Talks

Combinatorics has constantly evolved from the mere counting of classes of objects to the study of their underlying algebraic or analytic properties, such as symmetries or deformations. This was fostered by interactions with in particular statistical physics, where the objects in the class form a statistical ensemble, where each realization comes with some probability. Integrable systems form a special subclass: that of systems with sufficiently many symmetries to be amenable to exact solutions. In this talk, we explore various basic combinatorial problems involving discrete surfaces, dimer models of cluster algebra, or two-dimensional vertex models, whose (discrete or continuum) integrability manifests itself in different manners: commuting operators, conservation laws, flat connections, quantum Yang-Baxter equation, etc. All lead to often simple and beautiful exact solutions.

It is well known that the box-ball system discovered by Takahashi and Satsuma can be obtained by the ultra-discrete analogue of the discrete integrable system, including both the ultra-discrete analogue of the KdV lattice and the ultra-discrete analogue of the Toda lattice. This mini-course will demonstrate that it is possible to derive extended models of the box-ball systems related to the relativistic Toda lattice and the fundamental Toda orbits, which are obtained from the theory of orthogonal polynomials and their extensions. We will first introduce an elementary procedure for deriving box-ball systems from discrete KP equations. Then, we will discuss the relationship between discrete Toda lattices and their extensions based on orthogonal polynomial theory, and outline the exact solutions and ultra-discretization procedures for these systems. Additionally, we will introduce the box-ball system on R, which is obtained by clarifying its relationship with the Pitman transformation in ...

It is commonly believed that logical states of quantum error-correcting codes have to be highly entangled such that codes capable of correcting more errors require more entanglement to encode a qubit. Here we show that this belief may or may not be true depending on a particular code. To this end, we characterize a tradeoff between the code distance d quantifying the number of correctable errors, and geometric entanglement of logical states quantifying their maximal overlap with product states or more general "topologically trivial" states. The maximum overlap is shown to be exponentially small in d for three families of codes: (1) low-density parity check (LDPC) codes with commuting check operators, (2) stabilizer codes, and (3) codes with a constant encoding rate. Equivalently, the geometric entanglement of any logical state of these codes grows at least linearly with d. On the opposite side, we also show that this distance-entanglement tradeoff does not hold in general. For any constant d and k (number of logical qubits), we show there exists a family of codes such that the geometric entanglement of some logical states approaches zero in the limit of large code length. This work was done by the collaboration with Sergey Bravyi, Zhi Li, and Beni Yoshida. (https://arxiv.org/abs/2405.01332)

Our Cosmological Standard Model, LambdaCDM, is a remarkable success story. It describes our Universe’s evolution from the Big Bang until today in terms of only a small handful of parameters. Despite its many successes, LambdaCDM is not a fundamental theory. In particular, the microscopic origin of dark matter and dark energy remain among the greatest puzzles in modern physics. Of the two, dark energy poses a particularly vexing challenge, as we lack an understanding of the smallness of its value. At the same time, over the last decade, observations have revealed further cracks in the LambdaCDM model, manifesting as discrepancies between early and late universe determinations of its parameters. 
 

In this lecture, I will first review the LambdaCDM model and establish why it is considered our best model of the Universe. In the second part, I will discuss the intriguing possibility that the cosmic tensions, referring to the observational and theoretical challenges mentioned above, are...

It is well known that the box-ball system discovered by Takahashi and Satsuma can be obtained by the ultra-discrete analogue of the discrete integrable system, including both the ultra-discrete analogue of the KdV lattice and the ultra-discrete analogue of the Toda lattice. This mini-course will demonstrate that it is possible to derive extended models of the box-ball systems related to the relativistic Toda lattice and the fundamental Toda orbits, which are obtained from the theory of orthogonal polynomials and their extensions. We will first introduce an elementary procedure for deriving box-ball systems from discrete KP equations. Then, we will discuss the relationship between discrete Toda lattices and their extensions based on orthogonal polynomial theory, and outline the exact solutions and ultra-discretization procedures for these systems. Additionally, we will introduce the box-ball system on R, which is obtained by clarifying its relationship with the Pitman transformation in ...

Totally positive (TP) matrices are matrices in which each minor is positive. First introduced in 1930's by I. Schoenberg and F. Gantmakher and M. Krein, these matrices proved to be important  in many areas of pure and applied mathematics. The notion of total positivity was generalized by G. Lusztig in the context of reductive Lie groups and inspired the discovery  of cluster algebras by S. Fomin and A. Zelevinsky.

In this mini-course, I will first review some basic features of TP matrices, including their spectral properties and discuss some of their classical applications. Then I will focus on weighted networks parametrization of TP matrices due to A. Berenstein, S. Fomin and A. Zelevinsky. I will show how elementary transformations of planar networks lead to criteria of total positivity and important examples of mutations in the theory of cluster algebras. Finally, I will explain how particular sequences of mutations can be used to construct exactly solvable nonlinear dynamical sy...

In our study, we model the heart's biological system using percolation on a 2D lattice with three types of cells: Active, Waiting, and Inactive. The system incorporates inhibitory and refractory effects, influenced by two parameters: p_{act}, the probability of a Waiting cell becoming Active, and p_{switch}, the probability of an Inactive cell becoming Waiting. Inactive cells undergo a refractory period before reverting to Waiting.

Our findings show that inhibition raises the percolation threshold, slowing signal propagation. Conversely, reducing the refractory time lowers the threshold and speeds up signal transmission, but it can also trap the signal within the system. We analyzed the distribution of Inactive cells and critical exponents, observing that the Rushbrooke inequality is satisfied.

We consider \emph{bond percolation games} on the $2$-dimensional square lattice in which each edge (that is either between the sites $(x,y)$ and $(x+1,y)$ or between the sites $(x,y)$ and $(x,y+1)$, for all $(x,y) \in \mathbb{Z}^{2}$) has been assigned, \emph{independently}, a label that reads \emph{trap} with probability $p$, \emph{target} with probability $q$, and \emph{open} with probability $1-p-q$. Once a realization of this labeling is generated, it is revealed in its entirety to the players before the game starts. The game involves a single token, initially placed at the origin, and two players who take turns to make \emph{moves}. A \emph{move} involves relocating the token from where it is currently located, say the site $(x,y)$, to any one of $(x+1,y)$ and$(x,y+1)$. A player wins if she is able to move the token along an edge labeled a target, or if she is able to force her opponent to move the token along an edge labeled a trap. The game is said to result in a draw if it cont...

Totally positive (TP) matrices are matrices in which each minor is positive. First introduced in 1930's by I. Schoenberg and F. Gantmakher and M. Krein, these matrices proved to be important  in many areas of pure and applied mathematics. The notion of total positivity was generalized by G. Lusztig in the context of reductive Lie groups and inspired the discovery  of cluster algebras by S. Fomin and A. Zelevinsky.

In this mini-course, I will first review some basic features of TP matrices, including their spectral properties and discuss some of their classical applications. Then I will focus on weighted networks parametrization of TP matrices due to A. Berenstein, S. Fomin and A. Zelevinsky. I will show how elementary transformations of planar networks lead to criteria of total positivity and important examples of mutations in the theory of cluster algebras. Finally, I will explain how particular sequences of mutations can be used to construct exactly solvable nonlinear dynamical sy...