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The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator $L$ directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function $\zeta (s)=\mathrm{\text{Tr}}{L}^{-s}$ and the heat trace $\mathrm{\Theta }(t)=\mathrm{\text{Tr}}exp(-tL)$. The kernel $U(t;x,x^{\prime })$ of the heat semigroup $exp(-tL)$, called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as $\mathrm{\text{Tr}}f(tL)$, that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as $\mathrm{\text{Tr}}exp(-t{L}_{+})exp(-s{L}_{-})$, that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

We give a new and detailed proof of the variation formulas for the equivariant Ray–Singer metric, which are originally due to J. M. Bismut and W. Zhang.

Using the Oh–Schwarz spectral invariants and some arguments of Frauenfelder, Ginzburg and Schlenk, we show that the π₁-sensitive Hofer–Zehnder capacity of any subset of a closed symplectic manifold is less than or equal to its displacement energy. This estimate is sharp, and implies some new extensions of the Non-Squeezing Theorem.

We prove that a certain bilinear pairing (analogous to the Poincaré–Lefschetz intersection pairing) between filtered sub- and quotient complexes of a Floer-type chain complex and of its "opposite complex" is always nondegenerate on homology. This implies a duality relation for the Oh–Schwarz-type spectral invariants of these complexes which (in Hamiltonian Floer theory) was established in the special case that the period map has discrete image by Entov and Polterovich. The duality relation served as a key lemma in Entov and Polterovich's construction of a Calabi quasimorphism on certain rational symplectic manifolds, and the result that we prove here implies that their construction remains valid when the rationality hypothesis is dropped. Apart from this, we also use the nondegeneracy of the pairing to establish a new formula for what we have previously called the boundary depth of a Floer chain complex; this formula shows that the boundary depth is unchanged under passing to the opposite complex.

We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend from to Ham_c(M\U). Furthermore, we show that these invariants are continuous with respect to the C⁰-topology on Ham_c(M\U).

We apply the above results to Hofer geometry and establish unboundedness of the Hofer diameter of Ham_c(M\U) for stably displaceable U. We also answer a question of F. Le Roux about C⁰-continuity properties of the Hofer metric.

Calculating the spectral invariant of Floer homology of the distance function, we can find new superheavy subsets in symplectic manifolds. We show if convex open subsets in Euclidian space with the standard symplectic form are disjointly embedded in a spherically negative monotone closed symplectic manifold, their complement is superheavy. In particular, the $S^1$ bouquet in a closed Riemann surface with genus $g\ge 1$ is superheavy. We also prove some analogous properties of a monotone closed symplectic manifold. These can be used to extend Seyfaddni’s result about lower bounds of Poisson bracket invariant.

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [73], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a “classical” Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

We extend parts of the Lagrangian spectral invariants package recently developed by Leclercq and Zapolsky to the theory of Lagrangian cobordism developed by Biran and Cornea. This yields a nondegenerate Lagrangian “spectral metric” which bounds the Lagrangian “cobordism metric” (recently introduced by Cornea and Shelukhin) from below. It also yields a new numerical Lagrangian cobordism invariant as well as new ways of computing certain asymptotic Lagrangian spectral invariants explicitly.

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding $I:{ℝ}^n\to \mathrm{\text{Ham}}(M)$ with respect to the fragmentation norm on the group $\mathrm{\text{Ham}}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega )$. As an application, we provide an answer to Brandenbursky’s question on fragmentation norms on $\mathrm{\text{Ham}}({\mathrm{\Sigma }}_g)$, where ${\mathrm{\Sigma }}_g$ is a closed Riemannian surface of genus $g\ge 2$.