I will describe constraints on the topology of Lefschetz fibrations with 4-dimensional fibers (in the smooth category), obtained from Seiberg—Witten theory. Our main result relates the index of a Dirac operator on the total space with the monodromy of the fibration, and is reminiscent of the Grothendieck—Riemann—Roch formula in the holomorphic setting. As topological applications, we construct closed simply-connected symplectic 4-manifolds which admit Torelli symplectic diffeomorphisms that are not isotopic to products of squared Dehn twists on Lagrangian spheres, answering a question of Donaldson in the negative. Based on joint work with Hokuto Konno, Jianfeng Lin, and Anubhav Mukherjee.
We will consider random Schrödinger operators in 1d, both in the discrete and the continuum setting. It is known that for a large class of potentials, Anderson localisation holds: the spectrum is pure point and the eigenfunctions are exponentially localized. In the weak disorder limit, we will show that a random Dirac operator arises « generically ». This operator displays very nice properties. Joint work with Laure Dumaz (ENS).
A classical result by Maz'ya states that it is possible to bound from below the bottom of the spectrum of the Dirichlet-Bilaplacian of a general open set $\Omega \subseteq \mathbb{R}^N$ in terms of its inradius (namely, the radius of the largest ball inscribed in $\Omega$), provided that $N=2,3$. In a couple of recent papers, Bianchi $\&$ Brasco obtained two novel fractional spectral inequalities linked to this object, extending some classical results. We heavily rely on the ideas and the techniques developed in the above mentioned works, to infer some new optimal spectral bounds for higher-order fractional biharmonic operators.
Based on a work in progress with Francesca Bianchi and Lorenzo Brasco.
How to define an ellipse; its foci and excentricities; the construction of osculating circles; trajectories of billiards; Kepler’s laws about planets; linkages drawing ellipses.
Lattices in C; doubly periodic meromorphic functions; Liouville’s theorem; Weierstrass p-function; poles and zeros; functional algebraic equation between p(z) and p′(z).
Associativity of group law; Mordell’s theorem about finite generation; the rank of an elliptic curve; world records; theorem of Nagel-Lutz; automorphisms; j-invariant.
Integer and rational points on elliptic curves; congruent number problem; reduction modulo primes; applications to cryptography; modular forms; the conjecture of Birch-Swinnerton-Dyer, the ABC-conjecture.
In this talk, we will give an overview of several methods and ideas on how to find regular subgraphs in graphs with some average degree.
We will discuss the still widely opened Erdos-Rado Sunflower Conjecture and we will give a sketch of a recent proof of the Erdos-Szemeredi Conjecture. More precisely, we will show that any graph on $n$ vertices and average degree $C(k)\log(\log(n))$ contains a $k$-regular induced subgraph, which is tight up to the implied constant as shown by a classical construction of Erdos, Pyber and Szemeredi.
