Time: 5pm ~ 6:15pm (JST); 10am ~ 11:15am (CET); 4pm ~ 5:15pm (Taiwan)

Field: condensed matter physics
Keywords: topological superconductor, Van Hove singularity, Hubbard model, Kagome lattices

Abstract: Competing correlated electronic states are a central topic in condensed matter physics. A typical example is the close competition between spin density wave and d-wave superconductivity in the Hubbard model on the square lattice near half filling where the band structures have saddle points at which the Fermi surface topology changes from hole type to electron type. The saddle points are called van Hove singularity (VHS) points, and host diverging density of states with power-law behavior in the two dimensions. Recently, another type of VHS, namely the higher-order VHS was investigated in ABC-stacked trilayer graphene and twisted bilayer graphene. In this talk, I will first introduce the higher-order VHS, and make comparisons to the conventional VHS. Then I will discuss the enhanced nematicity driven by large flavor number with higher-order VHSs on the square and Kagome lattices. Finally, I will show that robust topological superconductivity can emerge on the square lattice due to interplay of spin-orbital coupling and higher-order VHSs.

At the crossroads of several approaches to quantum gravity, Spinfoams propose a discrete path integral for quantum general relativity built from topological field theory. With the spectrum of geometric operators directly read from the representation theory of the local symmetry group, they can be interpreted as a quantized version of Regge calculus and can be understood as implementing the dynamics of quantum states geometry in loop quantum gravity. I will explain the basics of the formalism, the motivations, the mathematical framework and the main tools. In three space-time dimensions, the spinfoam quantization of 3d gravity is given by the Turaev-Viro topological invariant, which is intimately related to the quantization of Chern-Simons theory. I will explain in particular how the spinfoam amplitudes solve the Wheeler-de Witt equation, implement the invariance under 3d diffeomorphisms (despite being formulated in a discretized space-time) and lead to a quasi-local version of holography. In four space-time dimensions, general relativity can be formulated as an almost-topological theory and I will explain how the existing spinfoam models introduce a sea of topological defects to re-create the gravitational degrees of freedom from a topological path integral. Finally, I will show how spinfoams are naturally defined in terms of group field theory, which are generalized tensor models, and the prospects that this opens. I will conclude with the main challenges and open lines of research of the field.

Program:

July 26
10:00 - 10:15 Registration and reception
10:15 - 11:45 Lecture 1
11:45 - 13:30 Lunch & coffee break
13:30 - 15:00 Lecture 2
15:00 - 16:00 Coffee break
16:00 - 17:00 Lecture 3
17:10 - 18:30 Short talk session

July 27
10:00 - 11:45 Lecture 4
11:45 - 13:30 Lunch & coffee break
13:30 - 15:00 Lecture 5
15:00 - 16:00 Coffee break
16:00 - 17:00 Lecture 6
17:30 - 20:00 Banquet

July 28
10:00 - 11:45 Lecture 7
11:45 - 13:30 Lunch & coffee break
13:30 - 15:00 Lecture 8
15:00 - 16:00 Coffee break
16:00 - 17:30 Lecture 9 & Closing

In these lectures, we will shed light on modern tools of higher algebra, where the traditional structures of algebra yield themselves only after controlled deformations. We will introduce infinity-categories, spectra, operads, and other standard tools of the last decade. The main applications will be to encode various higher-algebraic structures that inevitably arise in, and shed light on, geometry and topology. If time permits, we will illustrate how spectra naturally arise in geometric invariants.

The audience is imagined to consist of mathematicians interested in applications of infinity-categorical tools -- so a broad range of geometers (including topologists) and algebraists. From Lecture Two onward, I will assume basic knowledge of algebraic topology (e.g., the material of Hatcher) and homological algebra.

These lectures will be held between July 31 and August 10, each from 10:30 to 12:00, for a total of 8 lectures.

1st Week: Jul 31(mon), Aug 1(tue) - 3(thu)
- Introduction to ideas of higher algebra in geometry, for a general audience.
- Introduction to infinity-categories and to spectra.

2nd Week: Aug 7(mon) - 10(thu)
- Examples in geometry and topology, including invariants of Legendrian links and generating functions.
- Future Directions.

Profile:
Hiro Lee Tanaka is an assistant professor in the Department of Mathematics. After receiving his Ph.D. from Northwestern University and completing postdoctoral work at Harvard University, he conducted research at the Mathematical Sciences Research Institute in Berkeley, California, and at the Isaac Newton Institute in Cambridge, England. His research aims to fuse the higher structures in modern algebra with geometries emerging from both classical mechanics and supersymmetric field theories. Beyond research, Tanaka engages in efforts to create more equitable and supportive environments throughout the mathematics community.

Since the idea appeared in black hole physics, the concept of holography has become a guiding principle for quantum gravity. It is the notion that the dynamics of the geometry of a region of space-time can be entirely encoded in a theory living on its boundary. Although such holographic dualities have been well-developed in an asymptotical context, it remains a challenge to realize it exactly at finite distances. I will draw a possible route in 3d quantum gravity, by showing a duality between the Ponzano-Regge path integral for 3d quantum gravity as a topological field theory and the 2d (inhomogeneous) Ising model. This leads to an intriguing geometrical interpretation of the Ising critical couplings and opens the door to a possibly rich interplay between 3d quantum gravity and 2d condensed matter built out of holographic dualities.

Old isolated neutron stars have been gathering attention as targets to probe Dark Matter (DM) through temperature observations. DM will anomalously heat neutron stars through its gravitational capture and annihilation process, which predicts surface temperature as T_s ~ (1 − 3) × 10^3 K for t > 10^6 years. We may put constraints on DM-nucleon scattering cross section by finding even colder neutron stars.
This story, however, assumed that there is no relevant heating source for old neutron stars. In this talk, we discuss the creep motion of vortex lines in the neutron superfluid of the inner crust as the heating mechanism. This creep mechanism is inherent in the structure of neutron stars. The heating luminosity is proportional to the time derivative of the angular velocity of the pulsar rotation, and the proportional constant J has an approximately universal value for each neutron star. If this vortex creep heating is quantitatively relevant against DM heating, this mechanism may cause a serious background to probe DM.
The J parameter can be determined from the temperature observation of old neutron stars because the heating luminosity is balanced with the photon emission in the late time. We study the latest data of neutron star temperature observation and find that these data indeed give similar values of J, in favor of the assumption that these neutron stars are heated by the frictional motion of vortex lines. Besides, these values turn out to be consistent with the theoretical calculations of the vortex-nuclear interaction. Integarting all the results, we evaluate the vortex creep heating and conclude its significance against DM heating.

Cosmological observations have led to an extremely precise understanding of the large-scale structure of the Universe. A common assumption is to extrapolate large-scale properties to smaller scales; however, whether this is correct or not is unknown and many well-motivated early Universe scenarios predict substantially different structure formation histories. In this seminar I will discuss two scenarios where nonlinear structures form much earlier than is typically assumed. In the first case, the initial fluctuations are enhanced on small scales leading to either primordial black holes clusters or WIMP minihalos right after matter-radiation equality. In the second, I will show that an additional attractive dark force leads to structure formation even in the radiation dominated Universe. I will furthermore discuss possible observations of such early structure formation including changes to the cosmic microwave background, dark matter annihilation, and when the first galaxies form.