This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical ...

Using a codimension-1 algebraic cycle obtained from the Poincar e line bundle, Beauville de ned the Fourier transform on the Chow groups of an abelian variety A and showed that the Fourier transform induces a decomposition ...

A symplectic manifold is a smooth manifold M together with a choice of a closed non-degenerate two-form. Recent years have seen the importance of associating an A∞-category to M, called its Fukaya category, in helping ...

We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant ...

We prove the equality of two canonical bases of a rank 2 cluster algebra, the greedy basis of Lee–Li–Zelevinsky and the theta basis of Gross–Hacking–Keel–Kontsevich.

We prove that the half plane version of the uniform infinite planar triangulation (UIPT) is recurrent. The key ingredients of the proof are a construction of a new full plane extension of the half plane UIPT, based on a ...

We build the wrapped Fukaya category $\textit{W}$($\textit{E}$)for any monotone symplectic manifold $\textit{E}$, convex at infinity. We define the open-closed and closed-open string maps, OC : HH$_{*}$($\textit{W}$($\textit{E}$)) ...

A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph G. Let ...

We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology over fields of characteristic zero. The key ingredient is the construction of a degree-one ...

Let Y be a quartic hypersurface in P^4 with terminal singularities. The Grothendieck-Lefschetz theorem states that any Cartier divisor on Y is the restriction of a Cartier divisor on P^4 . However, no such result holds ...

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this property remains undecidable among the fundamental groups of compact, ...

The contributions of Lucio Russo to the mathematics of percolation and disordered systems are outlined. The context of his work is explained, and its ongoing impact on current work is described and amplified.

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, ...

The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories ...

In the post-war years, the theory of 3-dimensional manifolds has developed tremendously. On the one hand, Bing and Moise have proved that 3-manifolds can be triangulated, and that the Hauptvermutung (that any two triangulations ...

This thesis is chiefly concerned with a classical conjecture of Littlewood's regarding the L^1-norm of the Fourier transform, and the closely related idempotent theorem. The vast majority of the results regarding these ...

We prove that the topological cycles of an arbitrary infinite graph together with its topological ends form a matroid. This matroid is, in general, neither finitary nor cofinitary.