Talks

In this talk, I'll describe some recently discovered connections between one-dimensional interacting particle models (the ASEP and the TAZRP) and Macdonald polynomials and show the combinatorial objects that make these connections explicit. Recently, a new tableau formula was found for the modified Macdonald polynomial $\widetilde{H}_{\lambda}$ in terms of a queue inversion statistic that is naturally related to the dynamics of the TAZRP. We give a new compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ using the same queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula are the ASEP polynomials, which specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle, and the queue inversion statistic encodes to the dynamics of the ASEP.  Our tableaux are in bijection with Martin's multiline queues, from which we obtain an alternative multiline queue formula for $P_{\la...

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...

In this talk, the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice is revisited. A central role is played by a quantum oscillator-weighted five vertex model, which features an unusual weight conservation distinct from the conventional one. This approach clarifies the interrelations among several known results and refines their derivations, including the multiline queue construction and matrix product formulas. (Joint work with Masato Okado and Travis Scrimshaw)

The quantum K-theory of the flag variety is a ring defined by introducing a quantum product to the K-theory of the flag variety. Under appropriate localization, it is known that the following three rings (i), (ii), and (iii) are isomorphic, and this property allows for a detailed investigation of each ring: (i)the coordinate ring of the phase space of the relativistic Toda lattice, (ii) the quantum equivariant K-theory of the flag variety, and (iii) the K-equivariant homology ring of the affine Grassmannian.

The isomorphism between (i) and (ii) is derived from the Lax formalism of the relativistic Toda lattice [Ikeda-Iwao-Maeno]. The isomorphism between (ii) and (iii) is referred to as the K-Peterson isomorphism [Lam-Li-Mihalcea-Shimozono, Kato, Chow-Leung, Ikeda-Iwao-Maeno]. In this talk, I will outline how techniques from classical integrable systems, such as the construction of algebraic solutions and Bäcklund transformations, are applied to the study of geometry. This talk is ba...

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...