The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for desribing and analyzing non-smooth geometries. The dynamics is described by so-called causal variational principles.
The talk focuses on causal variational principles from the perspective of analysis and the calculus of variations. After a simple introduction to causal variational principles, I shall explain how Noether-like theorems can be formulated in this setting. The basic method is to work with so-called surface layer integrals, which I will introduce. The simplest version of a Noether-like theorem is proven. At the end of the talk, I will give an outlook on causal fermion systems and explain how our Noether-like theorems are related to the conservation laws for charge and energy-momentum in relativistic quantum theory.
This is joint work with Johannes Kleiner (Regensburg).
After giving an overview over some recently developed conformal methods in Riemannian and Lorentzian geometry, we focus on a result of a joint work with Marc Nardmann (Dortmund) stating that every conformal class contains a metric of bounded geometry. Finally, we sketch implications of the result in the theory of the Yamabe flow on noncompact manifolds.
Recently Gigli developed a Sobolev calculus on non-smooth spaces using module theory. He shows that the relaxed notion of gradient is sufficient to obtain “1-forms” and make it possible to define Sobolev differentials which resemble the ones in the smooth setting.
In this talk I will show that his theory fits nicely into the theory of Lipschitz differentiable spaces initiated by Cheeger, Keith and others. For this I present a new relaxation procedure for $L^p$-valued subadditive functionals and give a relationship between the module generated by a functional and the module generated by its relaxation.
In the framework of Lipschitz differentiable spaces, which include so called PI- and RCD(K,N)-spaces, the Lipschitz module is “pointwise” finite dimensional. A general renorming theorem together with the characterization above shows that the Sobolev spaces of such spaces are reflexive.
We study the space of minimal hyper-surfaces obtained by Almgren-Pitts min-max theory. The approach is by studying index and area bounds for such hyper-surfaces. It was proven by F.C.Marques and A. Neves that positively curved manifolds admit infinitely many embedded minimal hypersurfaces. We are able to show that these are indeed geometrically distinct embeddings.
Index theory over singular spaces is an active research area within global analysis and has many potential applications. For spaces which can be described by groupoids, the well-known index theorems for orbifolds indicate that the so-called inertia spaces encode the contribution of singularities to analytic indices. Moreover, the inertia space of a Lie groupoid encodes interesting topological, geometric, and analytic information about the original Lie groupoid. It is the goal of the talk to explain this point of view using non-commutative geometry as a unifying tool. In particular a Hochschild-Kostant-Rosenberg type theorem for the Hochschild homology of the convolution algebra of a proper Lie groupoid is indicated. The talk is based upon joint work, partially in progress, with H. Posthuma and X. Tang, as well as with C. Farsi and Ch. Seaton.
It is well known that the problem of determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a twelve-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is: the variation ensemble for that functional has never been satisfactorily determined. This is known as the $N$-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory), and remains without satisfactory solution. The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them, and historically the first, is density functional theory, based on a three-variable object for which $N$-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory.Ensemble representability for 1-RDMs was solved 50 years ago. However, only recently, thanks to outstanding work by Klyachko on generalized Pauli constraints, real progress has been made on pure representability for 1-RDM. These constraints determine small polytopes of admissible pure $N$-representable sets of 1-RDMs. Somewhat mysteriously, physical states seem to cling to the boundary of the polytopes. We speak of pinning when there are saturated constraints, implying strong selection rules which drastically simplify the configurations. Quasi-pinning appears to be ubiquitous. We review recent numerical evidence and theoretical justification for this phenomenon.
Topological fermion systems provide a general framework for desribing and analyzing non-smooth geometries. They can also be used to describe "quantum spaces" or "quantum space-times" as considered in quantum gravity. Moreover, they set the stage for the so-called fermionic projector formulation of relativistic quantum field theory.The aim of the talk is to give a simple introduction, with an emphasis on conceptual issues. Starting from a collection of functions on $\R^3$ (which can be thought of as Schrödinger wave functions), we ask the question whether the geometry of the Euclidean space is encoded in these functions. Bringing this question into a precise mathematical form leads us to the abstract definition of topological fermion systems. This definition will be illustrated by the examples of vector fields on the sphere, a vector bundle over a manifold, and a lattice system. As an example motivated from physics, we briefly consider Dirac spinors on a globally hyperbolic Lorentzian manifold and introduce the setting of causal fermion systems. The inherent geometric and analytic structures on a topological fermion system are introduced and explained. A brief outlook on the applications to quantum field theory is given.
In the talk, we explain how to formulate PDE's within the framework of jet spaces. This allows the definition of the so-called formal solution space of a non-linear PDE. In case the PDE is formally integrable, the formal solution space carries in a natural way the structure of a profinite dimensional manifold. We also explain the fundamentals of this particular category of infinite dimensional manifolds, and show that in many ways the profinite dimensional manifolds appearing as formal solution spaces of formally integrable PDE's are easier to deal with than the real solution spaces of these PDE's. In addition, we prove a new criterion for formal integrability formally integrable PDE's and derive from it that the Euler-Lagrange Equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.
The talk is on joint work with Batu Gueneysu, Humboldt University, Berlin.
The interior (or epsilon-regularity) estimates for two-dimensional variational problems has a long history, ranging from the classical minimal/CMC surface equations to harmonic maps and prescribed mean-curvature equations and more recently Willmore surfaces, $W^{2,2}$ conformal immersions and Dirac-harmonic maps. Here we will present some new estimates for a class of elliptic PDE with applications (in particular) to the free boundary problem for Dirac-harmonic maps, and global estimates for harmonic maps. This talk is comprised of separate joint works with Miaomiao Zhu and Tobias Lamm.
We study the $L^p$-spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum is $p$-independent. As a first example where $p$-independence fails we compute explicitly the $L^p$-spectrum for the hyperbolic space and its product spaces. Moreover, we give general results on the Green functions and the symmetry of the $L^p$-spectrum of Dirac operator. This is joint work with Bernd Ammann (Regensburg).
In this talk, we give an introduction of a deterministic discrete game interpretation of the mean curvature flow, proposed by R. V. Kohn and S. Serfaty, via the level set method. We show that the game value formally converges to the viscosity solution of mean curvature flow equation. We address some related problems and discuss properties of the solution by using the game.
The proof of Borell–Brascamp–Lieb (BBL) inequality for Riemannian manifolds by CorderoErausquin-McCann-Schmuckenschläger, and later for Finsler manifolds by Ohta, let Lott-Villani and Sturm to a new notion of a lower bound on the generalized Ricci curvature for general metric measure spaces, called curvature dimension. Both, the BBL inequality and the curvature condition, rely on geodesics in the 2-Wasserstein space, which was a natural candidate because of its connection to convex analysis in the Euclidean setting.
Based on Ohta's proof we show how to prove the BBL inequality via geodesics in the p-Wasserstein spaces for any p>1. Following Lott-Villani-Sturm, a new curvature condition can be defined via convexities along geodesics in the p-Wasserstein space and many known results like Poincaré, Bishop-Gromov follow by similar arguments.
As a "vertical dual" one can use the recent theory of the q-Cheeger energy (q is the Hölder conjugate of p) developed by Ambrosio-Gigli-Savaré to even get a q-Laplacian comparison, which however is equivalent to the usual one in the smooth setting. In a second talk given later we will study the gradient flow of the q-Cheeger energy, called q-heat flow, and use the "duality" and curvature condition to identify it with the gradient flow of the (3-p)-Renyi entropy (classical entropy in case p=2) in the p-Wasserstein space.
If times permits the Orlicz-Wasserstein space is introduced and the necessary adjustments to get the interpolation inequality and curvature condition are shown. However, by now there is no "vertical dual" to the theory of Orlicz-Wasserstein spaces, in particular there is no Orlicz-Cheeger energy and no Orlicz-Laplacian.
I would like to discuss some questions related to some semilinear equations driven by a nonlocal elliptic operator (for example, the Allen-Cahn equation, in which the classical Laplace operator is replaced by a fractional Laplacian). In particular, I would like to study the qualitative properties of the solutions, such as symmetry, density estimates of the level set, asymptotic behaviors, etc. The limit interfaces of these problems are related to both the local and the nonlocal perimeter functionals on this topic, I would like to discuss some recent rigidity and regularity results and to present some open problems.
Presenting a joint work with Hajlasz and Tyson, I will introduce some basics of the Heisenberg group. Since the Heisenberg-group is homeomorphic to the euclidean space, classical Homotopy groups are all zero, so it seems to be more reasonable to consider Lipschitz homotopy groups.
Then I will present an argument about non-triviality of certain Homotopy groups, by relating Homotopy groups of spheres and the Heisenberg group, and extending the Hopf invariant. As a consequence, one can also prove that Lipschitz mappings are not dense in certain the Sobolev spaces.