The purpose of this paper and its sequel is to develop the geometry built from the commutative algebras that naturally appear as the homology of differential graded algebras and, more generally, as the homotopy of algebras in spectra. The commutative algebras in question are those in the symmetric monoidal category of graded abelian groups, and, being commutative, they form the affine building blocks of a geometry, as commutative rings form the affine building blocks of algebraic geometry. We name this geometry Dirac geometry, because the grading exhibits the hallmarks of spin in that it is a remnant of the internal structure encoded by anima, it distinguishes symmetric and anti-symmetric behavior, and the coherent cohomology of Dirac schemes and Dirac stacks, which we develop in the sequel, admits half-integer Serre twists. Thus, informally, Dirac geometry constitutes a “square root” of $\mathbb {G}_m$-equivariant algebraic geometry.

We consider a theory of noncommutative Gröbner bases on decreasingly filtered algebras whose associated graded algebras are commutative. We transfer many algorithms that use commutative Gröbner bases to this context. As a result, we have a very efficient way to compute Ext groups for a large class of graded algebras. This has many applications especially in algebraic topology.

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We initiate an approach to simultaneously treat numerators and denominators of Green’s functions arising from quasi-periodic Schrödinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. Let $ (H_{\lambda ,\alpha ,\theta }u) (n)=u(n+1)+u(n-1)+ 2\lambda \cos 2\pi (\theta +n\alpha )u(n)$ be the almost Mathieu operator on $\ell ^2({\mathbb {Z}})$, where $\lambda , \alpha , \theta \in {\mathbb {R}}$. Let

We prove that for any $\theta $ with $2\theta \in \alpha {\mathbb {Z}}+{\mathbb {Z}}$, $H_{\lambda ,\alpha ,\theta }$ satisfies Anderson localization if $|\lambda |>e^{2\beta (\alpha )}$. This confirms a conjecture of Avila and Jitomirskaya (Ann Math (2) 170(1):303–342, 2009) and a particular case of the second spectral transition line conjecture of Jitomirskaya (XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, pp. 373–382, 1995).

We study the linear periods on ${{\,\textrm{GL}\,}}_{2n}$ twisted by a character using a new relative trace formula. We establish the relative fundamental lemma and the transfer of orbital integrals. Together with the spectral isolation technique of Beuzart-Plessis–Liu–Zhang–Zhu, we are able to compare the elliptic part of the relative trace formulae and to obtain new results generalizing Waldspurger’s theorem in the $n=1$ case.