Primordial black holes (PBHs) are thought to form through gravitational collapse of regions with excessively large density in the early universe, and they could serve as seeds for the formation of galaxies. They are also considered one of the important candidates for cold dark matter (DM). Detecting and constraining the abundance of PBHs can provide an effective constraint on realistic inflationary models. In this talk, I will combine inflation models with gravitational waves (GWs) to discuss cosmological phenomena related to primordial black holes. In particular, I will emphasize a simplified toy model of inflation, which naturally enhances the small-scale scalar perturbations by gluing together two linear potentials with different slopes. The enhanced perturbations can not only generate primordial black holes but also emit gravitational waves through higher-order perturbations. This research demonstrates the significant potential of primordial black hole studies, and it naturally leads to a crucial question of how to accurately estimate the PBH abundance. In the latter part of the talk, I will introduce how to use peaks theory to estimate the abundance of primordial black holes. Our new method works well for any form of the power spectrum, and considering the use of more systematic statistical methods, we believe it is currently the most precise approach in the academic community.

A theorem prover is a tool for the formalization of mathematics, that is, for rigorously expressing and verifying theorems and proofs on a computer. In recent years, the Lean theorem prover has seen progress in the formalization of a wide range of areas of mathematics. In this talk, I will explain formalization of mathematics in Lean from the basics and survey the formalized results achieved to date.

I will introduce the recent development of a method to calculate the (anti)symmetrized wave functions and energies of the ground and low-lying excited states using the unsupervised machine learning technique. I will also introduce the recent attempts to consider the spin-isospin degrees of freedom and extend them to the Dirac equation.

This seminar is co-hosted by GWX-EOS Working Group and iTHEMS-ABBL Joint Astro Study Group.

Abstract:
In the first part of my talk, I will review recent work on Bose-Fermi mixtures with an attractive interaction inducing pairing between bosons and fermions. After discussing a recent experiment on this system [1], which has confirmed predictions obtained by us some time ago within a many-body diagrammatic approach [2], I will present novel results for the compressibility [3] that suggest a metastable nature for the many-body phase observed in [1]. Then, I will discuss the extension of our calculations to two-dimensional Bose-Fermi mixtures [4,5]. The results obtained in 2D challenge previous beliefs formulated for 3D systems.
In the second part, I will discuss attractive polarized Fermi systems, for which the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase was proposed many years ago as a possible superfluid phase. I will discuss how significant precursor FFLO fluctuation effects appear already in the normal phase of polarized Fermi gases at finite temperature [6], and how they could be experimentally detected with ultracold gases. At zero temperature [7], I will discuss how the quasi-particle parameters of the normal Fermi gas change when approaching an FFLO quantum critical point, with a complete breakdown of the quasi-particle picture analogous to what found in heavy-fermion materials at an antiferromagnetic quantum critical point.
Finally, I will discuss a recent joint experimental-theoretical work on the motion of a vortex orbiting a pinned anti-vortex in a strongly interacting Fermi gas [8], highlighting the interplay between Andreev bound states in the vortex core and delocalized thermal excitations in shaping the vortex dynamics.

Recently some researchers gave many studies toward algebro-geometric foundation for tropical geometry. I focused on rational function semifields of tropical curves and characterized them. With this characterization, in this talk, I suggest a definition of ``rational function semifield of dimension one". This definition can write out weight in the term of $\boldsymbol{T}$-algebra homomorphism, and can write balancing condition together with harmonic functions, where both weight and balancing condition are fundamental concepts for tropical varieties and $\boldsymbol{T}$ is the tropical semifield $(\boldsymbol{R} \cup \{-\infty\}, \operatorname{max}, +)$.

Large language models such as ChatGPT are based on deep learning architectures known as Transformers. Owing to their remarkable performance and broad applicability, Transformers have become indispensable in modern AI development. However, it still remains an open question why Transformers perform so well and what the essential meaning of their unique structure is. One possible clue lies in the mathematical correspondence between Hopfield Networks and Transformers.

In this talk, I will first introduce the major developments over the past decade that have significantly increased the storage capacity of Hopfield Networks. I will then review the theoretical correspondence between Hopfield Networks and Transformers. Building on this background, I will present our recent findings: by extending this correspondence to include the hidden-state dynamics of Hopfield Networks, we discovered a new class of Transformers that can recursively propagate attention-score information across layers. Furthermore, we found, both theoretically and experimentally, that this new Transformer architecture resolves the “rank collapse” problem often observed in conventional multi-layer attention. As a result, when applied to language generation and image recognition tasks, it achieves performance surpassing that of existing Transformer-based models.

We propose a conceptual analogy in population genetics to the central dogma of molecular biology. While the central dogma describes the flow of information from DNA to RNA to protein, we posit that under neutrality, a population's demography shapes its underlying genealogy, which in turn determines patterns of genetic variation that give rise to phenotypic variation. At the center of this analogous dogma is the genetic genealogies. Recent advances in inferring the Ancestral Recombination Graph (ARG), a complete record of a population's genealogies, have enabled us to develop a suite of methods that interrogates each stage these fundamental and connected components:

Topological insulators are materials which do not conduct current inside but do conduct at the surface or the edge. The name "topological" comes from the fact that the "shape" of the wavefunction of electrons in topological insulators show non-trivial twist, which can be mathematically characterized by the language of topology. Alongside the development of the study of topological insulators in solids, analogous phenomena were found to exist also in other systems such as photonics, mechanics, geophysics, and active matter. In this seminar, I discuss how the underlying concept of "topology of states" can have a broad impact applicable to various areas in physics, with some emphasis on my own contribution to the field. I aim to structure the first half of my seminar to be accessible to those outside physics, and latter half to be more specialized, covering cutting-edge results.

The 4th quantum computing gathering organized by Quantum Computing Study Group

The ninth installment of the Intensive Lecture Series, organized by the Quantum Gravity Gatherings (QGG) study group at RIKEN iTHEMS, will feature Prof. Norio Kawakami from the Fundamental Quantum Science Program (FQSP) under RIKEN's Transformative Research Innovative Platform (TRIP). Over the course of two days, Prof. Kawakami will deliver a lecture series on quantum many-body systems.

In recent years, insights from quantum many-body physics have become central to research in quantum gravity, where correlation effects induced by gravity play nontrivial roles. By bridging perspectives from gravitational physics and quantum many-body dynamics, one hopes to understand how macroscopic spacetime and its geometric properties emerge from the collective behavior of quantum constituents at microscopic scales.

In this lecture series, Prof. Kawakami will introduce the fundamental properties of correlation effects through representative examples in condensed matter physics. A distinctive aspect of this event is its joint organization with the Fundamental Quantum Science Program (FQSP) at RIKEN. The goal is to further strengthen connections between the quantum gravity, condensed matter, and quantum information communities.

The lectures will be delivered in a blackboard-style format (in English), designed to foster interaction, active participation, and in-depth Q&A discussions. In addition, short talk sessions will be held, giving participants the opportunity to present briefly on topics of their choice. Through this informal and dynamic setting, we hope to spark active interactions among participants and create an environment where ideas can be shared openly and enthusiastically.

Abstract:
Some examples of theoretical methods to treat strongly correlated systems in condensed matter physics are explained. We start with the Kondo effect, which is one of the most fundamental quantum many-body problems and has been intensively studied to date in a wide variety of topics such as dilute magnetic alloys, heavy fermion systems, quantum dot systems, etc. Dynamical mean-field theory (DMFT) is then introduced, which enables us to systematically treat strongly correlated materials such as a Mott insulator. It is shown that the essence of DMFT is closely related to the Kondo effect. Furthermore, we explain how to apply conformal field theory (CFT) to treat correlation effects in one-dimensional electron systems.

Accurate prediction of quantum Hamiltonian dynamics and identification of Hamiltonian parameters are crucial for advancements in quantum simulations, error correction, and control protocols. This talk introduces a machine learning model with dual capabilities: it can deduce time-dependent Hamiltonian parameters from observed changes in local observables within quantum many-body systems, and it can predict the evolution of these observables based on Hamiltonian parameters. The model’s validity was confirmed through theoretical simulations across various scenarios and further validated by two experiments. Initially, the model was applied to a Nuclear Magnetic Resonance quantum computer, where it accurately predicted the dynamics of local observables. The model was then tested on a superconducting quantum computer with initially unknown Hamiltonian parameters, successfully inferring them. We believe that machine learning techniques hold great promise for enhancing a wide range of quantum computing tasks, including parameter estimation, noise characterization, feedback control, and quantum control optimization.