Sala Seminari (Dip. Matematica)
Abstract: In this talk I will review some new results on the depth of ordinary and symbolic powers as well as integral closures of powers of monomial ideals in order to show a deep connection between Commutative Algebra and some objects in Combinatorics such as simplicial complexes, integral points in polytopes and graphs.
Cryptography, Quantum Computation and Selberg's Sieve
We will discuss Shor's algorithm for factorization of integers using quantum computers. This algorithm employs Miller's algorithm, which we will also discuss. We will then explore an alternative, non-quantum, attempt to exploit Miller's algorithm. To do this we will introduce a new sort of discrete dynamical system, based on number theory. We will go on to begin analyzing this dynamical system using the theory of sieves.
Given a Heegaard splitting of a manifold M, one can consider simple
closed curves on the Heegaard surface as elements of its curve graph. I
will discuss the result that if some such curve K is far from both disk
sets (as measured in the curve graph), then the complement M-K is
hyperbolic. Moreover, there is a condition involving subsurface
projection that further ensures that M is obtained by long Dehn filling
of M-K, yielding that M is hyperbolic and (the geodesic representative
of) K is short.
In this talk I will describe some invariants for transverse links in
S^3 (endowed with the symmetric contact structure) arising from the
deformations of Khovanov sl_3 homology.
I will start with a brief introduction to the theory of transverse
links in S^3. Afterward, I will recall some known results concerning
transverse invariants in link homologies. In particular, I will focus
on the invariants coming from Khovanov-Rozansky homologies, and those
coming from the deformations of Khovanov homology.
Abstract: In the last decades a zoo of new periodic orbits for the n-body type problem appeared in literature as critical points of the Lagrangian action functional. However, in order to penetrate the intricate dynamics of this singular problem as well as to understand the topological properties of the loop space of the configuration manifold, one first step is to establish the relation (if any) between the Morse index and the stability property of the orbit as well as the index property of both colliding and unbounded motions.
Spesso in matematica capita che alcuni oggetti siano costruiti incollando un modello locale più semplice; ad esempio, le varietà lisce sono localmente R^n, e gli schemi sono localmente spettri di un anello.
Nella prima parte descriveremo il processo che porta dal modello locale all'oggetto globale per una categoria qualsiasi, in modo preciso. Mostreremo che ogni funtore (co)continuo si estende in maniera unica agli oggetti incollati. Questo ci permetterà di dimostrare dei teoremi non banali sulla rappresentabilita' di certi funtori.