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.

We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design. Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup. We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries. For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries. Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group. This work will open a new perspective into quantum information processing such as randomized benchmarking, and give a deep understanding to many-body systems such as monitored random circuits.

Information on the phase structure of strongly interacting matter at high baryon densities can be gained from observations of neutron stars and their detailed analysis. Bayesian inference methods are used to set constraints on the speed of sound in the interior of neutron stars, based on recent multimessenger data in combination with low-density constraints based on chiral effective field theory and perturbative QCD constraints at asymptotically high densities.
A detailed re-analysis is performed in order to clarify the influence of the latter constraints on the inference procedure. The impact of the recent new heavy (2.35 M_sol) black widow pulsar PSR J0952-0607 and of the unusually light supernova remnant HESS J1731-347 is inspected. One of the consequences of including PSR J0952-0607 in the database is a further stiffening of the equation-of-state, resulting in a 2.1 solar-mass neutron star in a reduced central density of less than five times the equilibrium density of normal nuclear matter.
A systematic Bayes factor assessment quantifies the evidence (or non-evidence) for small sound speeds, necessary for a strong first-order phase transition, within the range of densities realized in the core of neutron stars. Given the presently existing database, it can be concluded that the occurrence of a strong first-order phase transition in the core of even a 2.1 solar-mass neutron star is unlikely, while a continuous crossover cannot be ruled out.

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.

We introduce the first large-scale dataset, MNISQ, for both the Quantum and the Classical Machine Learning community during the Noisy Intermediate-Scale Quantum era. MNISQ consists of 4,950,000 data points organized in 9 subdatasets. Building our dataset from the quantum encoding of classical information (e.g., MNIST dataset), we deliver a dataset in a dual form: in quantum form, as circuits, and in classical form, as quantum circuit descriptions (quantum programming language, QASM). In fact, also Machine Learning research related to quantum computers undertakes a dual challenge: enhancing machine learning by exploiting the power of quantum computers, while also leveraging state-of-the-art classical machine learning methodologies to help the advancement of quantum computing. Therefore, we perform circuit classification on our dataset, tackling the task with both quantum and classical models. In the quantum endeavor, we test our circuit dataset with Quantum Kernel methods, and we show excellent results with up to 97% accuracy. In the classical world, the underlying quantum mechanical structures within the quantum circuit data are not trivial. Nevertheless, we test our dataset on three classical models: Structured State Space sequence model (S4), Transformer, and LSTM. In particular, the S4 model applied on the tokenized QASM sequences reaches an impressive 77% accuracy. These findings illustrate that quantum circuit-related datasets are likely to be quantum advantageous, but also that state-of-the-art machine learning methodologies can competently classify and recognize quantum circuits. We finally entrust the quantum and classical machine learning community.

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.