Related papers: A proof of the Fields Conjectures

We initiate the study of subgroups $H$ of the general linear group $GL_{\binom{n}{m}}(R)$ over a commutative ring $R$ that contain the $m$-th exterior power of an elementary group $\bigwedge^mE_n(R)$. Each such group $H$ corresponds to a…

Soergel bimodules and their Hochschild homology are known to be important in the context of link homology. In this article we observe that Soergel bimodules may be naturally identified as the cohomology of well-defined objects in the…

Extension conjecture states that if a simple module over an artin algebra has nonzero first self-extension group then it has nonzero i-th self-extension group for infinitely many positive integers i. It is shown by recollement of…

We study orbits of semigroups of $\text{SL}(2,\mathbb{Z})$, and demonstrate reciprocity obstructions: we show that certain such orbits avoid squares, but not as a consequence of obstructions inherited from an algebraic set, and not as a…

In 2010, V. Futorny and S. Ovsienko gave a realization of $U(\mathfrak{gl}_n)$ as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of $S_1\times S_2\times\cdots\times S_n$, where $S_j$ is…

We solve the first-order classification problem for rings $R$ of polynomials $F[x_1, \ldots,x_n]$ and Laurent polynomials $F[x_1,x_1^{-1}, \ldots,x_n,x_n^{-1}]$ with coefficients in an infinite field $F$ or the ring of integers $\mathbb Z$,…

The $G$-representation variety $R_G(\Sigma_g)$ parametrizes the representations of the fundamental groups of surfaces $\pi_1(\Sigma_g)$ into an algebraic group $G$. Taking $G$ to be the groups of $n \times n$ upper triangular or unipotent…

As a sequel to our proof of the analog of Serre's conjecture for function fields in Part I of this work, we study in this paper the deformation rings of $n$-dimensional mod $\ell$ representations $\rho$ of the arithmetic fundamental group…

Given a finite group G and a G-space X, we show that a direct sum $F_G (X) = \bigoplus_{n \geq 0}K_{G_n} (X^n) \bigotimes \C$ admits a natural graded Hopf algebra and $\lambda$-ring structure, where $G_n$ denotes the wreath product $G \sim…

We consider two dimensional $\mathcal{N}=(4,4)$ superconformal field theories in the moduli space of symmetric orbifolds of K3. We complete a classification of the discrete groups of symmetries of these models, conditional to a series of…

An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ is…

In the presence of an $\Omega$-deformation, local operators generate a chiral algebra in the topological-holomorphic twist of a four-dimensional $\mathcal{N} = 2$ supersymmetric field theory. We show that for a unitary $\mathcal{N} = 2$…

The van der Waerden's Conjecture states that the set $\mathscr{P}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f/\mathbb{Q}}$ of the splitting field…

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This…

We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4].…

A conjecture of Andersen, Masbaum and Ueno states that for any compact oriented surface $\Sigma_{g,n}$ and any pseudo-Anosov $f\in \mathrm{Mod}(\Sigma_{g,n}),$ the matrix $\rho_r(f)$ has infinite order for any large $r,$ where $\rho_r$ is…

Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular…

Given a finite subgroup $W \subset \GL(\fh)$ of the linear group of a finite-dimensional complex vector field $\fh$, it is a well-studied problem to describe the structure of the symmetric algebra $B= \sym(\fh^*)$ as a representation of…

On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra, A…

Consider the action of a subgroup $G$ of the permutation group on the polynomial ring $S := k[x_{1}, \ldots, x_{n}]$ via permutations. We show that if $k$ does not have characteristic two, then the following are independent of $k$: the…