Seminars and short courses

The aim of this seminar is to present an overview of the porous medium model and its hydrodynamic equation, the porous medium equation. We will focus on exploring the main characteristics of this equation and how we can see it from the particle system's point of view.

In this talk we will discuss the fundamental aspects of the theory of compensated compactness in the general A-free framework developed by Murat and Tartar, under the assumption that A has constant rank. We prove sharp weak continuity results for the compensated compactness quantities and we show that they are precisely the nonlinear quantities with Hardy space integrability, thus proposing an answer to a question raised by Coifman-Lions-Meyer-Semmes. This gives a link between compensated compactness and compensated regularity.

In this talk, I am going to review some aspects of the current state of the art of Integrability in the AdS/CFT correspondence and beyond. I will first review a general nonperturbative approach to compute multipoint correlation functions of local operators in the $N=4$ SYM theory which allows us to explore the theory even beyond the planar level. In the second part, I will describe my recent work about exploring deformations of $N=4$ SYM by irrelevant operators, which revives an old attempt of generalizing the AdS/CFT correspondence. Here integrability seems to also play an important role and opens the door for its application for non-conformal field theories.

This will be an informal talk on open problems in theoretical cosmology, suitable for a (relatively) general audience.

Ribbon categories are $3$-dimensional algebraic structures that control quantum link polynomials and that give rise to $3$-manifold invariants known as skein modules. I will describe how to use Khovanov-Rozansky link homology, a categorification of the $\operatorname{\mathfrak{gl}}(N)$ quantum link polynomial, to obtain a $4$-dimensional algebraic structure that gives rise to vector space-valued invariants of smooth $4$-manifolds. The technical heart of this construction is the newly established functoriality of Khovanov-Rozansky homology in the $3$-sphere. Based on joint work with Scott Morrison and Kevin Walker https://arxiv.org/abs/1907.12194.

Extremal black holes show the presence of an $\operatorname{AdS}_2$ factor as an universal feature. This fact provides a strong motivation for getting a deeper understanding of $\operatorname{AdS}_2$ spacetimes. In this talk, I will argue how $\operatorname{AdS}_2$ systems have a natural description in terms of $1d$ Calogero-type models and, in turn to SLE curves, which describe the geodesic motion of particles in $\operatorname{AdS}_2$. This treatment allows to compute the dimension of the phase space of these geodesics, linking it to the leading Bekenstein-Hawking black hole entropy and the black hole degeneracy.

There are several recent important results concerning the massless Einstein-Vlasov system such as the stability of Minkowski space or the stability of De Sitter spacetime. In this talk I will present some recent progress on the massless Einstein-Boltzmann system in both directions, initial singularity and long time behaviour:

• The massless Einstein-Boltzmann system with a conformal-gauge singularity in an FLRW background: https://arxiv.org/abs/1903.06067
• On the future of solutions to the massless Einstein-Vlasov system in a Bianchi I cosmology: https://arxiv.org/abs/1911.04937

This is joint work with Ho Lee and Paul Tod.

Geometric variational problems frequently lead to analytically extremely hard, non-linear partial differential equations, where the standard methods fail. Thus finding non-trivial solutions is challenging. The idea is to study solutions with a certain minimum level of symmetry (i.e. group actions with low cohomogeneity), and use the symmetry to reduce the original problem to systems of non-linear ordinary differential equations, typically with singular boundary values. In my talk I explain how to construct harmonic mappings between manifolds with a lot of symmetry (i.e. cohomogeneity one manifolds). If time permits, I will discuss applications of the developed methods.

Non-abelian statistics of anyons in two dimensions have attracted considerable interest in the past 20 years, in part due to the potential for realising fault-tolerant quantum computation. In comparison, in three dimensions there exists a no-go theorem for point particles realising non-abelian statistics. However, three-dimensional condensed matter systems naturally support spatially extended excitations, such as loops, which can admit non-abelian statistics.

In this talk I will give a brief overview of topological quantum computing with anyons, utilising the connections between the mathematics of topological quantum field theories and Hamiltonian models of topological phases of matter. Building on this connection I will then discuss the ongoing work of describing non-abelian exchange statistics of loop excitations in three dimensions and the potential applications to quantum computation.

The quantum double is a quasi-triangular Hopf algebra whose category of representations can be interpreted physically as describing the processes of fusion and braiding of anyons in the $2+1D$ Dijkgraaf-Witten TQFT. Motivated by the possibilities of topological quantum computing in $3+1D$, in this talk I will give an informal overview of my ongoing research towards understanding the categorified quantum double and its bicategory of $2$-representations. In particular, I will focus on the relation between such constructions and the Hamiltonian formulation of $3+1D$ Dijkgraaf-Witten TQFT in order to describe the braiding and fusion of extended excitations such as loops.

The algebraic structure of the moduli spaces of representations of surface groups (aka character varieties) has been widely studied due to their tight relation with moduli spaces of Higgs bundles. In particular, Hodge-type invariants, like the so-called E-polynomial, has been objective of intense research over the past decades. However, subtler algebraic invariants as their motivic classes in the Grothendieck ring of algebraic varieties remain unknown in the general case.

In this talk, we will construct a Topological Quantum Field Theory that computes the motivic classes of representation varieties. This tool gives rise to an effective computational method based on topological recursion on the genus of the surface. As application, we will use it to compute the motivic classes of parabolic $\operatorname{SL}(2,\mathbb{C})$-character varieties over any compact orientable surface.

In our journey towards homological mirror symmetry, we will resume our study of the derived category of coherent sheaves focusing on the functors involved in the definition of Fourier-Mukai transforms: push-forward, pull-back and tensor product. We will describe the associated derived functors and some relations among them, namely the projection formula and base change theorems.

Classically, there is a strong relationship between the shape of a Riemannian manifold and the spectrum of its Laplace-operator, and, more generally, with the spectrum of Laplace type operators. In particular, for the spectrum of the Laplacian, the presence of a large first nonzero eigenvalue is related to some concentration phenomena. In the first part of the talk, I will recall these classical relations. In the second part of the talk, I will introduce the Dirichlet-to-Neumann operator on a manifold with boundary. I will survey the same types of questions in this context (existence of large first nonzero eigenvalue, concentration phenomena) without going into the technical details. This second part corresponds to work in progress with Alexandre Girouard.

Since the 1980s, the study of invariants of 3-dimensional manifolds has benefited from the connections between topology, physics and number theory. Motivated by the recent discovery of a new homological invariant (corresponding to the half-index of certain $3d$ $N=2$ theories), in this talk I describe the role of quantum modular forms, mock and false theta functions in the study of $3$-manifold invariants. The talk is based on 1809.10148 and work in progress with Cheng, Chun, Feigin, Gukov, and Harrison.

I will introduce a measure of quantum-ness in quantum multi-parameter estimation problems. One can show that the ratio between the mean Uhlmann Curvature and the Fisher Information provides a figure of merit which estimates the amount of incompatibility arising from the quantum nature of the underlying physical system. This ratio accounts for the discrepancy between the attainable precision in the simultaneous estimation of multiple parameters and the precision predicted by the Cramér-Rao bound. We apply this measure to quantitatively assess the quantum character of phase transition phenomena in peculiar quantum critical models. We consider a paradigmatic class of lattice fermion systems, which shows equilibrium quantum phase transition and dissipative non-equilibrium steady-state phase transitions.

The aim of this work is the analysis of fluctuations around equilibrium for a diffusive and symmetric exclusion process with long jumps and infinitely extended reservoirs (introduced in [BGJO]). In particular we study how the parameters characterizing the model change the behavior of the stochastic fluctuations of the macroscopic density of particles around the equilibrium. We will see how the SPDE involved will pass from the one which solution is a generalized Ornstein-Uhlenbeck process when the reservoirs are weak, to an SPDE without diffusive term when the reservoirs are so strong that the fluctuations are only caused by their action.

Joint work with C. Bernardin, P. Gonçalves and M. Jara.

Reference

[BGJO] Bernardin, C., Gonçalves, P. and Oviedo Jimenez, B. Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps.

Motivated by its role in homological mirror symmetry, this talk will introduce the derived category of coherent sheaves on a smooth projective variety. We will see that many of the typical operations on coherent sheaves can be lifted to derived functors, respecting the triangulated structure of the derived category. This paves the way for the systematic study of these functors as Fourier-Mukai transforms.

The aim of this talk is to discuss the problem of classifying Kaehler-Einstein manifolds which admit an isometric and holomorphic immersion into the complex projective space. We start giving an overview of the problem focusing in particular on the Ricci-flat case. Ricci-flat non-flat Kaehler manifolds are conjectured to be not projectively induced. Next, we give evidence to this conjecture for Calabi’s Ricci-flat metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds.

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows for the explicit construction of solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results. As an illustration, we show that by factorizing a simple rational diagonal matrix, we explicitly obtain a class of Weyl metrics which includes, in particular, the solution describing the interior region of the Schwarzschild black hole, a cosmological Kasner solution and the Rindler metric.

I’ll say a few words about some homotopical (“higher”) methods to study knot spaces and diffeomorphism groups. A fascinating appearance, and one of my current obsessions, is made by the absolute Galois group of the rationals.

We describe an approach of how to find a $2$-group generalization of the spin-network basis from Loop Quantum Gravity. This gives a basis of spin-foam functions which are generalizations of the Wilson surface holonomy invariant for the $3$-dimensional Euclidean $2$-group.

This will be an introductory talk for the Working Seminar on Mirror Symmetry on the Hitchin System. During this minicourse, organized by T. Sutherland and myself, we aim to understand Mirror Symmetry on Higgs moduli spaces as a classical limit of the Geometric Langlands program. In this talk I will briefly describe the geometrical objects involved in this program and provide a motivation for it coming from mathematical physics. The structure of the working seminar will also be discussed.

This will be an introductory talk for the Working Seminar on Mirror Symmetry on the Hitchin System. During this minicourse, organized by T. Sutherland and myself, we aim to understand Mirror Symmetry on Higgs moduli spaces as a classical limit of the Geometric Langlands program. In this talk I will briefly describe the geometrical objects involved in this program and provide a motivation for it coming from mathematical physics. The structure of the working seminar will also be discussed.

I will explain the notion of coherence for duals and adjoints in higher categories. I will discuss a strategy for using knowledge of coherence data and the cobordism hypothesis to give presentations of fully extended bordism categories and mention some progress in dimension $3$.

My recent work has concerned two projects, both concerned with higher structures on supermanifolds. I will probably focus on my work extending the classification theorems for gerbes to supermanifolds, with an eye to examples on super Lie groups. My other project, that I will probably not discuss, concerns applying cyclic cohomology to super-Yang-Mills theory in the so-called superspace formalism (i.e., formulated as a theory on a supermanifold).

The $2$-representation theory of $2$-categories is a categorical analogue of the representation theory of algebras. In my talk, I will recall its origins, its purposes, its basic features and explain some important examples. This talk is based on joint work with Mazorchuk, Miemietz, Tubbenhauer and Zhang.

I’ll try to give a brief description of the categorification of colored HOMFLY polynomial for knots, and discuss various open problems related with these invariants, including their definition, properties, relationship with string theory, as well as surprising combinatorial identities and properties of these invariants in special cases of torus knots and rational knots.

We look into persistent tangles i.e., tangles such that its presence in a diagram implies that diagram is knotted and remark their prevalence. We also look into hyperfinite knots i.e., limits of sequences of classes of knots and relate them to wild knots.

In previous works we considered a model for topological recursion based on the Hopf Algebra of planar binary trees of Loday and Ronco and showed that extending this Hopf Algebra by identifying pairs of nearest neighbour leaves and thus producing graphs with loops we obtain the full recursion formula of Eynard and Orantin. We also discussed the algebraic structure of the spaces of correlation functions in $g=0$ and in $g\gt 0$. By taking a classical and a quantum product respectively we endowed both spaces with a ring structure. Here we will show that the extended algebra of graphs is in fact a Hopf algebra and can be seen as a sort of quantization of the Loday-Ronco Hopf algebra. This is work in progress.

Under the heading higher gauge theory I will briefy discuss:

• gerbes on supergroups (to be described by John Huerta in his talk).
• the relation between 2-group (discretized) connections and transports (with Jeff Morton).
• surface transport in 3D quantum gravity (with Jeanette Nelson).

I will then focus on two projects:

• development of the Eisermann invariant of knots (based on previous work with João Faria Martins).
• (higher) gauge theoretical models for anyons in topological quantum computation.

In this seminar we consider last passage percolation on $\mathbb{Z}^2$, a model in the Kardar–Parisi–Zhang (KPZ) universality class. We will investigate the universality of the limit distributions of the last passage time for different settings. In the first part we analyze the correlations of two last passage times for different ending points in a neighbourhood of the characteristic. For a general class of random initial conditions, we prove the universality of the first order correction when the two observation times are close. In the second part we consider a model of last passage percolation in half-space and we obtain the distribution of the last passage time for the stationary initial condition.

In this talk, rather than presenting new results, I will discuss relatively recent results in condensed matter which bring geometry and topology to the realm of quantum matter. In particular, I will focus on systems of free fermions. The ground state of a gapped translation invariant charge conserving free fermion Hamiltonian on a $d$-dimensional lattice can be described by a smooth map from a $d$-dimensional torus of quasi-momenta to a Grassmannian manifold. This map gives rise to a vector bundle over the torus whose isomorphism class determines the topological phase of the system. In particular, in $d = 2$, the Chern class can be naturally associated with the transverse Hall conductivity of the system. By considering families of systems of free fermions one can see phase transitions associated to the gap closing points in momentum space. The change in topology can be understood in terms of a transversal crossing of the image of the Hamiltonian in the space of Hermitian matrices with the subvariety formed by those matrices having multiple eigenvalues. Further geometric aspects will be discussed if time permits to do so.

The aim of this talk is to present a few models in the Kardar–Parisi–Zhang (KPZ) universality class, a class of stochastic growth models that have been widely studied in the last 30 years. We will focus in particular on last passage percolation (LPP) models. They provide a physical description of combinatorial problems, such as Ulam's problem, in terms of zero temperature directed polymers; but also a geometric interpretation of an interacting particle system, the totally asymmetric simple exclusion process (TASEP); and of a system of queues and servers. Moreover, in the large time limit, they share statistical features with certain ensembles of random matrices.

Large deviations have importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations correspond to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lower bound large deviations.

In 1952, Dirac proved that any graph on $n$ vertices with minimum degree $n/2$ contains a Hamiltonian cycle, i.e. a cycle which passes through every vertex of the graph exactly once. We prove a resilience version of Dirac’s Theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\epsilon \gt 0$, a.a.s. the following holds: let $G_0$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2 + \epsilon)d$. Then, $G_0$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved. This is joint work with Padraig Condon, Alberto Espuny Díaz, Daniela Kuhn and Deryk Osthus.

Large deviations have importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations correspond to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lower bound large deviations.

In this talk, we are going to present a user-friendly approach to derived geometry. One of our goals is to convince the audience that the notions of derived manifold / scheme / space / stack are just as natural as their classical counterparts. After having introduced the basic techniques, we will apply them to study certain moduli spaces of geometrical origin. Derived geometry also has recently found applications in arithmetics, which we will try to explain in the last part of the talk.

I will give an overview of some aspects of $3d$ TFT, from the Turaev-Viro and Reshetikin-Turaev invariants of oriented $3$-manifolds, to the more recent classifications of fully extended theories in terms of fusion categories and once extended theories in terms of Modular Tensor Categories.

I will discuss BPS invariants associated with quantum line operators in certain supersymmetric quantum field theories. Such operators can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR they are described by representation theory data. I will discuss the associated BPS spectral problem and the relevant indices.

Large deviations have importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations correspond to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lowerbound large deviations.

The decay of solutions to the Klein-Gordon equation is studied in two expanding cosmological spacetimes, namely the de Sitter universe in flat Friedmann-Lemaître-Robertson-Walker (FLRW) form, and the cosmological region of the Reissner-Nordström-de Sitter (RNdS) model. Using energy methods, for initial data with finite higher order energies, decay rates for the solution are obtained. Also, a previously established decay rate of the time derivative of the solution to the wave equation, in an expanding de Sitter universe in flat FLRW form, is improved, proving Rendall's conjecture. A similar improvement is also given for the wave equation in the cosmological region of the RNdS spacetime.

In this talk I will cover the recent work of M. Mariño and I (https://arxiv.org/abs/1905.09569, https://arxiv.org/abs/1905.09575) about an application of resurgence to superconductive quantum many-body systems. I will start by introducing the core idea of resurgence. Then, I will overview how to use the TBA to find the perturbative series of the ground-state of the Gaudin-Yang model (and other integrable models) to all orders. Finally, I will show how a resurgence analysis of such series connects to superconductivity and renormalon effects, leading to a concrete conjecture linking the Borel-summability of the perturbative series to the superconductor energy gap.

The idea of this informal seminar is to present, in cherry-picking fashion, some fascinating work by Ciaglia, Ibort and Marmo, who formulate the Schwinger approach to quantum theory using the contemporary language of groupoids and $2$-groupoids. By bringing in the notion of quantum measures, due to Sorkin, they achieve an elegant description of examples like the qubit or the two-slit experiment.

The aim is for the seminar to be comprehensible also to students with some awareness of quantum theory, and hopefully will be followed up by future seminars around quantum maths topics such as topological quantum computation and entanglement.

Main references

Ciaglia, Ibort, Marmo: https://arxiv.org/abs/1905.12274, https://arxiv.org/abs/1907.03883, https://arxiv.org/abs/1909.07265

We say that a Higgs bundle $E$ over a projective variety $X$ is curve semistable if for every morphism $f : C \to X$, where $C$ is a smooth irreducible projective curve, the pullback $f^\ast E$ is semistable. We study this class of Higgs bundles, reviewing the status of a conjecture about their Chern classes.

We use the theory of nonlinear fluctuating hydrodynamics to study stochastic transport far from thermal equilibrium in terms of the dynamical structure function which is universal at low frequencies and for large times and which encodes whether transport is diffusive or anomalous. For generic one-dimensional systems we predict that transport of mass, energy and other locally conserved quantities is governed by mode-dependent dynamical universality classes with dynamical exponents $z$ which are Kepler ratios of neighboring Fibonacci numbers, starting with $z = 2$ (corresponding to a diffusive mode) or $z = 3/2$ (Kardar-Parisi-Zhang (KPZ) mode). If neither a diffusive nor a KPZ mode are present, all modes have as dynamical exponent the golden mean $z=(1+\sqrt 5)/2$. The universal scaling functions of the higher Fibonacci modes are Lévy distributions. The theoretical predictions are confirmed by Monte-Carlo simulations of a three-lane asymmetric simple exclusion process.

The apparently random pattern of the non-trivial zeroes of the Riemann zeta function (all on the critical line, according to the Riemann hypothesis) has led to the suggestion that they may be related to the spectrum of an operator. It has also been known for some time that the statistical properties of the eigenvalue distribution of an ensemble of random matrices resemble those of the zeroes of the zeta function. With the objective to identify a suitable operator, we start by assuming the Riemann hypothesis and construct a unitary matrix model (UMM) for the zeta function. Our approach, however, could be termed piecemeal, in the sense that, we consider each factor (in the Euler product representation) of the zeta function to get a UMM for each prime, and then assemble these to get a matrix model for the full zeta function. This way we can write the partition function as a trace of an operator. Similar construction works for a family of related zeta functions.

In this talk I will describe how to construct the algebra of observables of a quantized scalar field when the spacetime metric is not smooth. I will show the main difference with the smooth case, the technical difficulties that arise and how we addressed them.

This is joint work with G. Hörmann, C. Spreitzer and J. Vickers.

It is well known from linear algebra that a family of mutually commuting normal operators on a finite dimensional complex inner product space can be simultaneously diagonalised. Strongly related to this is the fact that a representation of a commutative C*-algebra on a Hilbert space is generated by a so-called spectral measure, taking its values in the orthogonal projections. A result by Ruoff and the lecturer asserts that a similar phenomenon occurs for positive representations of algebras of continuous functions on a substantial class of Banach lattices.

Recently, it has become clear that these two facts for Hilbert spaces and Banach lattices can be understood from one underlying general theorem for positive homomorphisms of algebras of continuous functions into partially ordered algebras. This result can be proved by purely order-theoretic methods.

In this lecture, we shall explain this theorem and how it relates to the two special cases mentioned above. We shall also sketch the theory of measure and integration in partially ordered vector spaces that is necessary to formulate and establish it.

This is joint work with Xingni Jiang.

In this talk I will give a gentle introduction to Poisson manifolds, which can be thought of as (singular) symplectic foliations. As an illustration of the kind of problems one deals in Poisson geometry, I will discuss and give some results on stability of symplectic leaves.

The Reeb flow of a contact form is a generalisation of Hamiltonian flows on energy hypersurfaces in classical mechanics. In this talk I shall address the question of how "complicated" such flows can be. Among other things, I plan to discuss a construction of Reeb flows with a global surface of section on which the Poincaré return map is a pseudorotation. This is joint work with Peter Albers and Kai Zehmisch

The study of Legendrian submanifolds in contact geometry presents some similarities with knot theory. In particular, invariants are needed to distinguish Legendrian isotopy classes. Linearized Legendrian contact homology is one of these, and is based on the count of holomorphic curves. It is obtained after linearizing a differential graded algebra using an augmentation. A bilinearized version using two augmentations was introduced with Chantraine.

After a self-contained introduction to this context, the geography of these invariants will be described. In the linearized case, it was obtained with Sabloff and Traynor. The bilinearized case turned out to be far more general and was studied with Galant.

Second class particle; multispecies exclusion process; hierarchical matrix solution and proof; queueing interpretation.

Second class particle; multispecies exclusion process; hierarchical matrix solution and proof; queueing interpretation.

Complex zeros of nonequilibrium partition function; open ASEP phase transitions; continuous and discontinuous transitions; coexistence line.

Matrix product ansatz; proof of stationarity; computation of partition function $Z_L$; large $L$ asymptotics of $Z_L$; current and density profile; combinatorial approaches.

The asymmetric simple exclusion process (ASEP) has been studied in probability theory since Spitzer in 1970. Remarkably a version with open boundaries had already been introduced as a model for RNA translation in 1968. This “open ASEP” has since the 1990’s been widely studied in the theoretical physics community as a model of a nonequilibrium system, which sustains a stationary current. In these lectures I will introduce and motivate the model then present a construction — the matrix product ansatz — which yields the exact stationary state for all system sizes. I will derive the phase diagram and analyse the nonequilibrium phase transitions. Finally I will discuss how the approach generalises to multispecies systems.

In this first lecture I will introduce the motivations; correlation functions; mean-field theory and hydrodynamic limit; dynamical mean-field theory; domain wall theory.

While the title of my talk is a riff on the famous monograph Random walks and electric networks by Doyle and Snell, the contents of my talk are very much inspired by the book. I'll discuss how the concept of electrical resistance can be applied to the analysis of interacting particle systems on a weighted graph. I will start by summarizing the results of Caputo-Liggett-Richthammer, myself, and Hermon-Salez connecting the many-particle stochastic process to the one-particle random walk process on the level of Dirichlet forms. Then I will explain how to use this type of energy inequality to bound the cost of transporting particles by the effective resistance, and to perform coarse-graining on a class of state spaces which are bounded in the resistance metric. This new method plays a crucial role in the proofs of scaling limits of boundary-driven exclusion processes on the Sierpinski gasket.

In this talk, we will discuss existence, regularity, and qualitative properties of solutions to the Hamiltonian elliptic system $$ -\Delta u = |v|^{q-1} v\ \ \ \text{in} \ \Omega,\quad -\Delta v = |u|^{p-1} u\ \ \ \text{in} \ \Omega,\quad \partial_\nu u=\partial_\nu v=0\ \ \ \text{on} \ \partial\Omega,$$with $\Omega\subset \mathbb R^N$ bounded, both in the sublinear $pq< 1$ and superlinear $pq>1$ problems, in the subcritical regime. In balls and annuli, we show that least energy solutions are not radial functions, but only partially symmetric (namely foliated Schwarz symmetric). A key element in the proof is a new $L^t$-norm-preserving transformation, which combines a suitable flipping with a decreasing rearrangement. This combination allows us to treat annular domains, sign-changing functions, and Neumann problems, which are nonstandard settings to use rearrangements and symmetrizations. Our theorems also apply to the scalar associated model, where our approach provides new results as well as alternative proofs of known facts.

Generally, it is very difficult to compute nonequilibrium stationary states of a particle system. It turns out that, in some cases, you can find a solution with a quite interesting structure. The goal of this first part of the seminar is to present the structure of this solution — known as matrix product solution (or matrix product ansatz) — using the totally asymmetric exclusion process (TASEP) as a toy model.

The propagation of probe fields around black hole geometries is an interesting tool for the investigation of two important aspects: the stability of these geometry and the quasinormal modes spectra. In general, for simple enough metrics (e. g. spherically symmetric), the field equation reduces to a wave-like form with a specific potential. This turns the problem of integration into a wave scattering problem with a potential barrier similar to that of a Schrödinger equation in quantum mechanics. With the proper boundary conditions the spectrum is that of damped waves, the quasinormal modes. These represents the characteristic response of black holes in general to linear perturbations and give information about the stability of the geometry. In this talk I will refer some examples of quasinormal modes and destabilization of geometries in black holes in which a non-minimally coupled scalar field act, in the context of Horndeski theory. By using the same tools of field propagation, I still present some of our previous results in relation to the violation of strong cosmic censorship in charged geometries.

In this talk I will present the basic ideas of the D(iscontinuous)-P(etrov)-G(alerkin) method with optimal test functions which was introduced by Demkowicz and Gopalakrishnan in 2009. We will have a look at the abstract theory and in particular at the concept of optimal test functions. Then, I will define an ultraweak formulation of Poisson's equation and explain how to apply the abstract ideas of DPG methods to this particular case. Finally, I will discuss several problems where the DPG method has been used successfully. These applications range from singular perturbation problems to plate bending problems.

We discuss various versions of the weak gravity conjecture.

I will explain how to make use of geometric methods to obtain three related flavors of contact homology, a Floer theoretic contact invariant. In particular, I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over $\mathbb{Z}$ and capture interesting torsion information. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our constructions.

Aldous’ spectral gap conjecture asserted that on any graph the random walk process and the interchange process have the same spectral gap. In this talk I will review the work in collaboration with T. M. Liggett and T. Richthammer from 2009, in which we proved the conjecture by means of a recursive strategy. The main idea, inspired by electric network reduction, was to reduce the problem to the proof of a new comparison inequality between certain weighted graphs, which we referred to as the octopus inequality. The proof of the latter inequality is based on suitable closed decompositions of the associated matrices indexed by permutations. I will first survey the problem, with background and consequences of the result, and then discuss the recursive approach based on network reduction together with some sketch of the proof. I will also present a more general, yet unproven conjecture.

By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate $1$ the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and mixing time and show that the total variation distance to the uniform distribution displays a cutoff phenomenon. The results are extended to a family of log-concave distributions obtained by replacing the uniform sampling by a symmetric Beta distribution. This is joint work with Cyril Labbe' and Hubert Lacoin.

I will begin by discussing the two standard prototype random matrix models, one for Hermitian matrices and one for general matrices. For large matrices, the eigenvalues follow the "semicircular law" in the first case and the "circular law" in the second case. Furthermore, there is a simple relationship between these two laws.

I will then discuss two "multiplicative" analogs of these models, in which the random matrices are chosen from the unitary group and the general linear group, respectively. In the unitary case, the limiting eigenvalue distribution was computed by Biane and exhibits an interesting phase transition when a certain scaling parameter equals 4. I will then describe recent results of mine with Driver and Kemp on the general linear case. The limiting distribution again undergoes a phase transition and turns out to have a remarkably simple structure. The talk will be self-contained with lots of pictures and possibly even a few jokes.

I will describe the Segal-Bargmann transform for compact Liegroups, with emphasis on the case of the unitary group $U(N)$. In this case, the transform is a unitary map from the space of $L^2$ functions on $U(N)$ to the space of $L^2$ holomorphic functions on the "complexified" group $\operatorname{GL}(N;\mathbb{C})$. I will then discuss what happens in the limit as $N$ tends to infinity. Finally, I will describe an application to the eigenvalues of random matrices in $\operatorname{GL}(N;\mathbb{C})$. The talk will be self-contained and have lots of pictures.

I will discuss two recent theorems and one recent conjecture about maximizing or minimizing the first three eigenvalues of the Robin Laplacian of a simply connected planar domain. Conformal mappings and winding numbers play a key role in the geometric constructions. In physical terms, these eigenvalues represent decay rates for heat flow assuming a “partially insulating” boundary.

We consider the modified Korteweg-de Vries equation over $\mathbb{R}$ $$ u_t + u_{xxx}=\pm(u^3)_x. $$ This equation arises, for example, in the theory of water waves and vortex filaments in fluid dynamics. A particular class of solutions to (mKdV) are those which do not change under scaling transformations, the so-called self-similar solutions. Self-similar solutions blow-up when $t\to 0$ and determine the asymptotic behaviour of the evolution problem at $t=+\infty$. The known local well-posedness results for the (mKdV) fail when one considers critical spaces, where the norm is scaling-invariant. This means that self-similar solutions lie outside of the scope of these results. Consequently, the dynamics of (mKdV) around self-similar solutions are currently unknown. In this talk, we will show existence and uniqueness of solutions to the (mKdV) lying on a critical space which includes both regular and self-similar solutions. Afterwards, we present several results regarding global existence, asymptotic behaviour at $t=+\infty$ and blow-up phenomena at $t=0$. This is joint work with Raphaël Côte and Luis Vega.

The incompressible Navier–Stokes–Maxwell system is a classical model describing the evolution of a plasma (i.e. an electrically conducting fluid). Although small smooth solutions to this system (in the spirit of Fujita–Kato) are known to exist, the existence of large weak solutions (in the spirit of Leray) in the energy space remains unknown. This defect can be attributed to the difficulty of coupling the Navier–Stokes equations with a hyperbolic system. In this talk, we will describe recent results aiming at building solutions to Navier–Stokes–Maxwell systems in large functional spaces. In particular, we will show, for any initial data with finite energy, how a smallness condition on the electromagnetic field alone is sufficient to grant the existence of global solutions.

The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic equation on the sphere with various singularities and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar Schwarzschild solutions. Those describe, for example, the initial data for the interior of black holes. We construct the space of initial data for such solutions and show that the metric at the event horizon is constrained to the global attractors of such parabolic equations. Lastly, some properties of those attractors and its solutions are explored.

The usual KPZ equation is the scaling limit of weakly asymmetric microscopic models with one conserved quantity. In this talk I will present some weakly asymmetric microscopic models with several conserved quantities for which it is possible to derive macroscopic SPDEs with a KPZ flavor.

Joint work with R. Ahmed, T. Funaki, P. Gonçalves, S. Sethuraman and M. Simon.

Identifying patterns in data enables us to formulate questions that can lead to exact results. Since many of the patterns are subtle, machine learning has emerged as a useful tool in discovering these relationships. We show that topological features of Calabi–Yau geometries are machine learnable. We indicate the broad applicability of our methods to existing large data sets by finding relations between knot invariants, in particular, the hyperbolic volume of the knot complement and the Jones polynomial.

We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities. Our result implies that smooth, physically and dynamically reasonable ultracompact objects are not viable as observational alternatives to black holes whenever these instabilities occur on astrophysically short time scales. The proof of the theorem has two parts: (i) We show that light rings always come in pairs, one being a saddle point and the other a local extremum of an effective potential. This result follows from a topological argument based on the Brouwer degree of a continuous map, with no assumptions on the spacetime dynamics, and hence it is applicable to any metric gravity theory where photons follow null geodesics. (ii) Assuming Einstein's equations, we show that the extremum is a local minimum of the potential (i.e., a stable light ring) if the energy-momentum tensor satisfies the null energy condition.

Random Walks in Cooling Random Environments (RWCRE), a model introduced by L. Avena, F. den Hollander, is a dynamic version of Random Walk in Random Environment (RWRE) in which the environment is fully resampled along a sequence of deterministic times, called refreshing times. In this talk I will consider effects of the ressampling map on the fluctuations associated with the annealed law and the Large Deviation principle under the quenched measure. I conclude clarifying the paradox of different fluctuations and identical LDP for RWCRE and RWRE. This is a joint work with L. Avena, Y. Chino, and F. den Hollander.

In this talk, I shall explain how the classical theory of Stone duality may be generalized to a non-commutative setting. This theory has connections with étale groupoids, quantales, groups and inverse semigroups.

Some of the work was joint with Alina Vdovina. I shall assume no prior exposure to this theory.

Let \((M,\omega)\) be a compact symplectic manifold of dimension \(2n\) endowed with a Hamiltonian circle action with only isolated fixed points. Whenever \(M\) admits a toric \(1\)-skeleton \(\mathcal{S}\), which is a special collection of embedded \(2\)-spheres in \(M\), we define the notion of equivariant pseudo-index of \(\mathcal{S}\): this is the minimum of the evaluation of the first Chern class \(c_1\) on the spheres of \(\mathcal{S}\).

In this talk we will discuss upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of \(M\) are unimodal, we prove that it is at most \(n+1\) . Moreover, when it is exactly \(n+1\), \(M\) must be homotopically equivalent to \(\mathbb{C}P^n\).

We describe stable cup-$i$ products on the cochain complex with $\mathbb{F}_2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of any link.

I will review basic notions and results in topological quantum field theory and discuss its orbifolds, with the aim to apply them in the context of topological quantum computation.

Non-relativistic string theory is described by a sigma model that maps a two dimensional string worldsheet to a non-relativistic spacetime geometry. We discuss recent developments in understanding the spacetime geometry of non-relativistic string theory trying to provide several new insights. We show that the non-relativistic string action admits a surprisingly large number of symmetries. We introduce a non-relativistic limit to obtain the non-relativistic string action which also provides us the non-relativistic T-duality transformation rules and spacetime equations of motion.

In 1983, Grothendieck wrote a letter to Faltings in which he formulated a conjecture for hyperbolic curves over fields which are finitely generated over the rationals. Remaining open to date, it carries the study of rational points on an algebraic variety to the realm of profinite groups. Assuming only a working knowledge of basic Algebraic Geometry, we formulate and motivate the Section Conjecture and outline some modern attempts to tackle it.

The goal of this talk is to present the construction of the global attractor for a genuine nonautonomous variant of the Chafee-Infante parabolic equation in one spatial dimension. In particular, the attractor consists of asymptotic profiles (which correspond to the equilibria in the autonomous counterpart) and heteroclinic solutions between those profiles. We prove the existence of heteroclinic connections between periodic and almost periodic asymptotic profiles, yielding the same connection structure as the well-known Chafee-Infante attractor. This work is still an ongoing project with Alexandre N. Carvalho (ICMC - Universidade de São Paulo).

This talk will be concerned with handling problems about embedding Lagrangians in symplectic four-manifolds where the target manifold is rational. In particular, we will determine those three-fold blow-ups of the symplectic ball which admit an embedded Lagrangian projective plane.

The Entropy Method introduced by Guo, Papanicolaou and Varadhan (1988) has been used with great sucess to derive the scaling hydrodynamic behavior of wide ranges of conserved lattice gases (CLG). It requires to estimate the entropy of the measure of the studied process w.r.t. some good, usually product measure. In this talk, I will present an exclusion model inspired by a model introduced by Gonçalves, Landim, Toninelli (2008), with a dynamical constraint, where a particle at site $x$ can only jump to $x+\delta$ iff site $x-\delta$ is occupied as well. I will give some insight on the different microscopic and macroscopic situations that can occur for this model, and briefly describe the steps to derive the hydrodynamic limit for this model by adapting the Entropy Method to non-product reference measures. I will also expand on the challenges and question raised by this model and on some of its nice mapping features. Joint work with O. Blondel, M. Sasada, and M. Simon.

The Bethe Ansatz equations were initially conceived as a method to solve some particular Quantum Integrable Models (IM), but are nowadays a central tool of investigation in a variety of physical and mathematical theories such as string theory, supersymmetric gauge theories, and Donaldson-Thomas invariants. Surprisingly, it has been observed, in several examples, that the solutions of the same Bethe Ansatz equations are provided by the monodromy data of some ordinary differential operators with an irregular singularity (ODE/IM correspondence).

In this talk I will present the results of my investigation on the ODE/IM correspondence in quantum $g$-KdV models, where $g$ is an untwisted affine Kac-Moody algebra. I will construct solutions of the corresponding Bethe Ansatz equations, as the (irregular) monodromy data of a meromorphic $L(g)$-oper, where $L(g)$ denotes the Langlands dual algebra of $g$.

The talk is based on:

1. D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case. Comm. Math. Phys. (2016)
2. D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections II: The nonsimply-laced case. Comm. Math. Phys. (2017)
3. D Masoero, A Raimondo, Opers corresponding to Higher States of the $g$-Quantum KdV model. arXiv 2018.

In Last Passage Percolation(LPP) we assign i.i.d Exponential weights on the lattice points of the first quadrant of $\mathbb{Z}^2$. We then look for the up-right path going from $(0,0)$ to $(n,n)$ that collects the most weights along the way. One is then often interested in questions regarding (1) the total weight collected along the maximal path, and (2) the behavior of the maximal path. It is known that this path's fluctuations around the diagonal is of order $n^{2/3}$. The proof, however, is only given in the context of integrable probability theory where one relies on some algebraic properties satisfied by the Exponential Distribution. We give a probabilistic proof for this phenomenon where the main novelty is the probabilistic proof for the lower bound. Joint work with Marton Balazs and Timo Seppalainen

In this talk, we investigate the amenability of algebras from algebraic and analytical viewpoints.

We also consider its relationship with the

1. semi-simplicity of operator algebras and
2. crossed product Banach algebras associated with a class of $C^\ast$-dynamical systems.

Extending the colloquium lecture, which essentially deals with $\operatorname{spec} \mathbb{Z}$ we discuss the general case of our construction of dynamical systems for arithmetic schemes. Functoriality and the relation to rational Witt vectors and Fontaine's $p$-adic period ring $A_\inf$ will also be explained if time permits.

We study the free boundary problem for the "hard phase" material introduced by Christodoulou, both for rods in $(1+1)$-dimensional Minkowski spacetime and for spherically symmetric balls in $(3+1)$-dimensional Minkowski spacetime. Unlike Christodoulou, we do not consider a "soft phase", and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks, defined as hypersurfaces where the material's density, pressure and velocity are discontinuous, must be null hypersurfaces. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

This talk is an introductory overview of my research topic: Sensitivity of Networks.

We address the following questions: How does a dynamical network respond to perturbations of equilibrium - qualitatively? How does a perturbation of a targeted component spread in the network? What is the sign of the response?

In more detail, we consider general systems of differential equations inspired from chemical reaction networks: $dx/dt = S r(x)$. Here, $x$ might be interpreted as the vector of the concentrations of chemicals, $S$ is the stoichiometric matrix and $r(x)$ is the vector of reaction functions, which we consider as positive given parameters. Abstractly - for a given directed network: the vector $x$ represents the vertices, the matrix $S$ is the incidence matrix and the vector $r(x)$ refers to the directed arrows.

Sensitivity studies the response of equilibrium solutions to perturbations of reaction rate functions, using the network structure as ONLY data. We give here an introduction of the results and techniques developed through this structural approach.

In this talk we show how equidimensional algebraic correspondences between complex algebraic varieties can be used to construct pull-backs and transforms of a class of currents representable by integration. As a main application we exhibit explicit formulas at the level of complexes for a regulator map from the Higher Chow groups of smooth quasi-projective complex algebraic varieties to Deligne-Beilinson with integral coefficients.

We exhibit a few examples and indicate how this can be applied to Voevodsky’s motivic complexes. This is joint work with Pedro dos Santos and Robert Hardt.

The construction of initial data for the Cauchy problem in General Relativity is an interesting problem from both the mathematical and physical points of view. As such, there have been numerous methods studied in the literature the "Conformal Method" of Lichnerowicz-Choquet-Bruhat-York and the "gluing" method of Corvino-Schoen being perhaps the best-explored. In this talk I will describe an alternative, perturbative, approach proposed by A. Butscher and H. Friedrich, and show how it can be used to construct non-linear perturbations of initial data for spatially-closed analogues of the $k = -1$ FLRW spacetime. Time permitting, I will discuss possible renements/extensions of the method, along with its generalisation to the full Conformal Constraint Equations of H. Friedrich.

A Riemannian manifold $(M,g)$ of dimension $n \ge 3$ is called an Einstein space if $\operatorname{Ric} = \lambda \cdot \operatorname{id}$, where trivially $\lambda = \kappa$, with $\kappa$ the (normalized) scalar curvature; in this case one easily proves that $\lambda = \kappa = \operatorname{constant}$.

We recall the fact that any $2$-dimensional Riemannian manifold satisfies the relation $\operatorname{Ric} = \lambda \cdot \operatorname{id}$, but the function $\lambda= \kappa $ is not necessarily a constant. It is well known that any $3$-dimensional Einstein space has constant sectional curvature. Thus the interest in Einstein spaces starts with dimension $n=4$.

Singer and Thorpe discovered a symmetry of sectional curvatures which characterizes $4$-dimensional Einstein spaces. Later, this result was generalized to Einstein spaces of even dimensions $n = 2k \ge 4.$ We established curvature symmetries for Einstein spaces of arbitrary dimension $n \ge 4$.

There are certain compact 4-manifolds, such as real and complex hyperbolic 4-manifolds, 4-tori, and K3, where we completely understand the moduli space of Einstein metrics. But there are vast numbers of other 4-manifolds where we know that Einstein metrics exist, but cannot currently determine whether or not there are other Einstein metrics on them that are quite different from the currently-known ones. In this lecture, I will first present a characterization of the known Einstein metrics on Del Pezzo surfaces which I proved several years ago, and then describe an improved version which I obtained only quite recently.

I plan to cover the following topics: Euler equations; Burger's equation; $p$-system; symmetric hyperbolic PDEs; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

Bibliography

• Joel Smoller, Shock waves and Reaction Diffusion Equations.
• Constantine Dafermos, Hyperbolic Conservation Laws in Continuum Physics.
• Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics.
• Lecture notes of Blake Temple.

Metrics of Poincaré type are Kähler metrics defined on the complement $X\setminus D$ of a smooth divisor $D$ in a compact Kähler manifold $X$ which near $D$ are modeled on the product of a smooth metric on $D$ with the standard cusp metric on a punctured disk in $\mathbb{C}$. In this talk I will discuss an Arezzo-Pacard type theorem for such metrics. A key feature is an obstruction which has no analogue in the compact case, coming from additional cokernel elements for the linearisation of the scalar curvature operator. I will discuss that even in the toric case, this gives an obstruction to blowing up fixed points (which is unobstructed in the compact case). This additional condition is conjecturally related to ensuring the metrics remain of Poincaré type.

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.

We present a classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for any given number of rings. We demonstrate the use of this new classification scheme for three and four qubits and its potential in the context of qubit networks.

The presence of symmetric and more generally $k$-jet differentials on surfaces $X$ of general type play an important role in constraining the presence of entire curves (nonconstant holomorphic maps from $\mathbb{C}$ to $X$). Green-Griffiths-Lang conjecture and Kobayashi conjecture are the pillars of the theory of constraints on the existence of entire curves on varieties of general type.

When the surface as a low ratio $c_1^2/c_2$ a simple application of Riemann-Roch is unable to guarantee abundance of symmetric or $k$-jet differentials.

This talk gives an approach to show abundance on resolutions of hypersurfaces in $P^3$ with $A_n$ singularities and of low degree (low $c_1^2/c_2$). A new ingredient from a recent work with my student Michael Weiss gives that there are such hypersurfaces of degree $8$ (and potentially $7$). The best known result till date was degree $13$.

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_g\phi=0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A \in H^3(\Sigma)$ (where $\Sigma=\mathbb{R}^n$, $\mathbb{S}^n$ or $\mathbb{H}^n$ models the spatial hypersurfaces) there exists a unique solution $\phi$ of the wave equation converging to $A$ in $H^1(\Sigma)$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(\Sigma)$.

In this talk I with describe how to make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. I will consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3-dimensional Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions. I will also discuss extensions of this analysis to the case of the Einstein equations coupled with various tracefree matter models. This is work in collaboration with Diego Carranza.

The analytically extended Kerr and Reissner-Nordström metrics, describing respectively spinning or spherical charged black holes (BHs), reveal a traversable passage through an inner horizon (IH) to another external universe. Consider a traveler intending to access this other universe. What will she encounter along the way? Is her mission doomed to fail? Does this other external universe actually exist?

Answering these questions requires one to understand the manner in which quantum fields influence the internal geometry of BHs. In particular, this would include the computation of the renormalized stress-energy tensor (RSET) on BH backgrounds - primarily near the IH. Although a theoretical framework for such a computation does exist, this has been a serious challenge for decades (partially due to its inevitable numerical implementation). However, the recently developed pragmatic mode-sum regularization method has made the RSET computation more accessible.

In this talk, we will first consider the computation of the simpler quantity $\langle\phi^2\rangle_{ren}$, for a minimally-coupled massless scalar field inside a (4d) Reissner-Nordström BH. We shall then proceed with the long sought-after RSET, focusing on the computation of the semi-classical fluxes near the IH. Our novel results for the latter will be presented, with a closer look at the extremal limit. Finally - we will discuss possible implications to the fate of our traveler.

We consider the wave equation, $\square_g\psi=0$, in fixed flat Friedmann-Lemaitre-Robertson-Walker and Kasner spacetimes with topology $\mathbb{R}_+\times\mathbb{T}^3$. We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface $\{t=0\}$. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate $L^2$-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the $L^2(\mathbb{T}^3)$ norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

I will explain how three integration techniques for producing cohomology classes — Chen integrals for loop spaces, Bott-Taubes integrals for knots and links, and Kontsevich integrals for configuration spaces — come together in the computation of the cohomology of spaces of braids. The relationship between various integrals is encoded by certain graph complexes. I will also talk about the generalizations to other spaces of maps into configuration spaces (of which braids are an example). This will lead to connections to spaces of link maps and, from there, to other topics such as rope length, manifold calculus of functors, and a conjecture of Koschorke, all of which I will touch upon briefly. This is joint work with Rafal Komendarczyk and Robin Koytcheff.

I will first recall Lusztig's asymptotic Hecke algebra and its categorification, a fusion category obtained from the perverse homology of Soergel bimodules. For example, for finite dihedral Coxeter type this fusion category is a 2-colored version of the semisimplified quotient of the module category of quantum $\operatorname{sl}(2)$ at a root of unity, which Reshetikhin-Turaev and Turaev-Viro used for the construction of 3-dimensional Topological Quantum Field Theories.

In the second part of my talk, I will recall the basics of 2-representation theory and indicate how the fusion categories above can conjecturally be used to study the 2-representation theory of Soergel bimodules of finite Coxeter type.

This is joint work with Mazorchuk, Miemietz, Tubbenhauer and Zhang.

In this talk I will discuss the existence of complete extremal metrics on the complement of simple normal crossings divisors in compact Kähler manifolds, and stability of pairs, in the toric case.

Using constructions of Legendre and Apostolov-Calderbank-Gauduchon, we completely characterize when this holds for Hirzebruch surfaces. In particular, our results show that relative stability of a pair and the existence of extremal Poincaré type/cusp metrics do not coincide. However, stability is equivalent to the existence of a complete extremal metric on the complement of the divisor in our examples. It is the Poincaré type condition on the asymptotics of the extremal metric that fails in general.

This is joint work with Vestislav Apostolov and Lars Sektnan.