This thesis investigates the zeros of entire harmonic mappings on Rn. The starting point of this work is the failure to generalize Bézout’s theorem to entire holomorphic mappings, known as the Transcendental Bézout Problem, due to the instability of classical zero counting highlighted by the Cornalba and Shiffman counterexample [CS72]. Drawing inspiration from recent work in [Buh et al., 24b] on the Persistent Transcendental Bézout Theorem, and from the real harmonic analogue of the Cornalba and Shiffman counterexample presented in [Sto24], we extend their approach to this setting, a research direction explicitly suggested in [Buh et al., 24b, section 1.5.3]. The main objective is to obtain a bound on the growth rate of the number of connected components containing at least one zero within a ball as a function of its radius. We prove that this number is polynomially bounded by the growth rate of the maximum modulus of the mapping. The proof adapts the coarse nodal counting techniques of [Buh et al., 24a] and the methodology of [Buh et al., 24b]. The first chapter presents the properties of harmonic mappings and establishes Cauchytype estimates. The second chapter introduces a simplified version of stratified Morse theory adapted to manifolds with corners. This allows relating the local behavior of the mapping to the global topology of its sublevel sets on the closed ball via critical points. The third chapter presents persistence and its stability theorem. The final chapter proves the main result by combining these tools. Polynomial approximation allows applying Milnor’s algebraic complexity bounds, while persistent homology ensures the transfer of these bounds to the transcendental mapping.

A classical question in topology asks when a homeomorphism of smooth manifolds can be uniformly approximated by diffeomorphisms. While the answer is known for all dimensions but n = 4, it is natural to ask wether such approximations can preserve additional geometric structures, such as volume. In this master’s thesis, we study the approximation of volume-preserving homemorphisms of smooth, compact oriented manifolds without boundaries by smooth diffeomorphisms. We prove that if a volume-preserving homeomorphisms can be uniformly approximated by a diffeomorphisms sequence, then the approximating sequence can be chosen to preserve volumes. We provide a complete proof by following the constructive approach of Sikorav and Lefebvre, making all technical steps explicit. The proof combines smooth triangulations, Hodge theory and Moser-type isotopies to construct volume-preserving diffeomorphisms uniformly approximating a given homeomorphism. This constructive method also suggests potential extension to the setting of flux-preserving homeomorphism. In Chapter 1, we review the basic elements of simplicial geometry and introduce the foundations of simplicial homology and cohomology. We focus on the geometric realization of simplicial complexes in Euclidean spaces, the construction of smooth triangulations of arbitrarily fine mesh, and the simplicial machinery needed to state the de Rham Theorem. Chapter 2 introduces the necessary elements of Hodge theory, including the Hodge-de Rham Laplacian and its Green operator, which provide the canonical primitives of exact forms with controlled regularity. Finally, in Chapter 3, we construct the approximating sequence of volume-preserving diffeomorphisms, combining the tools developed in the previous chapters to give a complete proof of the main theorem.

The 2000s saw the emergence of persistence homology, a tool for constructing the homology of a manifold using techniques from data science. One of the advantages of this construction, which is of an algebraic nature, is that it can be linked to another construction that is easier to interpret from an analytical and geometric point of view: a barcode. On a smooth manifold $M$, this takes the form of a collection of intervals, whose infinite intervals allow us to read the homology of $M$. It is therefore useful to be able to count bars longer than a fixed number $\delta>0$. Such a barcode can be generated by a smooth function $f:M\to \mathbb{R}$. The recent article \cite{Egor} by Buhovsky, Payette, Polterovich, Polterovich, Shelukhin, and Stojisavljević establishes an inequality relating bars of length greater than a number $\delta>0$ in the barcode generated by $f$ and the volume of the graph of this function, viewed as a submanifold of the cotangent bundle $T^*M$. This inequality takes the form \[\mathcal{N}_{\delta}(f)\leq C(\delta) \cdot vol(graph(df))\] where the left-hand side is the number of bars longer than $\delta$, $vol(graph(df))$ is the volume of the graph of $df$ viewed as a submanifold of $T^*M$, equipped with the Sasaki metric, and $C(\delta)$ is a constant depending on $\delta$ and some structural information on the manifold $M$. The purpose of this thesis is to explain the dependence of the number $C(\delta)$ on $\delta$. We will achieve this by using some integral geometry techniques, used in a similar way in the paper \textit{Topological entropy of hamiltonian diffeomorphisms: a persistence homology and floer theory perspective} by Çineli, Ginzburg and Gürel.

Most of the results of the (four) articles included in this thesis focus on Floer theory and numerical invariants extracted from the filtered Floer homologies. A brief description of the results from each article is provided below. The first article (Chapter 1) is about the group of diffeomorphisms that preserve the form ωk. It is shown that the identity component of the group of diffeomorphisms of a symplectic manifold (M2n, ω) preserving the form ωk for k < n coincides with the identity component of the group of symplectomorphisms. This is not the case for k = n due to the famous symplectic non-squeezing theorem of Gromov [Gro85]. The second article (Chapter 2), concerns Lagrangian intersections. Let L be a closed rational Lagrangian submanifold of a compact convex symplectic manifold W with a rationality constant ρ. We prove that for every compactly supported Hamiltonian diffeomorphism f of W with γ(f) < ρ(L) one has #f(L) ∩ L ≥ cl(L,Z/2) + 1. This generalized the results of Polterovich [Pol93], Chekonov [Che98] and Kislev-Shelukhin [KS21]. In the third article (Chapter 3), we show that the spectral capacity of a Darboux ball is equal to its standard capacity, πr2 where r is the radius of the ball. We compute the spectral capacities of ellipsoids and polydisks. We show that if two disjoint balls with capacities a and b symplectically embed in an open set U of an aspherical symplectic manifold, then the spectral capacity of U is at least a + b. This gives an alternative proof of Gromov’s two ball-packing theorem. Analogous results are obtained for complex projective spaces. The fourth article (Chapter 4), studies Hamiltonian fragmentation in dimension four. We prove that for a given ϵ > 0 every C0-small enough Hamiltonian diffeomorphism of D × D can be decomposed into N = N(ϵ) Hamiltonian diffeomorphisms, each supported in D × D, for some topological disk D, can be different for each fragment, with area smaller than ϵ. This is the first four-dimensional Hamiltonian fragmentation result subsequent to the two-dimensional results [Fat80, LR10, EPPK12]. We prove the C0-continuity of Polterovich-Shelukhin spectral estimators and detect an isometric embedding of C∞ c (0,b) with C0-distance into the group Ham(D × D) equipped with the Hofer distance.

In this master’s thesis, we delve into the study of the group of volume-preserving diffeomor- phisms of the cube, equipped with the L2 metric. These functions represent configurations of an incompressible fluid within a cube. The distance between two configurations in this context is defined by the minimal kinetic energy required to transition from one to the other. We will present a result by Shnirelman establishing that the diameter of this group is finite. This means that for any pair of configurations, the minimal energy required to transition from one to the other is bounded. Shnirelman also shows that there exist configurations for which there is no minimizing path between them in terms of kinetic energy. In the first chapter, we define the objects of study that will be used throughout the thesis. The second chapter presents the main result of the paper, which is a bound on the distance between two configurations, and we demonstrate how this result leads to the conclusion that the diameter of the group is finite. Additionally, in this chapter, we construct a configuration for which the length with respect to the identity cannot be minimized. Chapter 3 is devoted to discrete flows, while Chapter 4 explains how certain configurations can be approximated by discrete flows. Finally, Chapter 5 deals with the proof of the main result.

This thesis comprises four articles that study rigidity phenomena of Hamiltonian transfor- mations of symplectic manifolds. The first article, co-authored with Egor Shelukhin, examines obstructions to the existence of Hamiltonian symmetries of finite order on a closed symplectic manifold (M,ω); Hamil- tonian torsion. In other words, we study the finite subgroups of the group of Hamiltonian diffeomorphisms Ham(M, ω). We identify three primary sources of obstructions: Topological constraints. Inspired by a result of Polterovich showing that symplectically aspherical symplectic manifolds do not admit Hamiltonian torsion, we establish that the presence of a non-trivial finite subgroup of Ham(M,ω) implies that there exists a sphere A ∈ π2(M) with ⟨[ω],A⟩ > 0 and ⟨c1(M),A⟩ > 0. In particular, symplectically Calabi-Yau, and spherically negative-monotone symplectic manifolds do not admit Hamiltonian torsion. The presence of J-holomorphic curves. For general closed symplectic manifolds, there are plenty of examples of Hamiltonian torsion, for instance, any rotation of the two-sphere by an irrational fraction of π. When (M, ω) is spherically positive-monotone, we show the existence of Hamiltonian torsion imposes geometrical uniruledness, which implies that non-constant J-holomorphic spheres are ubiquitous. This phenomenon was predicted in a list of problems contained in the introductory monograph of McDuff and Salamon. The spectral metric rigidity. Our study reveals that for spherically positive-monotone (M, ω), there exists a neighbourhood of the identity in Ham(M,ω), in the topology induced by the spectral metric, that does not contain any non-trivial finite subgroup. The main result of the second article establishes that for a broad class of symplectic manifolds the flux of a loop of symplectic diffeomorphisms is completely determined by the homotopy class of its orbits. As an application, we obtain a new vanishing result for the flux group and new instances where the existence of a fixed point of a symplectic circle action implies that it is Hamiltonian. Moreover, we obtain obstructions to the existence of non-trivial elements of Symp0(M,ω) that have finite order. The third article, co-authored with Han Lou, proves a version of the Hofer-Zehnder conjec- ture for closed semipositive symplectic manifolds whose quantum homology is semisimple; this result generalizes the groundbreaking work of Shelukhin in the spherically positive- monotone setting. The result shows that a Hamiltonian diffeomorphism possessing more contractible fixed points, counted homologically, than the total Betti number of the mani- fold, must have infinitely many periodic points. The key component of the proof is a new study of the effect of reduction modulo a prime on the bounds on filtered Floer homology that arise from semisimplicity. It relies on the theory of algebraic extensions of non-Archimedean normed fields. The fourth article, co-authored with Habib Alizadeh and Dylan Cant, investigates the dis- placeability of a closed Lagrangian submanifold L of a convex-at-infinity symplectic manifold by a compactly supported Hamiltonian diffeomorphism. We conclude that a Hamiltonian diffeomorphism φ whose spectral norm is smaller than some ħ(L) > 0, depending only on L ⊂ W , cannot displace L. Furthermore, we establish a cup-length estimate for the number of action values; when L is rational, this implies a cup-length estimate on the number of intersection points L ∩ φ(L). As a corollary, we demonstrate that the number of fixed points of a Hamiltonian diffeomorphism of a closed rational symplectic manifold (M,ω), whose spectral norm is smaller than the rationality constant, is bounded below by one plus the cup-length of M.

This dissertation explores the quantitative and qualitative properties of the cotangent bundles T ∗M of a closed smooth manifolds M , from the symplectic point of view. Quantitative aspects involve packing the open neighborhood W of the zero section with symplectic balls. We introduce a distance-like function ρW on the zero section M using the symplectic packing of two balls. In the case when W is the unit disc-cotangent bundle associated to the Riemannian metric g, we show how to recover the metric g from ρW . As an intermediate step, we construct a symplectic embedding from the ball B2n(2/√π) of capacity 4 to the product of Lagrangian unit discs Bn(1) × Bn(1). Such a construction implies the strong Viterbo conjecture for Bn(1) × Bn(1). We also give a bound on the relative Gromov width Gr(M, W) when M admits a non-contractible S1-action. The bound is given in terms of the symplectic action of the lift of non-contractible orbits of the S1-action. We also provide examples of when such a bound is sharp. This result is part of the joint work with Dylan Cant. The second part of this joint work is related to the qualitative aspects. We show the existence of periodic orbits of Hamiltonian systems on T ∗M for a large class of Hamiltonians. Another qualitative aspect is proof of the Arnol’d chord conjecture for conormal Legendrians in the co-sphere bundle S∗M . This part of the dissertation is joint work with Dylan Cant and Egor Shelukhin. We show that for a given closed submanifold N ⊂ M there exists a non-constant Reeb chord in (S∗M, α) with endpoints on ΛN := ν∗N ∩ S∗M, for arbitrary contact form α on S∗M which induces standard contact structure.

The group of compactly supported Hamiltonian diffeomorphisms of a symplectic manifold is endowed with a natural bi-invariant distance, due to Viterbo, Schwarz, Oh, Frauenfelder and Schlenk, coming from spectral invariants in Hamiltonian Floer homology. This distance, called the spectral norm, has found numerous applications in symplectic topology. However, its diameter is still unknown in general. In fact, for closed symplectic manifolds there is no unifying criterion for the diameter to be finite or infinite. It has been conjectured that for closed symplectically aspherical manifolds, the spectral norm has infinite diameter. In this thesis, we prove that for any Liouville domain the spectral norm has infinite diameter if and only if its symplectic cohomology does not vanish. This generalizes a result of Monzner-Vichery-Zapolsky and has applications in the setting of closed symplectic manifolds. For instance, we show that the product of two closed symplectically aspherical manifold has an infinite spectral diameter . More generally, we prove that any symplectically aspherical manifold which contains an incompressible Liouville domain of codimension zero with non-vanishing symplectic cohomology must have infinite spectral diameter.

This master’s thesis is concerned with mathematical billiards and distribution of reflection points of periodic trajectories of a strictly convex billiard table. A mathematical billiard is a dynamical system generated by the free motion of a particle inside of a domain with a perfectly reflecting boundary. A question of particular interest in the study of mathematical billiards is that of its periodic trajectories. We consider the case of planar strictly convex billiards. It is known that the reflection points of periodic trajectories of period n making one turn around the table are equidistributed with respect to a natural measure on the boundary. We show this result by a new and relatively elementary method using Lazuktin’s theory [12]. In the first chapter, we give a precise description of billiard dynamics and a brief introduction of Lazuktin’s theory, twist mappings and caustics. In Chapter 2 to 4, we elaborate each of the previous concepts and explain how they are related to billiards. Chapter 5 is dedicated to the proof of our main result, divided into two parts. We conclude by giving an appendix about measure theory.

Using methods of Chern-Weil Theory, Reznikov provides a sufficient condition for the non-triviality of the projectivization \( \mathbb{P}(E) \) of a complex vector bundle \( E \) as a Hamiltonian fibration. In the setting of geometric quantization, Savelyev and Shelukhin introduce a new invariant of Hamiltonian fibrations and a K-theoretic lift of Reznikov's result. This invariant is given by the Atiyah-Singer index of a family of \( \text{Spin}^{c} \)-Dirac operators. In this thesis, we consider Hamiltonian fibrations given by the Cartesian product and the fiber product of a collection of complex projective bundles \( \mathbb{P}(E_1), \cdots, \mathbb{P}(E_r) \). Using the same methods as Savelyev and Shelukhin, we define a family of \( \text{Spin}^{c} \)-Dirac operators acting on sections of a canonical Dirac bundle with values in a suitable prequantum fibration. The family index gives then an invariant of Hamiltonian fibrations with fibers given by a product of complex projective spaces and allows to construct examples of non-trivial Hamiltonian fibrations.

In this master's thesis, we construct a new invariant of Lagrangian cobordisms. This construction is inspired by the recent works of Solomon in which an extension of the Calabi homomorphism to exact Lagrangian paths is given. Solomon's extension was motivated by the fact that the graph of any Hamiltonian isotopy is an exact Lagrangian path. We use the Lagrangian suspension construction, which associates to every exact Lagrangian path a Lagrangian cobordism, to extend Solomon's invariant to Lagrangian cobordisms. In the first chapter, we give a brief introduction to the elementary properties of symplectic manifolds and their Lagrangian submanifolds. In the second chapter, we present an introduction to the group of Hamiltonian diffeomorphisms and discuss the fundamental properties of the Calabi homomorphism. Chapter 3 is dedicated to Lagrangian paths, Solomon's invariant and its critical points. In the last chapter, we introduce the notion of Lagrangian cobordism and we construct the new invariant. We analyze its critical points and evaluate it on the trace of the Lagrangian surgery of two curves on the torus. In this setting we further bound the new invariant in terms of the shadow of the cobordism, a notion recently introduced by Cornea and Shelukhin.