Related papers: Ramanujan complexes and Golden Gates in PU(3)
We study the spectra of cyclic signatures of finite graphs and the corresponding cyclic lifts. Starting from a bipartite Ramanujan graph, we prove the existence of an infinite tower of $3$-cyclic lifts, each of which is again Ramanujan.
We prove \cite[Conjecture~5.17]{Clausen} on the local light--profinite structure of smooth $p$-adic analytic Artin stacks. The argument proceeds in several reductions. First, by proving a generalization of van~Dantzig theorem for groupoids,…
The Bruhat-Tits theory is a key ingredient in the construction of irreducible smooth representations of $p$-adic reductive groups. We describe generalizations to arbitrary such representations of several results recently obtained in the…
Consider $(\mathbb{C}^*)^k$ acting on $\mathbb{C}^N$ satisfying certain 'quasi-symmetric' condition which produces a class of toric Calabi-Yau GIT quotient stacks. Using subcategories of $Coh([\mathbb{C}^N / (\mathbb{C}^*)^k])$ generated by…
This paper lays out the foundations of graded $K$-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification. For a row-finite $k$-graph…
This work concerns the study of properties of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain…
Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a…
We prove the 3-fold DT/PT correspondence for K-theoretic vertices via wall-crossing techniques. We provide two different setups, following Mochizuki and following Joyce; both reduce the problem to q-combinatorial identities on word…
We apply the Fixed Point Theorem for the actions of finite groups on Bruhat-Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable,…
Inspired by Manin's approach towards a geometric interpretation of Arakelov theory at infinity, we interpret in this paper non-Archimedean local intersection numbers of linear cycles in $P^{n-1}$ with the combinatorial geometry of the…
N=2 Seiberg-Witten theories allow an interesting interplay between the Argyres-Douglas loci, singularity structures and wall-crossing formulae. In this paper we investigate this connection by first studying the singularity structures of…
Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half…
We fit the three finestructure constants of the Standard Model with three, in first approximation theoretically estimable parameters, 1) a "unifiedscale",turning out not equal to the Planck scale and thus only estimable by a very…
In this paper we prove the Mumford-Tate conjecture in degree 2 for the product of an abelian surface $A$ and a K3 surface $X$ over a finitely generated field $K \subset \mathbb{C}$. The Mumford-Tate conjecture is a precise way of saying…
High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are…
A recent result of Cioab\u{a}, Dewar and Gu implies that any $k$-regular Ramanujan graph with $k\geq 8$ is globally rigid in $\mathbb{R}^2$. In this paper, we extend these results and prove that any $k$-regular Ramanujan graph of…
This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that preprint; more precisely, Proposition II…
The Kumjian-Pask algebra of a higher-rank graph generalises the Leavitt path algebra of a directed graph. We extend the definition of Kumjian-Pask algebra to row-finite higher-rank graphs $\Lambda$ with sources which satisfy a…
The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars-Schneider III(b) system from a quasi-Hamiltonian reduction of the internally fused double SU(n) x SU(n). In particular, the…
Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…