Past

I will present some results from the newly developed theory of wavelets; based on the joint work with the following authors:

P.M. Luthy, H.Šikić, F.Soria, G.L.Weiss, E.N.Wilson.One-DimensionalDyadic Wavelets.Mem. Amer. Math. Soc. 280 (2022), no 1378, ix+152 pp.

About two and a half decades ago and based on the influential book by Fernandez and Weiss, an approach was developed to study wavelets from the point of view of their connections with Fourier analysis. The idea was to study wavelets and other reproducing function systems via the four basic equations that characterized various properties of wavelet systems, like frame and basis properties, completeness, orthogonality, etc. Despite hundreds of research papers and the impressive development of the theory of such systems, some questions remain open even in the basic case of dyadic wavelets on the real line. Among the most thorough treatments that we provide in this lengthy paper is the issue of the understanding of the low-pass filters associated with the MRA structure. In this talk, I will focus on some of these results. As it turned out, a more general and abstract approach to the problem, using the study of dyadic orbits and the newly introduced Tauberian function, reveals several interesting properties and opens an interesting context for some older results

For many spaces of interest to number theorists one can construct cycles which in some ways behave like the coefficients of modular forms. The aim of this talk is to give an introduction to this idea by focusing on examples coming from modular curves and Heegner points and the relevant work of Zagier, Gross-Kohnen-Zagier and Borcherds. If time permits I will discuss generalizations to other spaces.

In this talk, we will present a loop expansion for lattice gauge theories and its application to prove ultraviolet stability in the Abelian Higgs model. We will first describe this loop expansion and how it relates to earlier works of Brydges-Frohlich-Seiler. We will then show how the expansion leads to a quantitative diamagnetic inequality, which in turn implies moment estimates, uniform in the lattice spacing, on the Holder-Besov norm of the gauge field marginal of the Abelian Higgs lattice model. Based on Gauge field marginal of an Abelian Higgs model, which is joint work with Ajay Chandra.

We apply recent ideas in Floer homotopy theory to some questions in symplectic topology. We show that Floer homology can detect smooth structures of certain Lagrangians, as well as using this to find restrictions on symplectic mapping class groups. This is based on joint work-in-progress with Ivan Smith.

TBA

Speaker: Lasse Grimmelt (North Wing)
Title: Modular forms and the twin prime conjecture

Abstract: Modular forms are one of the most fruitful areas in modern number theory. They play a central part in Wiles proof of Fermat's last theorem and in Langland's far reaching vision. Curiously, some of our best approximations to the twin-prime conjecture are also powered by them. In the existing literature this connection is highly technical and difficult to approach. In work in progress on this types of questions, my coauthor and I found a different perspective based on a quite simple idea. In this way we get an easy explanation and good intuition why such a connection should exists. I will explain this in this talk.

Speaker: Yang Liu (South Wing)
Title: Obtaining Pseudo-inverse Solutions With MINRES

Abstract: The celebrated minimum residual method (MINRES) has seen great success and wide-spread use in solving linear least-squared problems involving Hermitian matrices, with further extensions to complex symmetric settings. Unless the system is consistent whereby the right-hand side vector lies in the range of the matrix, MINRES is not guaranteed to obtain the pseudo-inverse solution. We propose a novel and remarkably simple lifting strategy that seamlessly integrates with the final MINRES iteration, enabling us to obtain the minimum norm solution with negligible additional computational costs. We also study our lifting strategy in a diverse range of settings encompassing Hermitian and complex symmetric systems as well as those with semi-definite preconditioners.

Topological data analysis (TDA) is an emerging field of research, which considers the application of topology to data analysis. Recently, these methods have been successfully applied to research problems in the field of geographical information science (GIS). This includes the problems of Point of Interest (PoI), street network and weather analysis. In this talk I will describe how TDA can be used to provide solutions to these problems plus how these solutions compare to those traditionally used by GIS practitioners. I will also describe some of the challenges of performing interdisciplinary research when applying TDA methods to different types of data.

Chronic hepatitis B virus (HBV) infection leads to liver damage that increases the risk of hepatocellular carcinoma and liver cirrhosis. Individuals with chronic HBV infection are often either treated with interferon alpha or nucleoside reverse transcriptase inhibitors (NTRL). While these NTRLs inhibit de novo DNA synthesis, they do not represent a functional cure for chronic HBV infection and so must be taken indefinitely. The resulting life-long treatment leads to an increased risk of selection for treatment resistant strains of HBV. Consequently, there is increased interest in a novel treatment modality, capsid protein allosteric modulators (CPAMs), that blocks a crucial step in the viral life cycle. I'll discuss recent work that identifies HBV serum RNA as an informative biomarker of CPAM treatment efficacy, evaluates CPAMs as a potential functional cure for HBV infection, and illustrates the role of mechanistic modelling in trial design using an age structured, multi-scale mathematical model. 

I will show that the massless irreducible representations of the Poincare group are precisely Corrolian field living on I^+. I will also show that the analogous massless irreducible representation of E11 are just the degrees of freedom of maximal supergravity. Finally I will speculate how spacetime could emerge from an underlying fundamental theory.

The affine and periodic Temperley–Lieb algebras are families of infinite-dimensional algebras with a diagrammatic presentation. They have been studied in the last 30 years, mostly for their physical applications in statistical mechanics, where the diagrammatic presentation encodes the connectivity property of the models. Most of the relevant representations for physics are finite-dimensional. In this work, we define finite-dimensional quotients of these algebras, which we name uncoiled algebras in reference to the diagrammatic interpretation of the quotient, and construct a family of Wenzl–Jones idempotents, each of which projects onto one of the one-dimensional modules these algebras admit. We also prove that the uncoiled algebras are sandwich cellular and sketch some of the applications of the objects we defined. This is joint work with Alexi Morin-Duchesne.

This seminar is part of our Frontiers in Quantitative Finance. Attendance is free of charge but requires prior online registration.

Abstract
Empirical studies consistently find that the price impact of large trades approximately follows a nonlinear power law. Yet, tractable formulas for the portfolios that trade off predictive trading signals, risk, and trading costs in an optimal manner are only available for quadratic costs corresponding to linear price impact. In this paper, we show that the resulting linear strategies allow to achieve virtually optimal performance also for realistic nonlinear price impact, if the “effective” quadratic cost parameter is chosen appropriately. To wit, for a wide range of risk levels, this leads to performance losses below 2% compared to the numerical Viterbi algorithm of Kolm and Ritter (2014) run at very high accuracy. The effective quadratic cost depends on the portfolio risk, but can be computed without any sophisticated numerics by simply maximizing an explicit scalar function.
Read more on this work here.

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves contains a Zariski-dense set of special points if and only if it is special. I'll explain this result, and discuss an effective version that Gal Binyamini, Harry Schmidt, Margaret Thomas and I proved.

We use the term shape optimization when we want to find a minimizer of an objective function that assigns real values to shapes of domains. Solving shape optimization problems can be quite challenging, especially when the objective function is constrained to a PDE, in the sense that evaluating the objective function for a given domain shape requires first solving a boundary value problem stated on that domain. The main challenge here is that shape optimization methods must employ numerical methods capable of solving a boundary value problem on a domain that changes after each iteration of the optimization algorithm.

The first part of this talk will provide a gentle introduction to shape optimization. The second part of this talk will highlight how the finite element framework leads to automated numerical shape optimization methods, as realized in the open-source library fireshape. The talk will conclude with a brief overview of some academic and industrial applications of shape optimization.

In order for artificial neural networks to learn a task, one must solve an inverse design problem. What network will produce the desired output? We have harnessed AI approaches to design physical systems to perform functions inspired by biology, such as protein allostery. But artificial neural networks require a computer in order to learn in top-down fashion by the global process of gradient descent on a cost function. By contrast, the brain learns by local rules on its own, with each neuron adjusting itself and its synapses without knowing what all the other neurons are doing, and without the aid of an external computer. But the brain is not the only biological system that learns by local rules; I will argue that the actin cortex and the amnioserosa during the dorsal closure stage of Drosophila development can also be viewed this way.

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

It is a truth universally acknowledged, that an infinite group in possession of a good algebraic structure, must be in want of a hyperbolic space to act on. For the mapping class group of a surface, one of the most popular choices is the curve graph. This is a combinatorial object, built from curves on the surface and intersection patterns between them.
Hyperbolicity of the curve graph was proved by Masur and Minsky in a celebrated paper in 1999. In the same article, they showed how the geometry of the action on this graph reflects dynamical/topological properties of the mapping class group; in particular, loxodromic elements are precisely the pseudo-Anosov mapping classes.
In light of this, one would like to better understand distances in the curve graph. The graph is locally infinite, and finding a shortest path between two vertices is highly non-trivial. In this talk, we will see how to use the machinery of train tracks to overcome this issue and compute (approximate) distances in the curve graph. If time permits -- which, somehow, it never does -- we will also analyse this construction from an algorithmic perspective.

I will talk about a family of measures on partitions (specifically, a case of Okounkov's Schur measures) which are in one-to-one correspondence with models of random unitary matrices and lattice fermions. Under these measures, as the expected size of a partition goes to infinity, the first part of a random partition generically exhibits the same universal asymptotic fluctuations as the largest eigenvalue of a GUE random Hermitian matrix. First, I'll describe how we can tune these measures to exhibit new edge fluctuations at a smaller scale, which naturally generalise the GUE edge behaviour. These new fluctuations are universal, having previously been found for trapped fermions, and when a measure is tuned to have them, the corresponding unitary matrix model is "multicritical". Then, I'll describe how our measures can escape these more general universality classes, when tuned to have several cuts in a certain "Fermi sea". In this case, the breakdown in universality arises from an oscillation phenomenon previously observed in multi-cut Hermitian matrix models. Moreover, we have a one-to-one correspondence with multi-cut unitary matrix models. This is partly based on joint work with Dan Betea and Jérémie Bouttier. 

For almost 10 years, it has been known that if a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct a hyperbolic space where said element acts WPD. For a long time, the following question remained unanswered: if Morse is equivalent to strongly contracting, does there exist a space where all generalized loxodromics act WPD? In this talk, I will present a construction of a hyperbolic space, that answers this question positively.

Diffusion is one of the most common phenomenon in natural sciences and large part of applied mathematics have been interested in the tools to model it. Trying to study different types of diffusions, the mathematical ways to describe them and the numerical methods to simulate them is an appealing challenge, giving a wide range of applications. The aim of our work is the design of a finite-volume numerical scheme to model non-local diffusion given by the fractional Laplacian and to build numerical solutions for the Lévy-Fokker-Planck equation that involves it. Numerical methods for fractional diffusion have been indeed developed during the last few years and large part of the literature has been focused on finite element methods. Few results have been rather proposed for different techniques such as finite volumes.

 
We propose a new fractional Laplacian for bounded domains, which is expressed as a conservation law. This new approach is therefore particularly suitable for a finite volumes scheme and allows us also to prescribe no-flux boundary conditions explicitly. We enforce our new definition with a well-posedness theory for some cases to then capture with a good level of approximation the action of fractional Laplacian and its anomalous diffusion effect with our numerical scheme. The numerical solutions we get for the Lévy-Fokker-Planck equation resemble in fact the known analytical predictions and allow us to numerically explore properties of this equation and compute stationary states and long-time asymptotics.

We determine the minimal forbidden minors characterising the class of countable graphs that embed into some compact surface. We will also discuss Thomas’s conjecture that the Robertson—Seymour Graph Minor Theorem extends to countably infinite graphs. [https://arxiv.org/abs/2301.11042]

The theory of D-modules has found remarkable applications in various mathematical areas, for example, the representation theory of complex semi-simple Lie algebras. Two pivotal theorems in this field are the Beilinson-Bernstein Localisation Theorem and the Riemann-Hilbert Correspondence. This talk will explore a p-adic analogue. Ardakov-Wadsley introduced the sheaf D-cap of infinite order differential operators on a given smooth rigid-analytic variety to develop a p-adic counterpart for the Beilinson-Bernstein localisation. However, the classical approach to the Riemann-Hilbert Correspondence does not apply in the p-adic context. I will present an alternative approach, introducing a solution functor for D-cap-modules using new methods from p-adic Hodge theory.

Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in AdS3 pure gravity, which we formulate as a topological quantum field theory. We explain how gravitational entropy can be viewed as bulk entanglement entropy of gravitational edge modes.  These edge modes transform under a quantum group symmetry. This suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.  Time permitting we will discuss a proposal for how our bulk TQFT arises from an ensemble of approximate CFT’s, generalizing the relation between JT gravity and random matrix ensemble.

Following Rosati and Perkowski’s work on constructing the first version of a rough super Brownian motion, we generalize the rough super Brownian motion to the case when the branching mechanism has infinite variance. In both case, we can prove the compact support properties and the exponential persistence.

We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain, coupled to a Fokker-Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent.

The micro-macro interaction is reflected by the presence of a drag term in the Fokker-Planck equation and the divergence of a polymeric extra-stress tensor in the Navier-Stokes balance of momentum equation. In a simplified case where the drag term is corotational, we prove global existence of weak solutions and discuss some of their properties: we use the relative energy method to deduce a weak-strong uniqueness type result, and derive the macroscopic closure of the kinetic model: a corotational Oldroyd-B model with stress-diffusion.

In the general noncorotational case, we consider “generalised dissipative solutions” — a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation, which accounts for the lack of compactness in the polymeric extra-stress tensor. Joint work with Endre Suli (Oxford).

I will discuss a basic problem in analytic number theory which has appeared recently in my work. This will be a gentle introduction to the Gauss circle problem, hopefully with a discussion of some extensions and applications to understanding L-functions.

One way of studying infinite groups is by analysing
 their actions on classes of interesting spaces. This is the case
 for Kazhdan's property (T) and for Haagerup's property (also called a-T-menability),
 formulated in terms of actions on Hilbert spaces and relevant in many areas
(e.g. for the Baum-Connes conjectures, in combinatorics, for the study of expander graphs, in ergodic theory, etc.)
 
Recently, these properties have been reformulated for actions on Banach spaces,
with very interesting results. This talk will overview some of these reformulations
 and their applications. Part of the talk is on joint work with Ashot Minasyan and Mikael de la Salle, and with John Mackay.
 

We consider a model of network of interacting  neurons based on jump processes. Briefly, the membrane potential $V^i_t$ of each individual neuron evolves according to a one-dimensional ODE. Neuron $i$ spikes at rate which only depends on its membrane potential, $f(V^i_t)$. After a spike, $V^i_t$ is reset to a fixed value $V^{\mathrm{rest}}$. Simultaneously, the membrane potentials of any (post-synaptic) neuron $j$ connected to the neuron $i$ receives a kick of value $J^{i,j}$.

We study the limit (mean-field) equation obtained where the number of neurons goes to infinity. In this talk, we describe the long time behaviour of the solution. Depending on the intensity of the interactions, we observe convergence of the distribution to a unique invariant measure (small interactions) or we characterize the occurrence of spontaneous oscillations for  interactions in the neighbourhood of critical values.

There are two main methods of constructing compact manifolds with holonomy $G_2$, viz. resolution of singularities (first applied by Joyce) and twisted connect sum (first applied by Kovalev).  In the second case, there is a known invariant (the $\overline{\nu}$-invariant, introduced by Crowley–Goette–Nordström) which can, in many cases, be used to distinguish between different examples.  This invariant, however, has limitations; in particular, it cannot be computed on the $G_2$-manifolds constructed by resolution of singularities.

In this talk, I shall begin by discussing the notion of a $G_2$-manifold and the $\overline{\nu}$-invariant and its limitations.  In the context of this, I shall then introduce two new invariants of $G_2$-manifolds, termed $\mu$-invariants, and explain why these promise to overcome these limitations, in particular being well-suited to, and computable on, Joyce's examples of $G_2$-manifolds.  These invariants are related to $\eta$- and $\zeta$-invariants and should be regarded as the Morse indices of a $G_2$-manifold when it is viewed as a critical point of certain Hitchin functionals.  Time permitting, I shall explain how to prove a closed formula for the invariants on the orbifolds used in Joyce's construction, using Epstein $\zeta$-functions.

We will be joined by people with mentoring experience to discuss the importance of both having and being a good mentor.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. Despite intense interest in the higher-order connectivity of these networks, their Betti numbers at higher dimensions have been largely unexplored.

In this talk, after a brief survey on random topology, we study the clique complexes of preferential attachment graphs, and we prove the asymptotics of the expected Betti numbers. If time allows, we will briefly discuss their homotopy connectedness as well. This is joint work with Gennady Samorodnitsky, Christina Lee Yu and Rongyi He, and it is based on the preprint https://arxiv.org/abs/2305.11259

A major reason for the failure of cancer treatment and disease progression is the heterogeneous composition of tumor cells at the genetic, epigenetic, and phenotypic levels. Despite extensive efforts to characterize the makeup of individual cells, there is still much to be learned about the interactions between heterogeneous cancer cells and between cancer cells and the microenvironment in the context of cancer invasion. Clinical studies and in vivo models have shown that cancer invasion predominantly occurs through collective invasion packs, which invade more aggressively and result in worse outcomes. In vitro experiments on non-small cell lung cancer spheroids have demonstrated that the invasion packs consist of leaders and followers who engage in mutualistic social interactions during collective invasion. Many fundamental questions remain unanswered: What is the division of labor within the heterogeneous invasion pack? How does the leader phenotype emerge? Are the phenotypes plastic? What's the role of the individual "cheaters"? How does the invasion pack interact with the stroma? Can the social interaction network be exploited to devise novel treatment strategies? I will discuss recent modeling efforts to address these questions and hope to convince you that identifying and perturbing the "weak links" within the social interaction network can disrupt collective invasion and potentially prevent the malignant progression of cancer. 

The Celestial Holography program encompasses recent efforts to understand the flat space hologram in terms of a CFT living on the celestial sphere. Here we have fun relating various extrapolate dictionaries in CCFT and examining tools we can apply when perturbing around a 4D CFT in the bulk.

Cluster algebras and their quantum counterparts were invented in the early 2000s in an attempt to construct elements of dual canonical bases. This turned out to be a harder goal than first realised. In this talk I will aim to give an introduction and overview of the theory and display the wide range of interesting maths which has gone into making steps in this area. I will try to assume as little possible prior knowledge and instead focus on interesting questions which remain open in this area.

I will present a generalisation of the Elekes-Szabó result, that any ternary algebraic relation in characteristic 0 having large intersections with (certain) finite grids must essentially be the graph of a group law, to a version where one obtains an algebraic group action. In the end the conclusion will be similar, but with weaker assumptions. This is recent work with Tingxiang Zou.

We consider an offline learning problem for an agent who first estimates an unknown price impact kernel from a static dataset, and then designs strategies to liquidate a risky asset while creating transient price impact. We propose a novel approach for a nonparametric estimation of the propagator from a dataset containing correlated price trajectories, trading signals and metaorders. We quantify the accuracy of the estimated propagator using a metric which depends explicitly on the dataset. We show that a trader who tries to minimise her execution costs by using a greedy strategy purely based on the estimated propagator will encounter suboptimality due to spurious correlation between the trading strategy and the estimator. By adopting an offline reinforcement learning approach, we introduce a pessimistic loss functional taking the uncertainty of the estimated propagator into account, with an optimiser which eliminates the spurious correlation, and derive an asymptotically optimal bound on the execution costs even without precise information on the true propagator. Numerical experiments are included to demonstrate the effectiveness of the proposed propagator estimator and the pessimistic trading strategy.

Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x²+y²=z²)? This is one of the simplest questions in arithmetic Ramsey theory which is still open. I will present a recent partial result, showing that in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x²+y²=z² for some integer z which may be in a different cell. 

The proof consists, after some initial maneuvers inspired by ergodic theory, in controlling the behavior of completely multiplicative functions along certain quadratic polynomials. Considering separately aperiodic and "pretentious" functions, the last major ingredient is a concentration estimate for functions in the latter class when evaluated along sums of two squares.

The talk is based on joint work with Frantzikinakis and Klurman.

Alan Weinstein, introduced the concept of "fat bundle" as a tool to understand when the total space of a fiber bundle with totally geodesic fibers allows a metric with positive sectional curvature. 

In recent times, certain weaker notions than the condition of having a metric with positive sectional curvature have been studied due to the apparent scarcity of spaces that meet this condition. Positive k^th-intermediate Ricci curvature (Ric_k > 0) on a Riemannian manifold Mⁿ is a condition that bridges the gap between positive sectional curvature and positive Ricci curvature. Indeed, when k = 1, this condition corresponds to positive sectional curvature, and when k = n−1, it corresponds to positive Ricci curvature. 

In this talk, I will discuss an ongoing project with Miguel Domínguez Vázquez, David González-Álvaro, and Jason DeVito, which aims to create new examples of compact Riemannian manifolds with Ric₂ > 0. We achieve this by employing a certain generalisation of the "fat bundle" concept.

Optimization problems subject to PDE constraints constitute a mathematical tool that can be applied to a wide range of scientific processes, including fluid flow control, medical imaging, option pricing, biological and chemical processes, and electromagnetic inverse problems, to name a few. These problems involve minimizing some function arising from a particular physical objective, while at the same time obeying a system of PDEs which describe the process. It is necessary to obtain accurate solutions to such problems within a reasonable CPU time, in particular for time-dependent problems, for which the “all-at-once” solution can lead to extremely large linear systems.

In this talk we consider iterative methods, in particular Krylov subspace methods, to solve such systems, accelerated by fast and robust preconditioning strategies. In particular, we will survey several new developments, including block preconditioners for fluid flow control problems, a circulant preconditioning framework for solving certain optimization problems constrained by fractional differential equations, and multiple saddle-point preconditioners for block tridiagonal linear systems. We will illustrate the benefit of using these new approaches through a range of numerical experiments.

This talk is based on work with Santolo Leveque (Scuola Normale Superiore, Pisa), Spyros Pougkakiotis (Yale University), Jacek Gondzio (University of Edinburgh), and Andreas Potschka (TU Clausthal).

A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. In this talk, we will present a reduced-order dynamical system derived from the lubrication model based on the nearest-neighbour droplet interactions in the weak condensation limit. Dynamics for periodic arrays of identical drops and pairwise droplet interactions are investigated which provide insights to the coarsening dynamics of a large droplet system. Weak condensation is shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics for the droplets and the overall mass of the fluid in two additional regimes of long-time dynamics. This is joint work with Thomas Witelski.

The key role of physical and mechanical interactions in cancer emerges from a very large variety of data sources and methods - from genomics to bioimaging, from proteomics to clinical records. Thus, learning physics-driven relational information is crucial to characterize its progression at different scales.

In this talk I will discuss how mathematical and computational tools allow for learning  and better understanding of  the mechano-biology of cancer,  thanks to the integration of  patient-specific data and physics-based models. I will present a few applications developed in the last decade in which the development of  digital twins,  empowered by ad-hoc learning tools,  allows us to test new hypotheses,  to assess the model predictions against biological and clinical data, and to aid decision-making in a clinical setting.

Funding from MUR - PRIN 2020, Progetto di Eccellenza 2023-2027 and Regione Lombardia (NEWMED Grant, ID: 117599, POR FESR 2014-2020) is gratefully acknowledged.

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

The space of subgroups of a countable group is a compact topological space which encodes many of the properties of its non-free actions. We will discuss some approaches to studying the Cantor-Bendixson decomposition of this space in the context of hyperbolic groups and groups which act (nicely) on trees. We will also give some conditions under which the conjugation action on the perfect kernel is highly topologically transitive and see how this can be applied to find new examples of groups (including all virtually compact special groups) which admit faithful transitive amenable actions. This is joint work with Damien Gaboriau.