Professor Amir Ali Ahmadi will talk about; 'Disjunctive Sum of Squares'

We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach, where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares approach certifies nonnegativity using multiple algebraic identities. Our main result is a disjunctive Positivstellensatz showing that the degree of each algebraic identity can be kept as low as the degree of the polynomial whose nonnegativity is in question. Based on this result, we construct a semidefinite programming–based converging hierarchy of lower bounds for the problem of minimizing a polynomial over a compact basic semialgebraic set, in which the size of the largest semidefinite constraint remains fixed throughout the hierarchy. We further prove a second disjunctive Positivstellensatz, which leads to an optimization-free hierarchy for polynomial optimization. We specialize this result to the problem of proving copositivity of matrices. Finally, we describe how the disjunctive sum of squares approach can be combined with a branch-and-bound algorithm, and we present numerical experiments on polynomial, copositive, and combinatorial optimization problems. The talk is self-contained and assumes no prior background in sum of squares optimization.

Descriptive set theory provides a useful framework for studying the complexity of classification problems in operator algebras. In this talk I will discuss how C*-algebras can be encoded as points in a Borel space, and introduce several equivalent parametrizations, including a new one in terms of ideals of a universal C*-algebra. I will then discuss examples of natural classes of C*-algebras that form Borel sets, as well as a parametrization of *-homomorphisms and recent results on the Borelness of certain functors. Time permitting, I will introduce KK-theory and the Kasparov product, and explain a new result showing that the Kasparov product is Borel in a certain appropriate parametrized setting.

Gravitational instantons are complete 4-dimensional hyperkähler manifolds with square-integrable curvature tensor. I will address the question whether all gravitational instantons (of type ALG) can be obtained as Hitchin moduli spaces. In particular, I will explain how to compute the (hyperkähler) Torelli map for (weakly) parabolic Higgs bundles on the 4-punctured sphere. This is based on recent joint work with Fredrickson, Mazzeo and Swoboda.

We propose a novel Black-Scholes model under which the stock price processes are modeled by stochastic differential equations driven  by sub-diffusions. The new framework can capture the less financial activity phenomenon during the bear markets while having the classical Black-Scholes model as its special case. The sub-diffusive spot market is arbitrage-free but is in general incomplete. We investigate the pricing for European-style contingent claims under this new model. For this, we study the Girsanov transform for sub-diffusions and use it to find risk-neutral probability measures for the new Black-Scholes model. Finally, we derive the explicit formula for the price of European call options and show that it can be determined by a partial differential equation (PDE) involving a fractional derivative in time, which we coin a time-fractional Black-Scholes PDE.

Deformation theory studies how varieties and other algebro-geometric objects vary in families. A central part of the subject is formal deformation theory, where one deforms over an Artinian base; such deformation problems are governed by Lie algebraic models. 

We pose the question of deforming varieties over nilpotent but not necessarily Artinian bases. These turn out to be classified by the same Lie algebraic models plus some topological structure. More precisely, we will consider partition Lie algebras in the category of ultrasolid modules, a variation of the solid modules of Clausen and Scholze that give a well-behaved category akin to topological modules.

To approach this result, we decompose deformation problems into n-nilpotent layers. Each of these layers is individually easier to understand, and is classified by simpler variants of partition Lie algebras.

In this talk I will discuss recent work, with R. Tione, on the regularity of stationary points for a class of planar polyconvex integrands which are conformally-invariant, a natural assumption in view of geometric applications. We prove that, in two dimensions, stationary points are smooth away from a discrete set. We also show full C^1-regularity for orientation-preserving solutions, which appear naturally in minimization problems of Teichmüller type.

A spatial graph is a graph whose nodes and edges carry spatial attributes. It is a smart modelling choice for capturing the skeleton of a shape, a blood vessel network, a porous tissue, and many other data objects with intrinsically complex geometry, often resulting in graphs with a high node and edge count. In this talk, we introduce a topological spatial graph coarsening approach based on a new framework that balances graph reduction against the preservation of topological characteristics, essential for faithfully representing the underlying shape. To capture the topological information required to calibrate the reduction level, we adapt the construction of classical topological descriptors made for point clouds (the so-called persistence diagrams) to spatial graphs. This relies on a new filtration called triangle-aware graph filtration. Our coarsening approach is parameter-free and we prove that it is equivariant under rotations, translations, and scaling of the initial spatial graph. We evaluate the performance of our method on synthetic and real spatial graphs and show that it significantly reduces the graph sizes while preserving the relevant topological information.

A difficult question in the local Langlands framework is to understand the interplay between the characters of irreducible smooth representations of a reductive group over a local field and the geometry of the dual space of Langlands parameters. An important invariant of the character (viewed as a distribution, i.e, a continuous linear functional on the space of smooth compactly supported functions) is the wavefront set, a measure of its singularities along with their directions. Motivated by the work of Adams, Barbasch, and Vogan for real reductive groups, it is natural to expect that the wavefront set is dual (in a certain sense) to the geometric singular support of the Langlands parameter. Dan Ciubotaru will give an overview of these ideas and describe recent progress in establishing a precise connection for representations of reductive p-adic groups. 

Since the foundational works of Diaconis, pointwise character bounds of the form $\chi(\sigma) \le \chi(1)^\alpha$ have guided the study of growth in finite simple groups. However, this classical machinery hits an algebraic bottleneck when confronted with non-class functions and unstructured subsets.

In this talk, we bypass this barrier by replacing classical representation theory with discrete analysis. By decomposing functions as $f = \sum f_\rho$ and bounding the $L_2$ norm $\|f_\rho\|_2 \le \chi_\rho(1)^\alpha$ for each representation $\rho$, we develop a robust theory of Fourier anti-concentration. We will demonstrate how this resolves the Liebeck–Nikolov–Shalev (LNS) conjecture—proving a group can be expressed optimally as the product of conjugates of an arbitrary subset $A$—and discuss how applying Boolean function analysis tools like hypercontractivity pushes this philosophy even further.

The quasi-isometry group QI(X) of a metric space X is a natural group of automorphisms of the space that preserve its large-scale structure. The quasi-isometry groups of most familiar spaces are usually enormous and quite wild. Spaces X for which QI(X) is understood tend to exhibit a sort of rigidity phenomenon: every quasi-isometry of such spaces is close to an isometry. We exploit this phenomenon to address the question of which abstract groups arise as the quasi-isometry groups of metric spaces. This talk is based on joint work with Paula Heim and Joe MacManus.

The notion of invariant random subgroups (IRS) is a fruitful, well-studied concept in dynamics on groups. In this talk, Hanna Oppelmayer will explain what it is and how to extend this notion to group von Neumann algebras LG, where G is a discrete countable group. We call it invariant random sub-von Neumann algebra (IRA). As an application, Hanna will provide a result concerning amenable IRAs, which generalises (in the discrete setup) a theorem of Bader-Duchesne-Lécureux about amenable IRSs. This is joint work with Tattwamasi Amrutam and Yair Hartman.

I will present a generalisation of Mirzakhani’s recursion for the volumes of moduli spaces of bordered Klein surfaces, including non-orientable surfaces. On these moduli spaces, the top form introduced by Norbury diverges as the lengths of one-sided geodesics approach zero. However, integrating this form over Gendulphe’s regularised moduli space—where the systole of one-sided geodesics is bounded below by epsilon—yields a finite volume. Using Norbury’s extension of the Mirzakhani–McShane identities to the non-orientable setting, we derive an explicit formula for the volume of the moduli space of one-bordered Klein bottles, as well as a recursion for arbitrary topologies that fully captures the dependence on the geometric regularisation parameter epsilon. I will conclude with remarks on the relation to refined topological recursion, which leads us to a refinement of the Witten–Kontsevich recursion and of the Harer–Zagier formula for the orbifold Euler characteristic of the moduli space of curves of genus g with n marked points. Based on joint work with P. Gregori and K. Osuga; the final part reflects ongoing work with N. Chidambaram, A. Giacchetto, and K. Osuga.

TBA 

Problems with physical constraints, such as the incompressibility constraint for mass conservation in fluids or Gauss's laws for electric and magnetic fields, result in generalized saddle point systems. So-called structure-preserving finite elements respect the constraints pointwise, resulting in more physically accurate solutions that are typically robust with respect to some problem parameters. However, constructing these finite elements may involve complicated spaces for the Lagrange multiplier variables. Augmented Lagrangian methods (ALMs) provide one process to compute the solution without the need for an explicit basis for the Lagrange multiplier space. In this talk, we present new convergence estimates for a standard ALM method, sometimes called the iterated penalty method, applied to structure-preserving discretizations of linear saddle point systems.

Professor Tobias Weinzierl will be talking about: 'Modern tasking approaches to simulate black holes (and other interesting phenomena): How can we make them fit to modern hardware?'

Over the past decade, my team has developed a simulation code for binary black hole mergers that runs on dynamically adaptive Cartesian meshes. 
Its dynamic adaptivity, coupled with multiple numerical schemes operating at different scales and non-deterministic loads from puncture sources, makes task-based parallelisation a natural choice:
Task stealing across fine-grained work units balances the load across many CPU cores, while treating tasks as atomic compute units should---in theory---allow us to deploy seamlessly to accelerators. In practice, it is far from straightforward.

Fine-grained tasks clash with accelerators, which thrive on large, homogeneous data access patterns;
task bursts on the CPU overwhelm tasking systems and produce suboptimal execution schedules;
and when tasks span address spaces, expensive memory movements kill performance.
Surprisingly, many mainstream tasking frameworks even lack the features our domain demands, i.e. to express key task concepts.
Our application serves as a powerful lens for examining these challenges. 
While our code base extends to other wave phenomena, Lagrangian techniques, and multigrid solvers, they all reveal the same fundamental tension: 
modern hardware increasingly struggles to accommodate modern HPC concepts, and it even challenges the notion that one solution fits all hardware components.
The talk proposes practical workarounds and solutions to these shortcomings, while all solutions are designed, wherever possible, to be upstreamed into mainstream software building blocks or at least decoupled from our particular PDE solver, making them broadly applicable to the community.

This talk is hosted by Rutherford Appleton Laboratory and will take place @ Harwell Campus, Didcot, OX11 0QX
 

Reaction systems are continuos-time dynamical systems with polynomial right-hand side, and are very common in biochemistry, cell signaling, population dynamics, and many other biological applications. We discuss global stability (i.e., the existence of a globally attracting point) and persistence (i.e., robust absence of extinction) for large classes of reaction systems. In particular, we describe recent progress on the proof of the Global Attractor Conjecture (which says that vertex-balanced reaction systems are globally stable) and the Persistence Conjecture (which says that weakly-reversible reaction systems are persistent), and how these results can be extended outside their classical setting using the notion of “disguised reaction systems". We will also discuss analogous results for the case where reaction systems are replaced by generalized Lotka-Volterra systems of arbitrary degree. 

We consider the question of whether almost-homomorphisms are close to honest homomorphisms. I’ll survey a few historical results, with different source/target collections of algebras, and also consider what to take as the definition of “almost-homomorphisms”. If we end up having time, I will sketch an elementary proof that almost-characters from commutative C*-algebras are close to honest characters.

Recently a new inhabitant entered the zoo of convexity notions for vectorial variational problems: functional convexity. I would like to report of progress in understanding the corresponding integrands, but also new insight into fine properties of most general class of related integrands: It turns out that rank-one convex functions share surprisingly many pointwise differentiablity properties with ordinary convex functions.

The well-known "Dunkl operators" associated to a finite real reflection group constitute a commutative parameter family of deformations of the directional derivatives in Euclidean space. These operators are "differential-reflection" operators. Heckman and Cherednik have defined trigonometric versions of Dunkl's operators. The interest for these operators lies in their deep ties to Macdonald polynomials and hypergeometric functions, to the Calogero-Moser quantum integrable system, and to the representation theory of Hecke algebras. 

"Hypergeometric shift operators" are powerful tools to study Weyl group symmetric structures and functions in these contexts. In this talk, Eric Opdam presents a theorem of existence and uniqueness of ''nonsymmetric shift operators'' for the Dunkl operators. These are themselves differential reflection operators which "shift" the parameters of the Dunkl operators by integers by means of a "transmutation relation".

(Joint work with Valerio Toledano Laredo) 

to follow

TBA

Professor Po-Ling Loh will talk about; 'Private estimation in stochastic block models'

We study the problem of private estimation for stochastic block models, where the observation comes in the form of an undirected graph, and the goal is to partition the nodes into unknown, underlying communities. We consider a notion of differential privacy known as node differential privacy, meaning that two graphs are treated as neighbors if one can be transformed into the other by changing the edges connected to exactly one node. The goal is to develop algorithms with optimal misclassification error rates, subject to a certain level of differential privacy.

We present several algorithms based on private eigenvector extraction, private low-rank matrix estimation, and private SDP optimization. A key contribution of our work is a method for converting a procedure which is differentially private and has low statistical error on degree-bounded graphs to one that is differentially private on arbitrary graph inputs, while maintaining good accuracy (with high probability) on typical inputs. This is achieved by considering a certain smooth version of a map from the space of all undirected graphs to the space of bounded-degree graphs, which can be appropriately leveraged for privacy. We discuss the relative advantages of the algorithms we introduce and also provide some lower-bounds for the performance of any private community estimation algorithm.

This is joint work with Laurentiu Marchis, Ethan D'souza, and Tomas Flidr.

Prostate cancer progression and therapeutic resistance present significant clinical challenges, particularly in the transition to castration-resistant disease. Although androgen deprivation therapy and second-generation drugs have improved patient outcomes, resistance frequently develops, reflecting tumour heterogeneity and the influence of its microenvironment. This talk presents two interdisciplinary studies that address these issues through data-driven mathematical approaches. We show how integrating experimental data with mathematical and statistical modelling can improve our understanding of prostate cancer dynamics and inform more effective, context-specific therapeutic strategies. The first study examines drug resistance and tumour evolution under treatment. We develop a multi-scale hybrid modelling framework to capture processes occurring across different temporal scales. Partial differential equations describe the behaviour of drugs and other chemicals in the tumour microenvironment (over the ‘fast’ timescale), while a cellular automaton captures the dynamics of tumour cells (over the ‘slow’ timescale). Through computational analysis of the model solutions, we examine the spatial dynamics of tumour cells, assess the efficacy of different drug therapies in inhibiting prostate cancer growth, and identify effective drug combinations and treatment schedules to limit tumour progression and prevent metastasis. The second study focuses on the role of host–microbiome interactions in obesity-associated prostate cancer. Using data from experiments with the TRAMP mouse model, we apply statistical and machine learning methods, including generalised linear models, Granger causality, and support vector regression, to characterise microbial dynamics and their responses to treatment. These findings are incorporated into a dynamical systems framework that captures microbiome–tumour co-evolution under therapeutic and dietary perturbations, providing insight into how dietary fat and combination therapies involving enzalutamide and phytocannabinoids influence tumour progression and gut microbiota composition.

This week’s Fridays@2 session is intended to provide advice on exam preparation and how to approach the Part A, B, and C exams.  A panel consisting of past examiners and current students will answer any questions you might have as you approach exam season.

In a joint work with Yu Deng (University of Chicago) and Xiao Ma (University of Michigan), we extended Lanford’s theorem to long times—specifically, for as long as the solution of the Boltzmann equation exists. This allowed us to fully carry out Hilbert’s program and derive the fluid equations in the Boltzmann–Grad limit. The underlying strategy builds on earlier joint work with Yu Deng that resolved a parallel problem in which colliding particles are replaced by nonlinear waves, thereby establishing the mathematical foundations of wave turbulence theory. In this talk, we will review this progress and discuss some related problems and future directions. 

TBA

Associate Professor Nicolas Boumal will talk about: 'Smooth, globally Polyak-Łojasiewicz functions are nonlinear least-squares'

Polyak-Łojasiewicz (PŁ) functions abound in the literature, especially in nonconvex optimization. When they are also smooth, they become surprisingly simple---with an exotic twist. The plan is for us to discover the structure of those functions and of their sets of minimizers via gradient flow and fiber bundles.

Joint work with Christopher Criscitiello and Quentin Rebjock.

We study the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

One can learn a lot about a compact manifold if one can show that it fibres over the circle - in essence, this allows us to view a static n-dimensional manifold as a manifold of dimension n-1 that evolves in time.Being fibred (over the circle) is a relatively rare property. It is much more common to be virtually fibred, that is, to admit a finite cover that is fibred. For example, it was the content of a conjecture of William Thurston, now two theorems by Ian Agol and Dani Wise, that all finite-volume hyperbolic 3-manifolds are virtually fibred; in fact, this property is extremely common among irreducible 3-manifolds.The situation is less clear in higher dimensions. On the obstruction side, we know that virtually fibred manifolds must have vanishing Euler characteristic. This immediately shows that compact hyperbolic manifolds in even dimensions will not be virtually fibred. A more involved obstruction comes from L2-homology: virtually fibred manifolds must be L2-acyclic. The motivation behind the research I will present lies in trying to find situations in which the vanishing of L2-homology is is not only necessary, but also sufficient for virtual fibring. It turns our that a lot more can be said if we replace aspherical manifolds by their homological cousins: Poincare duality groups. Concretely, if G is an n-dimensional Poincare-duality group over the rationals, and if G satisfies the RFRS property, then G is L2-acyclic if and only if there is a finite-index subgroup G0 of G and an epimorphism from G0 onto the integers such that its kernel is a Poincare-duality group over the rationals of dimension n-1. (This last theorem is joint with Sam Fisher and Giovanni Italiano.)The RFRS property was introduced in Agol's work on Thurston's conjecture. A countable group is RFRS if and only if it is residually {virtually abelian and poly-Z}. All compact special groups in the sense of Haglund-Wise satisfy this property, so there is a ready supply of RFRS groups, also among fundamental groups of hyperbolic manifolds in high dimensions.

We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider on the discretized circle the rate equation for the process where pairs of particles sitting on the same side jump simultaneously to the two neighboring sites, and the reverse jump where a pair of particles sitting on a common site jump simultaneously to the side in the middle. Depending on the rates, in the vanishing mesh size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. We identify the limiting gradient structure to be driven by entropy with respect to a generalization of the diffusive transport type with nonlinear mobility via EDP convergence. Furthermore, the DLSS equation with nonlinear mobility of the power type shares qualitative similarities with the fast diffusion and porous medium equations, since we find traveling wave solutions with algebraic tails and polynomial compact support, respectively.    
       

Joint work with Alexander Mielke and Artur Stephan arXiv:2510.07149. The DLSS part is based on joints works with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré.