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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...

In our previous study (TI-Mucciconi-Sasamoto, Forum of Mathematics, Pi 11(e27) 1-101,2023) we introduced a deterministic time evolution of a pair of skew semistandard Young tableaux called the skew RSK dynamics.

In this talk we introduce a variant based on the column bumping in the RSK correspondence, which we call the column skew RSK dynamics. Utilizing (bi) crystal structure in the pair of skew tableaux, we show that the column skew RSK dynamics can be mapped to the single species box and ball system (BBS).  Using the mapping we obtain a relation between restricted Cauchy sums about the modified Hall-Littlewood polynomials and the skew Schur polynomials. This talk is based on the joint work with Matteo Mucciconi, Tomohiro Sasamoto and Travis Scrimshaw.

In [1] we introduced the skew RSK dynamics, which is a time evolution for a pair of skew Young tableaux (P,Q). This gives a connection between the q-Whittaker measure and the periodic Schur measure, which immediately implies a Fredholm determinant formula for various KPZ models[2]. The dynamics exhibits interesting solitonic behaviors similar to box ball systems (BBS) and is related to the theory of crystal. 

In this talk we explain basics of the skew RSK dynamics. The talk is based on a collaboration with T. Imamura, M. Mucciconi. 

[1] T. Imamura, M. Mucciconi, T. Sasamoto, 
Skew RSK dynamics: Greene invariants, affine crystals and applications to $q$-Whittaker polynomials, Forum of Mathematics, Pi (2023), e27 1–10. 

[2] T. Imamura, M. Mucciconi, T. Sasamoto, 
Solvable models in the KPZ, arXiv: 2204.08420

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...