We will discuss the recent work by Rifkind and Virag concerning the shape of eigenvectors for the one-dimensional critical random Schrodinger operator. 

https://arxiv.org/abs/1605.00118

Elliptic curve cryptography (ECC) is a widely used public key cryptosystem. The security of ECC relies on the difficulty of the elliptic curve discrete log problem (ECDLP). Isogenies are morphisms of curves that can be used to transfer instances of ECDLP between elliptic curves. Suppose that we suspect that some proportion of curves are "weak" in the sense that the ECDLP can be solved quickly. To avoid an attacker moving the ECDLP to a weak curve, we would want to use curves for which it difficult to transfer the ECDLP. In this talk we will introduce the notion of an "isolated" curve. These are curves which do not admit many computable isogenies which obstructs the transferring of the ECDLP.

K-theory is a means of probing geometric objects by studying their vector bundles, i.e. parametrized families of vector spaces.  Algebraic K-theory, the version applying to varieties and schemes, is a particularly deep and far-reaching invariant, but it is notoriously difficult to compute.  The primary means of computing it is through its "cyclotomic trace" map K→TC to another theory called topological cyclic homology.  However, despite the enormous computational success of these so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of the cyclotomic trace has remained mysterious.

In this talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry.  By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices.  No prior knowledge of algebraic K-theory or derived algebraic geometry will be assumed.

This represents joint work with David Ayala and Nick Rozenblyum.

We consider a family of quasiperiodic solutions of the nonlinear Schrodinger equation on the 2-torus, namely the family of finite-gap solutions (tori). These solutions are inherited by the 2D equation from its completely integrable 1D counterpart (NLS on the circle) by considering solutions that only depend on one variable. Despite being linearly stable, we prove that these tori (under some genericness conditions) are nonlinearly unstable in the following strong sense: there exists solutions that start very close to those tori in certain Sobolev spaces, but eventually become larger than any given factor at later times. This is the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed. (joint work with M. Guardia (UPC, Barcelona), E. Haus (University of Naples), M. Procesi (Roma Tre), and A. Maspero (SISSA))

 We consider a family of quasiperiodic solutions of the nonlinear Schrodinger equation on the 2-torus, namely the family of finite-gap solutions (tori). These solutions are inherited by the 2D equation from its completely integrable 1D counterpart (NLS on the circle) by considering solutions that only depend on one variable. Despite being linearly stable, we prove that these tori (under some genericness conditions) are nonlinearly unstable in the following strong sense: there exists solutions that start very close to those tori in certain Sobolev spaces, but eventually become larger than any given factor at later times. This is the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed. (joint work with M. Guardia (UPC, Barcelona), E. Haus (University of Naples), M. Procesi (Roma Tre), and A. Maspero (SISSA))