Selected Topics of Simons Semesters in Banach Center

16.12.2019 - 20.12.2019 | Warsaw


Paul Frank Baum, "Index theory on odd-dimensional manifolds"
On a closed (i.e. compact, no boundary) odd-dimensional smooth manifold the index of any elliptic differential operator is zero. Thus, on odd-dimensional manifolds, interesting examples can only be obtained by dropping either ``elliptic" or ``differential". This talk will explain the two cases. If ``differential" is dropped, then the examples are Toeplitz operators (which are elliptic pseudo-differential operators of order zero). A corollary is the result of Louis Boutet de Monvel about Toeplitz operators associated to the boundary of a strictly pseudo-convex domain. If ``elliptic" is dropped, then examples are the differential operators on contact manifolds studied by E. van Erp and Baum-van Erp.

Tomasz Bielecki, "Semimartingales and Shrinkage of Filtration"
Semimartingales constitute an important class of stochastic processes. Poisson process and Brownian motion are semimartingales. Asemimartingale is quite adequately described in terms of the triple ofsemimartingale charcacteristics, which, in general,  depend on the relevantinformation flow that is used for such description.  In the field ofstochastic processes, the information flow that one uses is typically givenin form of so called filtration. In this work we study the change ofsemimartingale characteristics under a change of filtration from a largerone to a smaller one. This is a joint work with Monique Jeanblanc, JacekJakubowski and Mariusz Nieweglowski.

Ralph Chill, "Weak and strong approximation of semigroups on Hilbert spaces"
We study several equivalent conditions for strong convergence of -semigroups on Hilbert spaces and Banach spaces, and we indicate applications to numerical analysis, domain convergence and homogenisation.

Gui-Qiang G. Chen, "Nonlinear Equations of Mixed Elliptic-Hyperbolic Type -- Analysis and Applications"
As is well-known, two of the basic types of linear partial differential equations (PDEs) are hyperbolic and elliptic types, following the classification for linear PDEs first proposed by Jacques Hadamard in the 1920s; and linear theories of PDEs of these two types have been well developed. On the other hand, many nonlinear PDEs arising in geometry, mechanics, and other areas naturally are of mixed elliptic-hyperbolic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear PDEs of mixed type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear PDEs of mixed elliptic-hyperbolic type with these longstanding problems and will then discuss some recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing and identifying unified approaches, ideas, and techniques for dealing with the mixed-type problems. Further perspectives, trends, and open problems in this direction will also be addressed.

Michael Eastwood, "Some special geometries in dimension five"
Our Simons Semester was entitled Symmetry and Geometric Structures. Its target was the various smooth geometric structures on manifolds having a most symmetrical incarnation with an abundant though finite-dimensional family of symmetries: in other words, a Lie group. Although Riemannian geometry falls into this category, there are much more interesting cases. Especially intriguing examples occur in dimension four and five: to be discussed in this talk. In particular, the exceptional Lie group G2 will emerge from thin air!

Jürgen Hausen, "Cox Rings"
We first recall the basics on Cox Rings and show how they can be used for an explicit approach to algebraic varieties. Then we survey recent applications of this approach to the classification of Fano varieties.

Arieh Iserles, "Fast approximation on the real line"
While approximation theory in an interval is thoroughly understood, the real line represents something of a mystery. In this talk we review the state of the art in this area. Motivating our interest by the design of spectral methods for dispersive PDEs, we commence from familiar Hermite functions and move to recent results characterising all orthonormal sets on that have a skew-symmetric (or skew-Hermitian) tridiagonal differentiation matrix and such that their first expansion coefficients can be calculated in operations. In particular, we describe the generalised Malmquist–Takenaka system. The talk concludes with a (too!) long list of open problems and challenges.

Nikon Kurnosov, "Cohomology of holomorphic symplectic manifolds"
In this talk I will survey some results on cohomology of holomorphic symplectic manifolds, both Kahler and non-Kahler, in the context of Beauville-Bogomolov-Fujiki form.

Thomas Mettler, "Minimal surfaces & hyperbolicity"
The Beltrami—Klein model provides a natural generalization of hyperbolic geometry within the framework of projective geometry. However, projective geometry is too strong a generalisation, so that important properties of hyperbolic geometry would still exist in general. In my talk I will discuss a generalisation of hyperbolic geometry in terms of the existence of certain Lagrangian minimal surfaces.

Giovanni Mongardi, "Hodge conjecture and rational curves on Holomorphic symplectic manifolds."
In this talk, I will survey a series of results about Algebraic cycles on Holomorphic symplectic manifolds, with an emphasys on rational curves and the integral Hodge conjecture for one cycles.

Katharina Neusser, "Symmetry and Geometric Rigidity"
In differential geometry many important geometric structures are geometrically rigid in the sense that their automorphism groups are finite-dimensional Lie groups. Prominent examples of such structures are Riemannian manifolds, conformal manifolds and projective structures. Generically these geometric structures have trivial automorphism groups and so the ones among them with large automorphism groups or special types of automorphisms are typically geometrically and topologically very constrained and hence can often be classified. Recall for instance that a Riemannian manifold with an isometry group of largest possible dimension is isometric to a space of constant curvature. In this talk I will discuss some classical as well as some new results along those lines, concerned with (local) automorphism groups of various geometric structures and local and global questions of geometric rigidity.

Benoit Perthame, "PDEs for neural assemblies; analysis, simulations and behaviour"
Neurons exchange informations via discharges, propagated by membrane potential, which trigger firing of the many connected neurons. How to describe large assemblies of such neurons? What are the properties of these mean-field equations? How can such a network generate a spontaneous activity? Such questions can be tackled using nonlinear integro-differential equations. These are now classically used in the neuroscience community to describe neuronal assemblies. Among them, the best known is certainly Wilson-Cowan's equation which describe spiking rates arising in different brain locations.

Goran Peskir, "Optimal Real-Time Detection of a Drifting Brownian Coordinate"
Consider the motion of a Brownian particle in three dimensions, whose two spatial coordinates are standard Brownian motions with zero drift, and the remaining (unknown) spatial coordinate is a standard Brownian motion with a (known) non-zero drift. Given that the position of the Brownian particle is being observed in real time, the problem is to detect as soon as possible and with minimal probabilities of the wrong terminal decisions, which spatial coordinate has the non-zero drift. We solve this problem in the Bayesian formulation, under any prior probabilities of the non-zero drift being in any of the three spatial coordinates, when the passage of time is penalised linearly. [This is joint work with P. E. Ernst and Q. Zhou.]

Mark Pollicott, "Dynamics, Dimension and some Number Theory"
The Zaremba conjecture in number theory is that every natural number occurs as the denominator of a rational number arising from a finite continued fraction with digits taking only values from {1,2,3,4,5}. This conjecture remains open, but it was shown by Bourgain-Kontorovich and Huang to be true for most natural numbers (in a density one sense). Interestingly, the proof relies on a specific Cantor set X in the real line having (Hausdorff) dimension greater than 5/6. We will describe how to get this rigorous bound using the simple dynamics of the Gauss map defined by T(x) = 1/x (mod 1). Time permitting, we will discuss other related applications.

David Seifert, "Energy decay of damped waves"
The past decade has seen a number of exciting developments in the study of energy decay of damped waves. Much of the recent progress has been driven by theoretical advances in the theory of strongly continuous operator semigroups on Banach and Hilbert spaces. In this talk I will aim to explain the semigroup approach to the damped wave equation, present some classical results in the area and provide a glimpse of some of the more recent developments in the quantified asymptotic theory of operator semigroups on Hilbert spaces.

Hendrik Suess, "Toric Geometry and Einstein metric"
Toric geometry studies vanishing sets of polynomials, which admit a strong symmetry in form of a faithful action of an algebraic torus of sufficiently high dimension. These vanishing sets of polynomials are known as toric varieties and their geometry is completely captured by the combinatorics of certain associated polytopes. This connection allows fruitful interactions between algebraic geometry, combinatorics and convex geometry. In my talk I will demonstrate how to make use of these interactions to address the existence problem for Einstein metrics on such varieties.

Edriss S. Titi, "Mathematical Analysis of Atmospheric Models with Moisture"
In this talk I will present some recent results concerning global regularity of certain geophysical models. This will include the three-dimensional primitive equations with various anisotropic viscosity and turbulence mixing diffusion, and certain tropical atmospheric models with moisture. In particular, we will also show the global regularity of the coupled three-dimensional primitive equations with phase change moisture model. Moreover, in the non-viscous (inviscid) case it can be shown that there is a one-parameter family of initial data for which the corresponding smooth solutions of the primitive equations develop finite-time singularities (blowup).
Capitalizing on the above results, we can provide rigorous justification of the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations, for vanishing small values of the aspect ratio of the depth to horizontal width. Specifically, we can show that the Navier-Stokes equations, after being scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and that the convergence rate is of the same order as the aspect ratio parameter.
Furthermore, we will also consider the singular limit behavior of a tropical atmospheric model with moisture, as ε → 0, where ε > 0 is a moisture phase transition small convective adjustment relaxation time parameter.

Balint Toth, "Invariance principle for the random Lorentz-gas beyond the kinetic limits"
I will present an invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius , placed according to a Poisson point process of density , in the limit , , up to time scales of order (annealed) or (quenched, work in progress). This goes far beyond the kinetic time scales () proved in the groundbreaking work of Gallavotti (1969), Spohn (1978) [annealed] and Boldrighini-Bunimovich-Sinai (1983) [quenched). The novelty is that the diffusive scaling of particle trajectory and the kinetic (low density, Boltzmann-Grad) limit are taken simulataneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling.

Benjamin Weiss, "Predictive sets and Riesz sets"
A set P contained in the positive integers, N, is called predictive if for any zero entropy finite-valued stationary process is measurable with respect to the sima-field generated by for i in P . The property of having zero entropy is exactly that N is a predictive set. I will discuss some sufficient conditions and a necessary one for a set to be predictive. I will also discuss linear predictivity, predictivity among Gaussian processes and relate these to Riesz sets which arise in harmonic analysis.

Bogusław Zegarliński, "Coercive inequalities of higher order"
In my talk I will review some results concerning coercive inequalities of higher order for finite and infinite dimensional systems.

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