Trinh Khanh Duy (Waseda University)

Title: Gaussian beta ensembles and orthogonal polynomials

Abstract: For Gaussian beta ensembles, generalizations of Gaussian orthogonal/unitary/symplectic ensembles in terms of the joint density, the parameter $\beta$ is viewed as the inverse temperature. As $N$ tends to infinity, when $\beta$ is fixed, or more generally, when $\beta N$ tends to infinity, their empirical distribution converges weakly to a semicircle distribution, almost surely. Here $N$ is the system size. Gaussian fluctuations around the limit have been well-studied in which Chebyshev polynomials are involved. This talk introduces such types of results in two more different regimes: a high temperature regime ($N$ tends to infinity with $\beta N$ staying bounded) and a freezing regime ($N$ is fixed and $\beta$ tends to infinity). Associated Hermite polynomials and duals of Hermite polynomials will naturally appear.

Reika Fukuizumi (Tohoku University)

Takahiro Hasebe (Hokkaido University)

Title: Time-inhomogeneous branching processes and Loewner chains

Abstract: Branching processes are a Markov process model for population. Discrete-state branching processes were introduced by Galton-Watson in 1874 (and independently by Bienayme in 1845) to describe extinction of families. Continuous-state branching processes were much later introduced by Jirina in 1958. Branching processes are well investigated under the assumption of time-homogeneity. This talk presents recent work of the speaker on the time-inhomogeneous case which is much less studied in the literature. The analysis method is quite a new and is based on ``reverse evolution families'' and / or ``Loewner chains'' developed in complex analysis. This is a joint work with Jose Luis Perez and Pavel Gumenyuk.

Seiichiro Kusuoka (RIMS)

Title: $\Phi ^4_3$-quantum field theory and a construction of a nontrivial and rotation invariant $\Phi^4_3$-measure

Abstract: In this talk we recall the background of the constructive $\Phi^4_3$-quantum field theory and introduce a result on a construction of a rotation invariant $\Phi ^4_3$-measure. Our construction is based on the stochastic quantization and the method of singular SPDEs. The difference from previous results is applying an approximation given by regularization and localization of the interaction. In particular, we do not apply approximations by enlargement of a torus. The choice of the approximation enables us to show the rotation invariance of the constructed measure. This talk is based on a joint work with Sergio Albeverio.

Hiroki Moriya (Tokyo Institute of Technology)

Title: Macroscopic approach for the large deviation of the symmetric simple exclusion process

Abstract: We study the current fluctuation of the symmetric simple exclusion process (SEP). To calculate its large deviation, we use the macroscopic fluctuation theory (MFT) that provides the path integral formulation for the diffusive model. The saddle point equation can be mapped into the classically integrable PDEs through a generalization of the canonical Cole-Hopf transformation. The inverse scattering method (ISM) enables us to analyze the solutions of this PDEs and we derive the large deviation function of the current starting from the step initial condition. It coincides with the formula obtained previously by microscopic calculations. This provides the first analytic confirmation of the validity of the MFT for an interacting model in the time dependent regime.

This talk is based on a joint work with Kirone Mallick and Tomohiro Sasamoto.

Kohei Motegi (Tokyo University of Marine Science and Technology)

Title: Last passage percolation models and refined dual Grothendieck polynomials

Abstract: Grothendieck polynomials, which are symmetric functions related with Schubert calculus, also appear as various objects in various types of integrable models. Recently, Yeliussizov discovered that the transition probabilities of some last passage percolation (LPP) model are expressed as the dual Grothendieck polynomials. We (M-Scrimshaw) extended Yeliussizov's correspondence to an inhomogeneous version of the LPP model, and showed the correspondence between transition probabilities and the skew and refined version of dual Grothendieck polynomials. Using this correspondence, we gave a probability theoretic derivation of finite version of Littlewood, Cauchy and integration formulas for refined dual Grothendieck polynomials. I will mainly explain this probability theory approach to study this symmetric function, and also briefly touch upon the ongoing research related to this topic, which is a joint work with Shinsuke Iwao (Keio) and Travis Scrimshaw (Hokkaido).

Shuta Nakajima (Meiji University)

Title: Sharp threshold sequence and universality for Ising perceptron models

Abstract: ​In this talk, we discuss a family of Ising perceptron models with {0,1}-valued activation functions. This includes the classical half-space models and some of the symmetric models considered in recent works. For each of these models, we show that the free energy is self-averaging, there is a sharp threshold sequence, and the free energy is universal concerning the disorder. A prior work by Xu (2019) used very different methods to show a sharp threshold sequence in the half-space Ising perceptron with Bernoulli disorder. Recent works of Perkins-Xu (2021) and Abbe-Li-Sly (2021) determined the sharp threshold and the limiting free energy in a symmetric perceptron model. The results apply in more general settings and are based on new "add one constraint" estimates extending Talagrand's estimates for the half-space model. This talk is based on joint work with Nike Sun.

Yoshiko Ogata (University of Tokyo)

Title: An invariant of symmetry protected topological phases with on-site finite group symmetry for two-dimensional Fermion systems

Abstract: We consider SPT-phases with on-site finite group G symmetry for two-dimensional Fermion systems. We derive an invariant of the classification. This invariant is different from what is predicted from quantum field theory. However, if we additionally require the CPT-symmetry, which is automatic in quantum field theory, our invariant reduces to the predicted one.

Shota Sakamoto (Tokyo Institute of Technology)

Title: A Cauchy problem of the Boltzmann equation without angular cutoff near an equilibrium in low-regularity spaces

Abstract: We consider a Cauchy problem of the Boltzmann equation, which is a fundamental model of the dynamics of a dilute gas. This equation is an integro-differential equation, and a kernel of the integral operator is known to have angular singularity coming from collisions of particles. We are interested in the case where this singularity governs the structure of the operator.

Fundamental results to this problem near an equilibrium were given in Sobolev spaces which enable us to use embedding inequalities. In our recent studies, we found a global-in-time solution in a space where we did not employ the embeddings. This is due to a difference of spatial structures, which will be explained in the talk.

This talk is based on two joint works, one is with R.-J. Duan (Chinese University of Hong Kong), S.-Q. Liu, (Central China Normal University) and R. Strain (University of Pennsylvania), and the other is with R.-J. Duan and Yoshihiro Ueda (Kobe University).

Hirohiko Shimada (National Institute of Technology, Tsuyama College )

Title: Four-point Functions in 1+1d Non-relativistic CFTs

Abstract: Scale-invariant field theories with dynamical exponent z=2, known as non-relativistic conformal field theories (NRCFT), are important as they describe universality classes in the cold atom systems and in non-equilibrium statistical physics. To date, only few exact results are known since they have only Schrödinger symmetry. From the RG perspective, however, it is possible that the theory space of the NRCFTs in 1+1d may be rich as in the 2d CFTs.

We use the path-integral to calculate the exact four-point functions of fundamental fields and extract the operator product expansions to show that there is a one-parameter family of NRCFTs, which may be closely related to the sine-Gordon model with z=1. We also discuss the Schrödinger block, an analogue of the conformal block, obtained as a function of the (intermediate) scaling dimension and three generalised cross-ratios, which may be of use in bootstrapping certain classes of non-equilibrium phase transitions. This is based on joint works with Hidehiko Shimada.

Hayate Suda (Keio University )

Title: Seat number configuration of the box-ball system

Abstract: The box-ball system (BBS) is a cellular automaton that exhibits the solitonic behavior, introduced by [Takahashi-Satsuma-90]. It is known that the dynamics of the BBS can be linearized by several methods. In this talk, we introduce the notion of the carrier process with seat numbers and the corresponding seat number configuration (SNC). We explain that SNC gives a new linearization method of the BBS, and that some explicit relationships between existing linearization methods are obtained via the language of SNC. We also explain an application of our results to the generalized hydrodynamic limit for the BBS with finite carrier capacity. This talk is based on the joint work with Matteo Mucciconi, Makiko Sasada and Tomohiro Sasamoto.

Taiji Suzuki (University of Tokyo)

Title: Convergence of mean field gradient Langevin dynamics for optimizing two-layer neural networks

Abstract: In this talk, we discuss optimization procedures of two-layer neural networks via the gradient Langevin dynamics in a mean field regime. For that purpose, we first introduce a theoretical guarantee of a linear convergence of the mean field gradient Langevin algorithm in the infinite width limit under a uniform log-Sobolev inequality condition. Next, we propose some specific optimization methods for a finite width and discrete time setting. The constructions of those methods are based on the convex optimization techniques for finite dimensional objectives. Finally, we discuss the linear convergence of a vanilla gradient Langevin dynamics without an infinite width limit under a bit stronger condition than the uniform log-Sobolev inequality.

Kazuyuki Wada (National Institute of Technology, Hachinohe College)

Title: Spectral theory and weak limit theorem for quantum walks in 1d

Abstract: We consider a discrete time two-state quantum walk (QW) in one-dimension. A time evolution operator associated with QW has a form of a product of a shift operator and a coin operator. If a coin operator is space-homogeneous, we can derive the limit distribution by the Fourier transformation. If a coin operator is space-inhomogeneous, spectral theory for unitary operator is valid to derive the limit distribution. In this talk, we focus on the long-range type quantum walks. This talk is based on joint works with Masaya Maeda (Chiba Univ.) and Akito Suzuki (Shinshu Univ.).

Hironobu Yoshida (University of Tokyo)

Title: Majorana reflection positivity in the attractive SU(N) Fermi-Hubbard Model

Abstract: Recent advances in experimental techniques have made it possible to simulate multicomponent fermions with ultracold atoms. In particular, N-component fermions with SU(N) symmetry in optical lattices have attracted attention because they are predicted to exhibit various exotic phases that do not appear in the SU(2) counterpart. These systems are well described by the SU(N) Fermi-Hubbard model, but this model is notoriously difficult to analyze mathematically and new theoretical tools must be developed.

In this talk, I would like to present theorems on the ground states of the attractive SU(N) Fermi-Hubbard model: the degeneracy, the SU(N) quantum number, and the long-range order. To prove the theorems, I make full use of a method of Majorana reflection positivity, inspired by recent progress in the sign-problem-free quantum Monte Carlo simulations.

Nobuyuki Yoshioka (University of Tokyo)

Title: Neural-net representation of quantum many-body states

Abstract: Achieving both quantitative description and qualitative understanding on quantum matters has long remained as one of the most intriguing yet challenging problems in quantum science. Although empirical/deductive approaches based on physical insights of scientists have been very successful, it has been nontrivial how to obtain quantitative description of strongly correlated materials, interfacial chemical reactions, and so on. In this talk, I aim to walk through the novel approach of utilizing artificial neural networks to represent quantum many-body wave functions, and overview its application in condensed matter physics, statistical physics, quantum chemistry, and quantum information. We first introduce the variational wave function form which is based on the (restricted) Boltzmann machine, and explain its characteristics and optimization. Then, we further explore the frontier of wider network structures including convolutional neural networks, autoregressive models, and more generalized ones. After overviewing the state-of-the-art achievement by neural quantum states in variational calculation, we also discuss the application of neural networks for quantum state tomography, quantum control, etc. Finally, we discuss the future direction of the entire research field.