Geometric objects that have been investigated in detail so far, such as closed Riemann surfaces, are sometimes locally homogeneous. Loosely speaking, their infinitesimal behavior is the same at each point. In this talk, I would like to explain the idea of investigating such objects using the Lie group theory.In the first part of the talk, I will recall the notions of Lie group actions and their quotient spaces with examples, and then explain the definitions of locally homogeneous spaces and their deformations (Teichmüller spaces). In the second part of the talk, I will consider anti-de Sitter manifolds as a special case, i.e., Lorentzian manifolds of negative constant curvature. As in the Riemannian case, a differential operator called the Laplacian (or the Klein-Gordon operator) is defined on Lorentzian manifolds. Unlike the Riemannian case, it is no longer an elliptic differential operator but a hyperbolic differential operator. In its spectral analysis, new phenomena different from those in the Riemannian case have been discovered in recent years, following pioneering works by Toshiyuki Kobayashi and Fanny Kassel. I would like to explain stable eigenvalues of the hyperbolic Laplacian of anti-de Sitter 3-manifolds with recent progress.

We are now living in the era of precision cosmology. The relic abundance of dark matter (DM) is now observationally well-determined, and its error is smaller than O(1)%. This means that the same or much higher precision is required when we make theoretical predictions.
Weakly Interacting Massive Particle (WIMP) has long been the leading candidate for DM because of its beautiful mechanism to predict the observed relic abundance. WIMP is in the same thermal bath as the Standard Model particles in the beginning. At a certain point when the temperature of the Universe is smaller than the DM mass, it decouples to fix its number density. The yield of the DM is determined by its annihilation cross-section to the Standard Model sector.
It seems that there is no ambiguity in the calculation of this process at first: the cross-section is purely theoretical and all the remainings are described in the Standard Model physics. However, the source of the uncertainty does remain in the Standard Model sector. The dilution of the number density of DM particle depends on the expansion rate of the Universe, which is determined by the Standard Model particles. The effective degree of freedom (d.o.f) of the relativistic species controls this factor. We have to deal with the non-equilibrium dynamics to precisely describe the time-evolution of the d.o.f, in which we need numerical approaches.
In this talk, he introduced his work to update these calculations. By implementing the latest findings in the non-equilibrium dynamics in i) the neutrino decoupling, ii) the QCD phase transition, iii) the electroweak phase transition, and iv) the perturbative calculations, they found that the final d.o.f is smaller than the previous estimate in more than 1%. This is larger than the level of precision in observations. It is also important that the uncertainty is quantified by them.
Another good news is that he makes the calculated d.o.f with its error publically available. With these updates, we now correctly know the points to probe DM!

Through the course of life, from the moment of birth till death, an organism will achieve various states of equilibrium or ‘homeostasis’ which will inevitably encounter perturbations. The processes of cell growth, differentiation, infection, mutation, evolution and adaptation work together as a coordinated ‘system’, described by mathematical models for population dynamics, to maintain a healthy state. Any disruptions to this system leads to disease including infection, allergy, cancer, and aging. We are conducting interdisciplinary research to elucidate “Quantitative Population Dynamics” through the course of life with original mathematical theory and computational simulation, which are both our CORE approach. Our mathematical model-based approach has quantitatively improved a current gold-standard approach essentially relying on the statistical analysis of “snapshot data” during dynamic interaction processes in life sciences research. In this talk, I will explain how our interdisciplinary approach extends our understanding for complicated clinical data and apply real world problem with an example of the Novel Coronavirus Disease, COVID-19.