The space of finitely additive measures on compact convex bodies (also called convex valuations) is an infinite-dimensional vector space that shares many properties with the cohomology algebra of a compact Kaehler manifold. In particular it is a graded algebra that satisfies a version of Poincaré duality. In a recent work with Jan Kotrbatý (Prague) and Thomas Wannerer (Jena) we prove a version of the mixed hard Lefschetz theorem and mixed Hodge-Riemann relations. The latter can be translated into new higher-order versions of the famous Alexandrov-Fenchel inequality for mixed volumes. The proof combines techniques from differential geometry and functional analysis.
 
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Séminaire Laurent Schwartz — EDP et applications
 

Séminaire Laurent Schwartz — EDP et applications
 

Séminaire Laurent Schwartz — EDP et applications
 

The perturbative expansion of quantum field theory associates numbers (or functions) to combinatorial graphs. These Feynman integrals are often transcendental and hard to evaluate. I will review various combinatorial invariants of graphs that behave similar to these integrals. In particular, I will explain a relation between spanning tree partitions and circuit partitions. It allows for efficient counting of these partitions, producing for every graph an integer sequence that determines the Feynman integral, conjecturally through an Apery-like limit. This is joint work with Francis Brown and Karen Yeats.
 
(based on https://arxiv.org/abs/2304.05299 and ongoing work)
 
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
I will present work in progress on asymptotic detector operators and their measurements in perturbative quantum gravity in asymptotically-flat spacetimes. The simplest example is the energy detector, which in gravity detects outgoing graviton radiation and records its energy. I will review more general detector operators in QFT, defined as light-ray operators, and construct their analogs in gravitational theories. Then I will review how to compute event shapes — correlation functions of detector operators in suitable states — in perturbation theory, repackaged in a modern language. Utilizing this, I will present the first computation of an energy-energy correlator (EEC) in perturbative quantum gravity, to leading nontrivial order in the gravitational coupling. One can think of such observables as capturing a universal part of quantum correlations between idealized LIGO detectors — as unlikely as it may be to be measured in the real world. As a bonus, I will present some results on the two-point correlators of detectors measuring powers of energy and mention work towards the three-point energy correlator (EEEC) in gravity. Along the way, I will comment on the « space of asymptotic detector operators » in a gravitational theory.
 
 
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We introduce a category of inscribed v-sheaves as a minimal differential extension of the theory of diamonds in p-adic geometry, then explain how to apply this theory to differentiate natural period maps in p-adic Hodge theory.
 
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In classical work, Bowen and Margulis proved the equidistribution of closed geodesics in any closed hyperbolic manifold. Together with Jeremy Kahn and Vladimir Marković, we asked ourselves what happens in a 3-manifold if we replace curves by surfaces. (Slightly different aspects of this question have also been studied by Calegari, Marques and Neves.) The natural analogue of a closed geodesic is then a minimal surface, as totally geodesic surfaces exist only very rarely. Nevertheless, it still makes sense (for various reasons, in particular to ensure uniqueness of the minimal representative) to restrict our attention to surfaces that are almost totally geodesic.
The statistics of these surfaces then depend very strongly on how we order them: by genus, or by area. If we focus on surfaces whose area tends to infinity, we conjecture that they do indeed equidistribute; we proved a partial result in this direction. If, however, we focus on surfaces whose genus tends to infinity, the situation is completely opposite: we proved that they then accumulate onto the totally geodesic surfaces of the manifold (if there are any).
 

We review some results and questions concerning sparse equidistribution results for horocycle flows in constant curvature, and present a recent result joint with Kanigowski and Radziwill that horocycle flows equidistribute for all points with infinite discrete orbit along times equal to products of two primes.
 

Séminaire de géométrie arithmétique
The universal Weil cohomology (obtained in a recent work jointly with B. Kahn) is taking values in an abelian Q-linear (Q is the field of rational numbers) tensor category M which is rigid but its Q-algebra E = End (1) of endomorphisms of the unit is not a field, a priori. André’s theory of motivated cycles MA, in characteristic zero, can be recovered via the universal Weil cohomology as a localisation of M; thus E = Q  if and only if M = MA is André’s category which is then universal for all Weil cohomologies taking values in abelian Q-linear rigid tensor categories. A similar picture holds true for the universal mixed Weil cohomology with values in MM with respect to Nori motives NM.
However, in any characteristic, this new Weil cohomology yields a universal homological equivalence hum and a canonical comparison faithful tensor functor F from Grothendieck motives (modulo hum) MG to M. This F is an equivalence if and only if MG is abelian, F is exact and the Grothendieck Lefschetz standard conjecture holds true. Moreover, F is an equivalence with M semi-simple if and only if hum is numerical equivalence. Therefore Grothendieck’s standard conjectures for the universal Weil cohomology or the stronger Voevodsky’s nilpotence conjecture (independently of any Weil cohomology) imply that E = Q.
 A standard hypothesis is then that this absolutely flat Q-algebra E is a domain hence a field. This hypothesis is equivalent to the property that M is Tannakian. Similarly, for MM. Note that, for every self correspondence, the trace and the Lefschetz number (as well as the coefficients of the characteristic polynomials) are defined over E. As a consequence, if E is a field all these are the same independently of the Weil cohomology.
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I will discuss Effective Field Theories that can originate from microscopic unitary theories, and their relation to moment theory. I will show that  theories with isolated massive higher-spin particles, and theories with very irrelevant interactions, don’t posses healthy UV completions, and I will show how Vector Meson Dominance follows from such first principles.
 
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Probability and analysis informal seminar
Schwarzian Theory is a quantum field theory which has attracted a lot of attention in the physics literature in the context of two-dimensional quantum gravity, black holes and AdS/CFT correspondence. It is predicted to be universal and arise in many systems with emerging conformal symmetry, most notably in Sachdev–Ye–Kitaev random matrix model and Jackie–Teitelboim gravity.In this talk we will discuss our recent progress on developing rigorous mathematical foundations of the Schwarzian Field Theory, including rigorous construction of the corresponding measure, calculation of both the partition function and a natural class of correlation functions, and a large deviation principle.
 
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