Related papers: Generalized 1-skeleta and a lifting result

We develop the theory of almost-holomorphic and quasimodular forms for orthogonal groups of a lattice of signature $(2,n)$ through orthogonal lowering and raising operators. The interactions with the regularized theta lift of Borcherds is a…

We prove a generalization of a result of Peres and Schlag on the dimensions of certain exceptional sets of projections and then apply it to a geometric problem.

Morita equivalence classes of Lie groupoids serve as models for differentiable stacks, which are higher spaces in differential geometry, generalizing manifolds and orbifolds. Representations up to homotopy of Lie groupoids provide a higher…

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds…

This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine…

We recently introduced a notion of tilings of geometric realizations of finite relative simplicial complexes and related those tilings to the discrete Morse theory of R. Forman, especially when they have the property of being shellable, a…

We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.

A key ingredient in the Taylor-Wiles proof of Fermat last theorem is the classical Ihara's lemma which is used to rise the modularity property between some congruent galoisian representations. In their work on Sato-Tate,…

This paper uses rigid Hecke eigensheaves, building on Yun's work on the construction of motives with exceptional Galois groups, to produce the first robust examples of `generalized Kuga-Satake theory' outside the Tannakian category of…

This note is devoted to the theory of projective limits of finite-dimensional Lie groups, as developed in the recent monograph ``The Lie Theory of Connected Pro-Lie Groups'' by K.H. Hofmann and S.A. Morris. We replace the original, highly…

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose…

Let $f$ be a polynomial function over the complex numbers and let $\phi$ be a smooth function over $\mathbb{C}$ with compact support. When $f$ is non-degenerate with respect to its Newton polyhedron, we give an explicit list of candidate…

An approach is shown that proves various theorems of plane geometry in an algorithmic manner. The approach affords transparent proofs of a generalization of the Theorem of Morley and other well known results by casting them in terms of…

We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak…

As a generalization of the classical killing-contractible-complexes lemma, we present algebraic Morse theory via homological perturbation lemma, in a form more general than existing presentations in the literature. Two-sided Anick…

We give an explicit representation for the sums of multiple zeta-star values of fixed weight and height in terms of Riemann zeta values.

Interpolated multiple $q$-zeta values are deformation of multiple $q$-zeta values with one parameter, $t$, and restore classical multiple zeta values as $t = 0$ and $q \to 1$. In this paper, we discuss generating functions for sum of…

Given a configuration $A$ of $n$ points in $\mathbb{R}^{d-1}$, we introduce the higher secondary polytopes $\Sigma_{A,1},\dots, \Sigma_{A,n-d}$, which have the property that $\Sigma_{A,1}$ agrees with the secondary polytope of…

Recently, Lin and Shinder constructed non-trivial homomorphisms from Cremona groups of rank >2 to \mathbb{Z} using motivic techniques. In this short note we propose an alternative perspective from median geometry on their theorem.

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and…