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For $N>1$, we constructed a canonical connected fundamental domain for $Γ_0(N)$ in [Nie, Parent], utilizing an interesting function $W: {\mathbb Z}/N\to {\mathbb N}$. In this paper, we further study the function $W$, prove some identities, and use it to match the cusps, with widths, produced by our connected fundamental domain with the known cusp classes of $Γ_0(N)$. Furthermore, we list the boundary arcs and the gluing patterns of our connected fundamental domain, a key step in understanding the modular curve $X_0(N)$ by this approach.

In this paper, we define the virtual fundamental cycle of a global Kuranishi chart as an element in the (analytic) orbispace K-homology of the virtual orbifold and verify that it defines the same invariants as those in \cite{Abouzaid23}.

The S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by V. Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial etale fundamental group.

In this essay, I argue that the idea that there is a most fundamental discipline, or level of reality, is mistaken. My argument is a result of my experiences with the "science wars", a debate that raged between scientists and sociologists in the 1990's over whether science can lay claim to objective truth. These debates shook my faith in physicalism, i.e. the idea that everything boils down to physics. I outline a theory of knowledge that I first proposed in my 2015 FQXi essay on which knowledge has the structure of a scale-free network. In this theory, although some disciplines are in a sense "more fundamental" than others, we never get to a "most fundamental" discipline. Instead, we get hubs of knowledge that have equal importance. This structure can explain why many physicists believe that physics is fundamental, while some sociologists believe that sociology is fundamental. This updated version of the essay includes and appendix with my responses to the discussion of this essay on the FQXi website.

We give an elementary criterion for the norm of the fundamental unit $\varepsilon_K$ of $K=\mathbb{Q}(\sqrt M)$, $M$ square-free. More precisely, if $\varepsilon_K = a+b\sqrt M$, $a,b \in \mathbb{Z}$ or $\frac{1}{2}\mathbb{Z}$, its norm ${\bf S}_K$ only depends on $m := {\bf gcd} \big(\frac{a+1}{{\bf gcd}(a+1,b)}, M\big)$ and $m' := {\bf gcd} \big(\frac{a-1}{{\bf gcd}(a-1,b)}, M\big)$ as follows when $-1$ is a global norm: ${\bf S}_K = -1$ if and only if $m=m'=1$ (resp. $m=m'=2$) for $M$ odd (resp. even) (Theorems 1.1 or 2.4).

The mainstream view of meaning is that it is emergent, not fundamental, but some have disputed this, asserting that there is a more fundamental level of reality than that addressed by current physical theories, and that matter and meaning are in some way entangled. In this regard there are intriguing parallels between the quantum and biological domains, suggesting that there may be a more fundamental level underlying both. I argue that the organisation of this fundamental level is already to a considerable extent understood by biosemioticians, who have fruitfully integrated Peirce's sign theory into biology; things will happen there resembling what happens with familiar life, but the agencies involved will differ in ways reflecting their fundamentality, in other words they will be less complex, but still have structures complex enough for what they have to do. According to one approach involving a collaboration with which I have been involved, a part of what they have to do, along with the need to survive and reproduce, is to stop situations becoming too chaotic, a concept that accords with familiar 'edge of chaos' ideas. Such an extension of sign theory (semiophysics?) needs to be

This article studies an extended Nori and local fundamental group schemes of Abelian varieties. We also discuss the birational invariance of these group schemes and study their behaviour under the Albanese and étale morphisms.

To truly eliminate Cartesian ghosts from the science of consciousness, we must describe consciousness as an aspect of the physical. Integrated Information Theory states that consciousness arises from intrinsic information generated by dynamical systems; however existing formulations of this theory are not applicable to standard models of fundamental physical entities. Modern physics has shown that fields are fundamental entities, and in particular that the electromagnetic field is fundamental. Here I hypothesize that consciousness arises from information intrinsic to fundamental fields. This hypothesis unites fundamental physics with what we know empirically about the neuroscience underlying consciousness, and it bypasses the need to consider quantum effects.

We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about $k$-graphs when we show that the abelianisation of fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

Theories that attempt to unify the four fundamental interactions and alternative theories of gravity predict time and/or spatial variation of the fundamental constants of nature. Different versions of these theories predict different behaviours for these variations. In consequence, experimental and observational bounds are an important tool to check the validity of such proposals. In this paper, we review constraints on the possible variation of the fundamental constants from astronomical observations and geophysical experiments designed to test the constancy of the fundamental constants of nature over different timescales. We also focus on the limits that can be obtained from white dwarfs, which can constrain the variation of the constants with the gravitational potential.

We initiate a study of the spectral theory of the locally symmetric space $X=Γ\backslash G/K$, where $G=SO(3,Complex)$, $Γ=SO(3,Z[i])$, $K=SO{3}$. We write down explicit equations defining a fundamental domain for the action of $Γ$ on $G/K$. The fundamental domain is well-adapted for studying the theory of $Γ$-invariant functions on $G/K$. We write down equations defining a fundamental domain for the subgroup $Γ_Z=\SO(2,1)_Z$ of $Γ$ acting on the symmetric space $G_R/K_R$, where $G_R$ is the split real form $\SO(2,1)$ of $G$ and $K_R$ is its maximal compact subgroup $\SO(2)$. We formulate a simple geometric relation between the fundamental domains of $Γ$ and $Γ_Z$ so described. We then use the previous results compute the covolumes of of the lattices $Γ$ and $Γ_Z$ in $G$ and $G_R$.

The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs, are given. In particular, it is shown that the spheres are the only ovaloids which are a critical point of the area functional of the second fundamental form under various constraints.

Five fundamental scales of mass follow from holographic limitations, a self-similar law for angular momentum and the basic scaling laws for a fractal universe with dimension 2. The five scales correspond to the observable universe, clusters, galaxies, stars and the nucleon. The fundamental scales form naturally a self-similar hierarchy, generating new relationships among the parameters of the nucleon,the cosmological constant and the Planck scale. There is implied a sixth fundamental scale thatcorresponds to the electrostatic force within an atom. Identifying the implied scale as such leads to new relationships among the fundamental charge, the mass of the electron and cosmological parameters. Theseconsiderations also suggest that structures on the scale of galaxies and larger must be bound by non-Newtonian forces.

Our collective views regarding the question "what is fundamental?" are continually evolving. These ontological shifts in what we regard as fundamental are largely driven by theoretical advances ("what can we calculate?"), and experimental advances ("what can we measure?"). Rarely (in my view) is epistemology the fundamental driver; more commonly epistemology reacts (after a few decades) to what is going on in the theoretical and experimental zeitgeist.

I present the case for fundamental physics experiments in space playing an important role in addressing the current "dark energy'' crisis. If cosmological observations continue to favor a value of the dark energy equation of state parameter w=-1, with no change over cosmic time, then we will have difficulty understanding this new fundamental physics. We will then face a very real risk of stagnation unless we detect some other experimental anomaly. The advantages of space-based experiments could prove invaluable in the search for the a more complete understanding of dark energy. This talk was delivered at the start of the Fundamental Physics Research in Space Workshop in May 2006.

Cosmic Probes of Fundamental Physics take two primary forms: Very high energy particles (cosmic rays, neutrinos, and gamma rays) and gravitational waves. Already today, these probes give access to fundamental physics not available by any other means, helping elucidate the underlying theory that completes the Standard Model. The last decade has witnessed a revolution of exciting discoveries such as the detection of high-energy neutrinos and gravitational waves. The scope for major developments in the next decades is dramatic, as we detail in this report.

The problem of fundamental units is discussed in the context of achievements of both theoretical physics and modern metrology. On one hand, due to fascinating accuracy of atomic clocks, the traditional macroscopic standards of metrology (second, metre, kilogram) are giving way to standards based on fundamental units of nature: velocity of light $c$ and quantum of action $h$. On the other hand, the poor precision of gravitational constant $G$, which is widely believed to define the ``cube of theories'' and the units of the future ``theory of everything'', does not allow to use $G$ as a fundamental dimensional constant in metrology. The electromagnetic units in SI are actually based on concepts of prerelativistic classical electrodynamics such as ether, electric permitivity and magnetic permeability of vacuum. Concluding remarks are devoted to terminological confusion which accompanies the progress in basic physics and metrology.

In this paper, we reject commonly accepted views on fundamentality in science, either based on bottom-up construction or top-down reduction to isolate the alleged fundamental entities. We do not introduce any new scientific methodology, but rather describe the current scientific methodology and show how it entails an inherent search for foundations of science. This is achieved by phrasing (minimal sets of) metaphysical assumptions into falsifiable statements and define as fundamental those that survive empirical tests. The ones that are falsified are rejected, and the corresponding philosophical concept is demolished as a prejudice. Furthermore, we show the application of this criterion in concrete examples of the search for fundamentality in quantum physics and biophysics.

Cosmology is intrinsically intertwined with questions in fundamental physics. The existence of non-baryonic dark matter requires new physics beyond the Standard Model of elemenatary-particle interactions and Einstein's general relativity, as does the accelerating expansion of the universe. Current tensions between various cosmological measurements may be harbingers of yet more new physics. Progress on understanding dark matter and cosmic acceleration requires long term, high-precision measurements and excellent control of systematics, demanding observational programs that are often outside the discovery/characterization mode that drives many areas of astronomy. We outline potential programs through which the Hubble Space Telescope (HST) could have a major impact on issues in fundamental physics in the coming years. To realize this impact, we suggest the introduction of a "HST Fundamental Physics" observational program that would be subject to a modified proposal and review process.