by presenting two seemingly unrelated problems: environmental-driven compartmentalisation of the intracellular space; and self-organisation during collective migration of multicellular communities. Our results reveal hidden connections between these two processes hinting at the general role that chemical regulation of physical interactions plays in controlling self-organisation across scales in living matter

For some values of degrees d=(d_1,...,d_c), we construct a compactification of a Hilbert scheme of complete intersections of type d. We present both a quotient and a direct construction. Then we work towards the construction of a quasiprojective coarse moduli space of smooth complete intersections via Geometric Invariant Theory.

I will discuss several interesting examples of classes of structures for which there is a sensible first-order theory of "almost all" structures in the class, for certain notions of "almost all". These examples include the classical theory of almost all finite graphs due to Glebskij-Kogan-Liogon'kij-Talanov and Fagin (and many more examples from finite model theory), as well as more recent examples from the model theory of infinite fields: the theory of almost all algebraic extensions and the universal/existential theory of almost all completions of a global field (both joint work with Arno Fehm). Interestingly, such asymptotic theories are sometimes quite well-behaved even when the base theories are not.

This paper shows that a simple sale contract with a collection of options implements the full-information first-best allocation in a variety of continuous-time dynamic adverse selection settings with news. Our model includes as special cases most models in the literature. The implementation result holds regardless of whether news is public (i.e., contractible) or privately observed by the buyer, and it does not require deep pockets on either side of the market. It is an implication of our implementation result that, irrespective of the assumptions on the game played, no agent waits for news to trade in such models. The options here do not play a hedging role and are, thus, not priced using a no-arbitrage argument. Rather, they are priced using a game-theoretic approach.

I will explain the content of Geometric Langlands (which is a theorem over the ground fields of characteristic 0 but still a conjecture in positive characteristic) and show how it implies a description of the space of automorphic functions in terms of Galois data. The talk will mostly follow a joint paper with Arinkin, Kazhdan, Raskin, Rozenblyum and Varshavsky from 2022.

Deflation is a technique to remove a solution to a problem so that other solutions to this problem can subsequently be found. The most prominent instance is deflation we see in eigenvalue solvers, but recent interest has been in deflation of rootfinding problems from nonlinear PDEs with many isolated solutions (spearheaded by Farrell and collaborators). 

In this talk I’ll show you recent results on deflation techniques for optimisation algorithms with many local minima, focusing on the Gauss—Newton algorithm for nonlinear least squares problems.  I will demonstrate advantages of these techniques instead of the more obvious approach of applying deflated Newton’s method to the first order optimality conditions and present some proofs that these algorithms will avoid the deflated solutions. Along the way we will see an interesting generalisation of Woodbury’s formula to least squares problems, something that should be more well known in Numerical Linear Algebra (joint work with Güttel, Nakatsukasa and Bloor Riley).

Main preprint: https://arxiv.org/abs/2409.14438. 

WoodburyLS preprint: https://arxiv.org/abs/2406.15120

A number of algorithms are now available---including Halko-Martinsson-Tropp, interpolative decomposition, CUR, generalized Nystrom, and QR with column pivoting---for computing a low-rank approximation of matrices. Some methods come with extremely strong guarantees, while others may fail with nonnegligible probability. We present methods for efficiently estimating the error of the approximation for a specific instantiation of the methods. Such certificate allows us to execute "responsibly reckless" algorithms, wherein one tries a fast, but potentially unstable, algorithm, to obtain a potential solution; the quality of the solution is then assessed in a reliable fashion, and remedied if necessary. This is joint work with Gunnar Martinsson. 

Time permitting, I will ramble about other topics in Randomised NLA. 

We consider two problems in fluid dynamics: the collective locomotion of flying animals and the interaction of vortex rings with fluid interfaces. First, we present a model of formation flight, viewing the group as a material whose properties arise from the flow-mediated interactions among its members. This aerodynamic model explains how flapping flyers produce vortex wakes and how they are influenced by the wakes of others. Long in-line arrays show that the group behaves as a soft, excitable "crystal" with regularly ordered member "atoms" whose positioning is susceptible to deformations and dynamical instabilities. Second, we delve into the phenomenon of vortex ring reflections at water-air interfaces. Experimental observations reveal reflections analogous to total internal reflection of a light beam. We present a vortex-pair--vortex-sheet model to simulate this phenomenon, offering insights into the fundamental interactions of vortex rings with free surfaces.

Originally posed in the 1950s, the Hadwiger-Nelson problem interrogates the ‘chromatic number of the plane’ via an infinite unit-distance graph. This question remains open today, known only to be 5,6, or 7. We may ask the same question of the hyperbolic plane; there the lack of homogeneous dilations leads to unique behaviour for each length scale d. This variance leads to other questions: is the d-chromatic number finite for all d>0? How does the d-chromatic number behave as d increases/decreases? In this talk, I will provide a summary of existing methods and results, before discussing improved bounds through the consideration of semi-regular tilings of the hyperbolic plane.

[This is the second in a series of two talks; the first talk will be in the Algebra Seminar of Tuesday Feb 4th https://www.maths.ox.ac.uk/node/70022]

In 2005, Faltings initiated a p-adic analogue of the complex Simpson correspondence, a theory that has since been explored by various authors through different approaches. In this two-lecture series (part I in the Algebra Seminar and part II in the Arithmetic Geometry Seminar), I will present a joint work in progress with Michel Gros and Takeshi Tsuji, motivated by the goal of comparing the parallel approaches we have developed and establishing a robust framework to achieve broader functoriality results for the p-adic Simpson correspondence.

The approach I developed with M. Gros relies on the choice of a first-order deformation and involves a torsor of deformations along with its associated Higgs-Tate algebra, ultimately leading to Higgs bundles. In contrast, T. Tsuji's approach is intrinsic, relying on Higgs envelopes and producing Higgs crystals. The evaluations of a Higgs crystal on different deformations differ by a twist involving a line bundle on the spectral variety.  A similar and essentially equivalent twisting phenomenon occurs in the first approach when considering the functoriality of the p-adic Simpson correspondence by pullback by a morphism that may not lift to the chosen deformations.
We introduce a novel approach to twisting Higgs modules using Higgs-Tate algebras, similar to the first approach of the p-adic Simpson correspondence. In fact, the latter can itself be reformulated as a twist. Our theory provides new twisted higher direct images of Higgs modules, that we apply to study the functoriality of the p-adic Simpson correspondence by higher direct images with respect to a proper morphism that may not lift to the chosen deformations. Along the way, we clarify the relation between our twisting and another twisting construction using line bundles on the spectral variety that appeared recently in other works.

Given two von Neumann algebras A,B with an action by a locally compact (quantum) group G, one can consider its associated equivariant correspondences, which are usual A-B-correspondences (in the sense of Connes) with a compatible unitary G-representation. We show how the category of such equivariant A-B-correspondences carries an analogue of the Fell topology, which is preserved under natural operations (such as crossed products or equivariant Morita equivalence). If time permits, we will discuss one particular interesting example of such a category of equivariant correspondences, which quantizes the representation category of SL(2,R). This is based on joint works with Joeri De Ro and Joel Dzokou Talla. 

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 $X_0$.

How to uniformly, or at least almost uniformly, choose an element from a finite group G? When G is too large to enumerate all its elements, direct (pseudo)random selection is impossible. However, if we have an explicit set of generators of G (e.g., as in the Rubik's cube group), several methods are available. This talk will focus on one such method based on the well-known product replacement algorithm. I will discuss how recent results on property (T) by Kaluba, Kielak, Nowak and Ozawa partially explain the surprisingly good performance of this algorithm.

Real-world networks have a complex topology with many interconnected elements often organized into communities. Identifying these communities helps reveal the system’s organizational and functional structure. However, network data can be noisy, with incomplete link observations, making it difficult to detect significant community structures as missing data weakens the evidence for specific solutions. Recent research shows that flow-based community detection methods can highlight spurious communities in sparse networks with incomplete link observations. To address this issue, these methods require regularization. In this talk, I will show how a Bayesian approach can be used to regularize flows in networks, reducing overfitting in the flow-based community detection method known as the map equation.

The normal covering number $\gamma(G)$ of a finite group $G$ is the minimal size of a collection of proper subgroups whose conjugates cover the group. This definition is motivated by number theory and related to the concept of intersective polynomials. For the symmetric and alternating groups we will see how these numbers are closely connected to some elementary (as in "relating to basic concepts", not "easy") problems in additive combinatorics, and we will use this connection to better understand the asymptotics of $\gamma(S_n)$ and $\gamma(A_n)$ as $n$ tends to infinity.