Past

Bacteria represent the major component of the world’s biomass. A number of these bacteria are motile and swim with the use of flagellar filaments, which are slender helical appendages attached to a cell body by a flexible hook. Low Reynolds number hydrodynamics is the key for flagella to generate propulsion at a microscale [1]. In this talk I will discuss two projects related to swimming of a model bacterium Escherichia coli (E. coli).

E. coli has many flagellar filaments that are wrapped in a bundle and rotate in a counterclockwise fashion (if viewed from behind the cell) during the so-called ‘runs’, wherein the cell moves steadily forward. In between runs, the cell undergoes quick ‘tumble’ events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. In the first part of the talk, I will discuss how cells reorient during tumble and the mechanical forces at play and show the predominant role of hydrodynamics in setting the reorientation angle [2].

In the second part, I will talk about hydrodynamics of bacteria near walls in visco-elastic fluids. Flagellar motility next to surfaces in such fluids is crucial for bacterial transport and biofilm formation. In Newtonian fluids, bacteria are known to accumulate near walls where they swim in circles [3,4], while experimental results from our collaborators at the Wu Lab (Chinese University of Hong Kong) show that in polymeric liquids this accumulation is significantly reduced. We use a combination of analytical and numerical models to propose that this reduction is due to a viscoelastic lift directed away from the plane wall induced by flagellar rotation. This viscoelastic lift force weakens hydrodynamic interaction between flagellated swimmers and nearby surfaces, which results in a decrease in surface accumulation for the cells. 

References

[1] Lauga, Eric. "Bacterial hydrodynamics." Annual Review of Fluid Mechanics 48 (2016): 105-130.

[2] Dvoriashyna, Mariia, and Eric Lauga. "Hydrodynamics and direction change of tumbling bacteria." Plos one 16.7 (2021): e0254551.

[3] Berke, Allison P., et al. "Hydrodynamic attraction of swimming microorganisms by surfaces." Physical Review Letters 101.3 (2008): 038102.

[4] Lauga, Eric, et al. "Swimming in circles: motion of bacteria near solid boundaries." Biophysical journal 90.2 (2006): 400-412.

On a smooth Riemannian manifold, the uniqueness of a geodesic given initial conditions follows from standard ODE theory. This is known to fail in the setting of RCD(K,N) spaces (metric measure spaces satisfying a synthetic notion of Ricci curvature bounded below) through an example of Cheeger-Colding. Strengthening the assumption a little, one may ask if two geodesics which agree for a definite amount of time must continue on the same trajectory. In this talk, I will show that this is true for RCD(K,N) spaces. In doing so, I will generalize a well-known result of Colding-Naber concerning the Hölder continuity of small balls along geodesics to this setting.

3d N=4 gauge theories can be studied on a circle times a closed Riemann surface. Their partition functions on this geometry, known as twisted indices, were computed some time ago using supersymmetric localisation on the Coulomb branch. An alternative perspective is to consider the theory as a supersymmetric quantum mechanics on S^1. In this talk I will review this point of view, which unveils interesting connection to topics in geometry such as wall-crossing and symplectic duality of quasi-maps.

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

We prove that with high probability $\mathbb{G}^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system for $n\equiv 1,3\pmod{6}$, establishing the exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.
This is joint work with Ashwin Sah and Michael Simkin. 

Characterising the statistical properties of high dimensional random functions has been one of the central focus of the theory of disordered systems, and notably spin glasses, over the last decades. Applications to machine learning via deep neural network has seen a resurgence of interest towards this problem in recent years. The simplest yet non-trivial quantity to characterise these landscapes is the annealed total complexity, i.e. the rate of exponential growth of the average number of stationary points (or equilibria) with the dimension of the underlying space. A paradigmatic model for such random landscape in the $N$-dimensional Euclidean space consists of an isotropic harmonic confinement and a Gaussian random function, with rotationally and translationally invariant covariance [1]. The total annealed complexity in this model has been shown to display a ”topology trivialisation transition”: for weak confinement, the number of stationary points is exponentially large (positive complexity) while for strong confinement there is typically a single stationary point (zero complexity).

In this talk, I will present recent results obtained for a distinct exactly solvable model of random lanscape in the $N$-dimensional Euclidean space where the random Gaussian function is replaced by a superposition of $M > N$ random plane waves [2]. In this model, we compute the total annealed complexity in the limit $N\rightarrow\infty$ with $\alpha = M/N$ fixed and find, in contrast to the scenario exposed above, that the complexity remains strictly positive for any finite value of the confinement strength. Hence, there is no ”topology trivialisation transition” for this model, which seems to be a representative of a distinct class of universality.

References:

[1] Y. V. Fyodorov, Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices, Phys. Rev. Lett. 92, 240601 (2004) Erratum: Phys. Rev. Lett. 93, 149901(E) (2004).

[2] B. Lacroix-A-Chez-Toine, S. Belga-Fedeli, Y. V. Fyodorov, Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity, arXiv preprint arXiv:2202.03815, submitted to J. Math. Phys.

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain or a parameter in the equation. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving an optimization problem constrained by an augmented system of equations that characterize the location of the branch points. The flexibility and robustness of the method also allow us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency. We will apply this technique on systems arising from biology, fluid dynamics, and engineering, such as the FitzHugh-Nagumo model, Navier-Stokes, and hyperelasticity equations.

This talk is about the mathematics behind an artistic project focusing on the vibrations of soap films.

In the past few years, the study of the geometric properties of random maps has been extended to a new regime, the "high genus regime", where we are interested in maps whose size and genus tend to infinity at the same time, at the same rate.
We consider here a slightly different case, where the genus also tends to infinity, but less rapidly than the size, and we study the law of simple cycles (with a well-chosen rescaling of the graph distance) in unicellular maps (maps with one face), thanks to a powerful bijection of Chapuy, Féray and Fusy.
The interest of this work is that we obtain exactly the same law as Mirzakhani and Petri who counted closed geodesics on a model of random hyperbolic surfaces in large genus (the Weil-Petersson measure). This leads us to conjecture that these two models are somehow "the same" in the limit. This is joint work with Svante Janson.

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

Detectors in collider experiments are modeled by light-ray operators in Quantum Field Theory. For example, energy detectors are certain null integrals of the stress-energy tensor, localized at an angle on the celestial sphere, where they collect quanta that escape in their direction. In this talk, I will discuss a series of work developing a nonperturbative, convergent operator product expansion (OPE) for light-ray operators in Conformal Field Theories (CFTs). Objects appearing in the expansion are more general light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. An important application is to event shapes in collider physics, which correspond to correlation functions of light-ray operators within the state created by the incoming particles. I will discuss some applications of the light-ray OPE in CFT, and mention some extensions to QCD which make contact with measurements at the LHC. Talk based primarily on [1905.01311] and [2010.04726].

Synchronization, epidemic processes and information spreading are natural processes that emerge from the interaction between people. Mathematically, all of them can be described by models that, despite their simplicity, they can predict collective behaviors. In addition, they have in common a very interesting feature: a phase transition from an active state to an absorbing state. For example, the spread of an epidemic is characterized by the infection rate, the control parameter, which basically determines whether the epidemic will spread in the network or, if this rate is very low, few people become infected and the system falls into an absorbing state where there are no more infected people. In this presentation we will present the analytical and computational approaches used to investigate these classical models of statistical physics that present phase transitions and we will also show how the network topology influences such dynamical processes. The behavior of such dynamics can be much richer than we imagine.

We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb R$, is realisable over the field $E$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $\rho$ over $\mathbb Q(\zeta_n)$ to one over $E$.

The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons.

Black hole horizons are normally at finite spatial distance from the exterior region, but when they are degenerate (or extreme as they are usually referred to in this case) the spatial distance becomes infinite. One can still fall into the black hole in finite proper time but the crossing sphere is replaced by an "internal infinity". Near to the horizon of an extreme Kerr black hole, the scattering properties of test fields bear some similarities to what happens at an asymptotically flat infinity. This observation triggered a natural question concerning the peeling behaviour of test fields near such horizons. A geometrical tool known as the Couch-Torrence inversion is particularly well suited to studying this question. In this talk, I shall recall some essential notions on the peeling of fields at an asymptotically flat infinity and describe the Couch-Torrence inversion in the particular case of extreme Reissner-Nordström black holes, where it acts as a global conformal isometry of the spacetime. I will then show how to extend this inversion to more general spherically symmetric extreme horizons and describe what results can be obtained in terms of peeling. This is a joint ongoing project with Jack Borthwick (University of Besançon) and Eric Gourgoulhon (Paris Observatory).

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Recently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université).

Via Poincaré duality, fundamental groups of aspherical manifolds have (appropriately shifted) isomorphisms between their homology and cohomology. In a 1973 Inventiones paper, Bieri and Eckmann defined a broader notion of a Duality Group, where the isomorphism between homology and cohomology can be twisted by what they called a Dualizing Module. Examples of these groups in geometric group theory (after passing to a finite-index subgroup) include $GL(n,\mathbb{Z})$, mapping class groups, and automorphism groups of free groups.

In work-in-progress with Thomas Wasserman we are looking into the following puzzle: the examples of duality groups that we know of that do not come from manifolds all have classifying spaces that satisfy a weaker local condition called the Cohen-Macaulay property. These spaces also satisfy weaker (twisted) versions of Poincaé duality via their local homology sheaves (or local cohomology cosheaves), and we are attempting to understand more about the links between these geometric versions of duality and the algebraic notion of a duality group. The goal of the talk is to explain more about the words used in the above paragraphs and say where we have got to so far.


 

I will introduce a class of mean-field games under forward performance and for general risk preferences. Players interact through competition in fund management, driven by relative performance concerns in an asset diversification setting. This results in a common-noise mean field game. I will present the value and the optimal policies of such games, as well as some concrete examples. I will also discuss the partial information case, i.e.. when the risk premium is not directly observed. 

A loop of Hamiltonian diffeomorphisms of a symplectic manifold $X$ defines, by clutching, a symplectic fibration over the two-sphere with fibre $X$.  We prove that the integral cohomology of the total space splits additively, answering a question of McDuff, and extending the rational cohomology analogue proved by Lalonde-McDuff-Polterovich in the late 1990’s. The proof uses a virtual fundamental class of moduli spaces of sections of the fibration in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for the definition of robust loss functions such as the square-root lasso.  Standard approaches to deal with non-smoothness leverage either proximal splitting or coordinate descent. The effectiveness of their usage typically depend on proper parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a different route. By over-parameterization and marginalising on certain variables (Variable Projection), we show how many popular non-smooth structured problems can be written as smooth optimization problems. The result is that one can then take advantage of quasi-Newton solvers such as L-BFGS and this, in practice, can lead to substantial performance gains. Another interesting aspect of our proposed solver is its efficiency when handling imaging problems that arise from fine discretizations (unlike proximal methods such as ISTA whose convergence is known to have exponential dependency on dimension). On a theoretical level, one can connect gradient descent on our over-parameterized formulation with mirror descent with a varying Hessian metric. This observation can then be used to derive dimension free convergence bounds and explains the efficiency of our method in the fine-grids regime.

The line operators in a 2+1D TQFT form an algebraic structure called a modular tensor category (MTC). There is a natural action of a Galois group on MTCs which maps a given TQFT to other 'Galois conjugate' TQFTs. I will describe this Galois action and give several examples of Galois conjugate TQFTs. Galois action on a unitary TQFT can result in a non-unitary TQFT. I will derive a sufficient condition under which unitarity is preserved. Finally, I will describe the invariance of 0-form and 1-form symmetries of TQFTs under Galois action.    

I'll review 2104.13932, where we analyze the supersymmetric index of N=4 SU(N) Super Yang-Mills using the Bethe Ansatz approach, expressing it as a sum and concentrating on some family of contributions to the sum. We show that in the large N limit each term in this family corresponds to the contribution of a different euclidean black hole to the partition function of the dual gravitational theory. By taking into account non-perturbative contributions (wrapped D3-branes), we further show a one to one match between the contributions of the gravitational saddles and this family of contributions to the index, both at the perturbative and non-perturbative levels. I'll end with some new results regarding the Bethe Ansatz expansion and the information one could extract from it.