Past

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. This is a continuation of the previous talk based on a recent preprint (https://arxiv.org/abs/2311.08366) with Harald Oberhauser.

In the past two decades, starting with the pioneering work of Bourgain, Brezis, and Mironescu, there has been widespread interest in characterizing Sobolev and BV (bounded variation) functions by means of non-local functionals. In my recent work I have studied two such functionals: a BMO-type (bounded mean oscillation) functional, and a functional related to the fractional Sobolev seminorms. I will discuss some of my results concerning the limits of these functionals, the concept of Gamma-convergence, and also open problems. 

Modular curves play a key role in the Langlands programme, being the simplest example of so-called Shimura varieties.  Their less famous cousins, Shimura curves, are also very interesting, and very concrete. 
In this talk I will give a gentle introduction to the arithmetic of Shimura curves, with lots of explicit examples. Time permitting, I will say something about recent work about intersection numbers of geodesics on Shimura curves.

Involutive knot Floer homology, a refinement of knot Floer theory, is a powerful knot invariant which was used to solve several long-standing problems, including the one-is-not-enough result for 4-manifolds with boundary. In this talk, we show that if the involutive knot Floer homology of a knot K admits an invariant splitting, then the induced splitting if the knot Floer homology of P(K), for any pattern P, can be made invariant under its \iota_K involution. As an application, we construct an infinite family of examples of pairs of exotic contractible 4-manifolds which survive one stabilization, and observe that some of them are potential candidates for surviving two stabilizations.
 

Mathematicians have been interested in the problem of compactifying the Jacobian variety of curves since the mid XIX century. In this talk we will discuss how all 'reasonable' compactified Jacobians of nodal curves can be classified combinatorically. This suffices to obtain a combinatorial classification of all 'reasonable' compactified universal (over the moduli spaces of stable curves) Jacobians. This is a joint work with Orsola Tommasi.

When the data is large, or comes in a streaming way, randomized iterative methods provide an efficient way to solve a variety of problems, including solving linear systems, finding least square solutions, solving feasibility problems, and others. Randomized Kaczmarz algorithm for solving over-determined linear systems is one of the popular choices due to its efficiency and its simple, geometrically intuitive iterative steps. 
In challenging cases, for example, when the condition number of the system is bad, or some of the equations contain large corruptions, the geometry can be also helpful to augment the solver in the right way. I will discuss our recent work with Michal Derezinski and Jackie Lok on Kaczmarz-based algorithms that use external knowledge about the linear system to (a) accelerate the convergence of iterative solvers, and (b) enable convergence in the highly corrupted regime.

Join us for a discussion about preparing for PhD and PostDoc Interviews. We will be talking to Melanie Rupflin and Mura Yakerson.

This talk concerns some ideas around the question of when a *-homomorphism into a quotient C*-algebra lifts. Lifting of *-homomorphisms arises prominently in the notions of projectivity and semiprojectivity, which in turn are closely related to stability of relations. Blackadar recently defined the notions of l-open and l-closed C*-algebras, making use of the topological space of *-homomorphisms from a C*-algebra A to another C*-algebra B, with the point-norm topology. I will discuss these properties and present new characterizations of them, which lead to solutions of some problems posed by Blackadar. This is joint work with Dolapo Oyetunbi.

We present a new formulation of string and particle amplitudes that emerges from simple one-dimensional models. The key is a new way to parametrize the positive part of Teichmüller space. It also builds on the results of Mirzakhani for computing Weil-Petterson volumes. The formulation works at all orders in the perturbation series, including non-planar contributions. The relationship between strings and particles is made manifest as a "tropical limit". The results are well adapted to studying the scattering of large numbers of particles or amplitudes at high loop order. The talk will in part cover results from arXiv:2309.15913, 2311.09284.

This event is free but requires prior registration. To register, please click here.

Abstract
In the contemporary AI landscape, Large Language Models (LLMs) stand out as game-changers. They redefine not only how we interact with computers via natural language but also how we identify and extract insights from vast, complex datasets. This presentation delves into the nuances of training and customizing LLMs, with a focus on their applications to quantitative finance.


About the speaker
Ioana Boier is a senior principal solutions architect at Nvidia. Her background is in Quantitative Finance and Computer Science. Prior to joining Nvidia, she was the Head of Quantitative Portfolio Solutions at Alphadyne Asset Management, and led research teams at Citadel LLC, BNP Paribas, and IBM T.J. Watson Research. She has a Ph.D. in Computer Science from Purdue University and is the author of over 30 peer-reviewed publications, 15 patents, and the winner of several awards for applied research delivered into products.
View her LinkedIn page

Frontiers in Quantitative Finance is brought to you by the Oxford Mathematical and Computational Finance Group and sponsored by CitiGroup and Mosaic SmartData.
 

One of the fundamental problems in representation theory is determining the representation type of algebras. In this talk, we will introduce the representation type of cyclotomic quiver Hecke algebras, also known as cyclotomic Khovanov-Lauda-Rouquier algebras, especially in affine type A and affine type C. Our main result relies on novel constructions of the maximal dominant weights of integrable highest weight modules over quantum groups. This talk is based on collaborations with Susumu Ariki, Berta Hudak, and Linliang Song.

Title: Elliptic curves and modularity

Abstract: The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

Pseudo metrics between persistence modules can be defined starting from Noise Systems [1].  Such metrics are used to compare the modules directly or to extract stable vectorisations. While the stability property directly follows from the axioms of Noise Systems, finding algorithms or closed formulas to compute the distances or associated vectorizations  is often a difficult problem, especially in the multi-parameter setting. In this seminar I will show how extra properties of Noise Systems can be used to define algorithms. In particular I will describe how to compute stable vectorisations with respect to Wasserstein distances [2]. Lastly I will discuss ongoing work (with D. Lundin and R. Corbet) for the computation of a geometric distance (the Volume Noise distance) and associated invariants on interval modules.

[1] M. Scolamiero, W. Chachólski, A. Lundman, R. Ramanujam, S. Oberg. Multidimensional Persistence and Noise, (2016) Foundations of Computational Mathematics, Vol 17, Issue 6, pages 1367-1406. doi:10.1007/s10208-016-9323-y.

[2] J. Agerberg, A. Guidolin, I. Ren and M. Scolamiero. Algebraic Wasserstein distances and stable homological invariants of data. (2023) arXiv: 2301.06484.

Sequential biomedical data is ubiquitous, from time-resolved data about patient encounters in the clinical realm to DNA sequences in the biological domain.  The talk will review our latest work in representation learning from longitudinal data, with a particular focus on finding optimal representations for complex and sparse healthcare data. We show how these representations are useful for comparing patient journeys and finding patients with similar health outcomes. We will also venture into the field of genome engineering, where we build models that work on DNA sequences for predicting editing outcomes for base and prime editors. 

I will discuss logarithmic corrections to various CFT partition functions in the context of the AdS4/CFT3 correspondence for theories arising on the worldvolume of M2-branes. I will use four-dimensional gauged supergravity and heat kernel methods and present general expressions for the logarithmic corrections to the gravitational on-shell action or black hole entropy for a number of different supergravity backgrounds. I will outline several subtleties and puzzles in these calculations and contrast them with a similar analysis of logarithmic corrections performed directly in the eleven-dimensional uplift of a given four-dimensional supergravity background. This analysis suggests that four-dimensional supergravity consistent truncations are not the proper setting for studying logarithmic corrections in AdS/CFT. These results have important implications for the existence of scale-separated AdS vacua in string theory and for effective field theory in AdS more generally.

Roughly speaking, class field theory for a number field K describes the abelianization of its absolute Galois group in terms of the idele class group of K. Geometric class field theory is what we get when K is instead the function field of a smooth projective geometrically connected curve X over a finite field. In this talk, I give a precise statement of geometric class field theory in the unramified case and describe how one can prove it by showing the Picard stack of X is the “free dualizable commutative group stack on X”. A key part is to show that the usual “divisor class group exact sequence“ can be done in families to give the adelic uniformization of the Picard stack by the moduli space of Cartier divisors on X. 

I will recall the Zilber-Pink conjecture for Shimura varieties and give my perspective on current progress towards a proof.

In this talk, we investigate distributionally robust optimization (DRO) in a dynamic context. We consider a general penalized DRO problem with a causal transport-type penalization. Such a penalization naturally captures the information flow generated by the models. We derive a tractable dynamic duality formula under a measure theoretic framework. Furthermore, we apply the duality to distributionally robust average value-at-risk and stochastic control problems.

A homogeneous space G/H is called a reductive symmetric space if G is a (real) reductive Lie group, and H is a symmetric subgroup of G, meaning that H is the subgroup fixed by some involution on G. The representation theory on reductive symmetric spaces was studied in depth in the 1990s by Erik van den Ban, Patrick Delorme, and Henrik Schlichtkrull, among many others. In particular, they obtained the Plancherel formula for the L^2 space of G/H. An important aspect is that this generalizes the group case, obtained by Harish-Chandra, which corresponds to the case when G = G' x G' and H is the diagonal subgroup.

In our collaborative efforts with A. Afgoustidis, N. Higson, P. Hochs, Y. Song, we are studying this subject from the perspective of noncommutative geometry. I will describe this exciting new development, with a particular emphasis on describing what is new and how this is different from the traditional group case, i.e. the reduced group C*-algebra of G.

In this talk, we present an algorithm to compute p-adic heights on hyperelliptic curves with good reduction. Our algorithm improves a previous algorithm of Balakrishnan and Besser by being considerably simpler and faster and allowing even degree models. We discuss two applications of our work: to apply the quadratic Chabauty method for rational and integral points on hyperelliptic curves and to test the p-adic Birch and Swinnerton-Dyer conjecture in examples numerically. This is joint work with Steffen Müller.

Richard Hamilton introduced the Ricci flow as a way to study the Poincaré conjecture, which says that every simply connected, compact three-manifold is homeomorphic to the three-sphere. In this talk, we will introduce the Ricci flow in a way that is accessible to anyone with basic knowledge of Riemannian geometry. We will give some examples, discuss finite time singularities, and give an application to a theorem of Hamilton which says that every compact Riemannian 3-manifold with positive Ricci curvature admits a metric of constant positive sectional curvature.

This talk concerns the design and analysis of adaptive FEM-based solution strategies for partial differential equations (PDEs) with uncertain or parameter-dependent inputs. We present two conceptually different strategies: one is projection-based (stochastic Galerkin FEM) and the other is sampling-based (stochastic collocation FEM). These strategies have emerged and become popular as effective alternatives to Monte-Carlo sampling in the context of (forward) uncertainty quantification. Both stochastic Galerkin and stochastic collocation approximations are typically represented as finite (sparse) expansions in terms of a parametric polynomial basis with spatial coefficients residing in finite element spaces. The focus of the talk is on multilevel approaches where different spatial coefficients may reside in different finite element spaces and, therefore, the underlying spatial approximations are allowed to be refined independently from each other.

We start with a more familiar setting of projection-based methods, where exploiting the Galerkin orthogonality property and polynomial approximations in terms of an orthonormal basis facilitates the design and analysis of adaptive algorithms. We discuss a posteriori error estimation as well as the convergence and rate optimality properties of the generated adaptive multilevel Galerkin approximations for PDE problems with affine-parametric coefficients. We then show how these ideas of error estimation and multilevel adaptivity can be applied in a non-Galerkin setting of stochastic collocation FEM, in particular, for PDE problems with non-affine parameterization of random inputs and for problems with parameter-dependent local spatial features.

The talk is based on a series of joint papers with Dirk Praetorius (TU Vienna), Leonardo Rocchi (Birmingham), Michele Ruggeri (University of Strathclyde, Glasgow), David Silvester (Manchester), and Feng Xu (Manchester).

In the 1960s, Lynden-Bell, studying the dynamics of galaxies around steady states of the gravitational Vlasov-Poisson equation, described a phenomenon he called "violent relaxation," a convergence to equilibrium through phase mixing analogous in some respects to Landau damping in plasma physics. In this talk, I will discuss recent work on this gravitational Landau damping for the linearised Vlasov-Poisson equation and, in particular, the critical role of regularity of the steady states in distinguishing damping from oscillatory behaviour in the perturbations. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.

Recent advances in experimental techniques have enabled remarkable discoveries and insight into how the dynamics of thin gas/vapour films can profoundly influence the behaviour of liquid droplets: drops impacting solids can “skate on a film of air” [1], so that they can “bounce off walls” [2,3]; reductions in ambient gas pressure can suppress splashing [4] and initiate the merging of colliding droplets [5]; and evaporating droplets can levitate on their own vapour film [7] (the Leidenfrost effect). Despite these advances, the precise physical mechanisms governing these phenomena remains a topic of debate.  A theoretical approach would shed light on these issues, but due to the strongly multiscale nature of these processes brute force computation is infeasible.  Furthermore, when films reach the scale of the mean free path in the gas (i.e. ~100nm) and below, new nanoscale physics appears that renders the classical Navier-Stokes paradigm inaccurate.

In this talk, I will overview our development of efficient computational models for the aforementioned droplet dynamics in the presence of gas nanofilms into which gas-kinetic, van der Waals and/or evaporative effects can be easily incorporated [8,9].  It will be shown that these models can reproduce experimental observations – for example, the threshold between bouncing and wetting for drop impact on a solid is reproduced to within 5%, whilst a model excluding either gas-kinetic or van der Waals effects is ~170% off!  These models will then be exploited to make new experimentally-verifiable predictions, such as how we expect drops to behave in reduced pressure environments.  Finally, I will conclude with some exciting directions for future wor

[1] JM Kolinski et al, Phys. Rev. Lett.  108 (2012), 074503. [2] JM Kolinski et al, EPL.  108 (2014), 24001. [3] J de Ruiter et al, Nature Phys.  11 (2014), 48. [4] L Xu et al, Phys. Rev. Lett. 94 (2005), 184505. [5] J Qian & CK Law, J. Fluid. Mech. 331 (1997), 59.  [6] KL Pan J. Appl. Phys. 103 (2008), 064901. [7] D Quéré, Ann. Rev. Fluid Mech. 45 (2013), 197. [8] JE Sprittles, Phys. Rev. Lett.  118 (2017), 114502.  [9] MV Chubynsky et al, Phys. Rev. Lett.. 124 (2020), 084501.

For a complete theory T, Lascar associated with it a Galois group which we call the Lacsar group. We will talk about some of my work on recovering the Lascar group as the fundamental group of Mod(T) and some recent progress in understanding the higher homotopy groups.

The mapping class group of a surface has a hierarchical structure in which the geometry of the group can be seen by examining its action on the curve graph of every subsurface. This behavior was one of the motivating examples for a generalization of hyperbolicity called hierarchical hyperbolicity. Hierarchical hyperbolicity has many desirable consequences, but the definition is long, and proving that a group satisfies it is generally difficult. This difficulty motivated the introduction of a new condition called combinatorial hierarchical hyperbolicity by Behrstock, Hagen, Martin, and Sisto in 2020 which implies the original and is more straightforward to check. In recent work, Hagen, Mangioni, and Sisto developed a method for building a combinatorial hierarchically hyperbolic structure from a (sufficiently nice) hierarchically hyperbolic one. The goal of this talk is to describe their construction in the case of the mapping class group and illustrate some of the parallels between the combinatorial structure and the original. 

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.