Talks

The generation of large scales white noise is a generic property of the dynamics of physical systems described by local non-linear partial differential equations. Non-linearities prevent the small scale dynamics to be erased by smoothing. Unresolved small scale dynamics act as an uncorrelated (white or Poissonian) noise (seemingly stochastic but actually deterministic) contribution to large scale dynamics. Such is the case for cosmic inhomogeneities. In the standard model of cosmology the primordial density power spectrum is taken to be sub-Poissonian and subsequent non-linear evolutions will inevitably produce white noise which will dominate on the largest scales. Non-observation of white noise on the Hubble scale precludes a power law extrapolation of the power spectrum below one comoving parsec and places severe constraints on a wide variety of phenomena in the early universe, including phase transitions, vorticity and gravitational radiation.

I will discuss the relation between non-compact spin chains and the zero-range processes introduced by Sasamoto-Wadati, Povolotsky and Barraquand-Corwin. The main difference compared to the prime examples of integrable particle processes, namely the SSEP and the ASEP, is that for the models discussed in this talk several particles can occupy one and the same site. Guided by the desire to maintain the integrable structure, I will introduce boundary conditions for these models that are obtained from the boundary Yang-Baxter equation. This allows to define analogues of the open SSEP and ASEP with boundary reservoirs. Some recent exact results concerning these types of integrable non-equilibrium zero-range processes will be discussed.

In this series of lectures, I will give an introduction to the theory of moments of L-functions. I will focus on important examples, such as the moments of the Riemann zeta function and Dirichlet L-functions, as well as some GL_2 families. I will also present some of the important tools for understanding moments, as well as applications of moments.

The notion of congruence (modulo an integer q) was formalised by C. F. Gauss in his Disquisitiones arithmeticae. This is a basic yet fundamental concept in all aspects of number theory. Indeed congruences allow to evaluate and compare integers in way considerably richer than the archimedean order alone permits.

In analytic number theory, several outstanding question -starting with Dirichlet’s theorem on primes in arithmetic progressions- reduce to the of measuring whether some classical arithmetic function (say the characteristic function of prime numbers) correlate with suitable q periodic functions for instance Gauss sums, Jacobi sums or Kloosterman sums. It turns out that these functions, when the modulus q is a prime (to which one can reduce via the Chinese Reminder Theorem) can be recognised as « trace functions». The study of trace functions was initiated by A. Weil in the 1940’s and was pursued by A. Grothendieck in the second half of the century with his refoundation of alge...

The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet known. In this talk, I will propose a new method to rigorously compute the exact degree of each iterate for lattice equations. The strategy is to extend Halburd's method, which is a novel approach to computing the exact degree of each iterate for mappings (ordinary difference equations) from the singularity structure, to lattice equations.

First, I will illustrate, without rigorous discussion, how to calculate degrees for lattice equations using the lattice version of Halburd's method and discuss what problems we need to solve to make the method rigorous. Then, I will provide a framework to ensure that all calculations are accurate and rigorous. If time permits, I would also like to discuss how to detect the singularity structure of a lattice equati...

Many interesting spaces arise as Coulomb branches of 3d N=4 quiver gauge theories, including nilpotent orbit closures and affine Grassmannian slices. These interesting spaces often admit interesting embeddings into one another. For example, one nilpotent orbit closure might be contained inside another. That said, it is much less clear how to describe or construct such an embedding from a purely Coulomb branch perspective. I will discuss joint work with Dinakar Muthiah, where we describe a natural Coulomb branch connection with Coulomb branches: for finite ADE types, the embeddings respect monopole operators thought of as functions on the Coulomb branch. This perspective also allows us to generalize the story, and construct embeddings for arbitrary quivers which have the same property.