Related papers: Upper bounds for dimensions of singularity categor…

We study several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. We prove new bounds on these dimensions and raise several questions for further investigation. For instance, we…

We show that all closed $2$-dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subspace of small dimension. The entropy and dimension of the…

We consider weighted Hardy inequalities involving the distance function to the boundary of a domain in the $N$-dimensional Euclidean space with nonempty boundary. We give a lower bound for the corresponding best Hardy constant for a domain…

Let $\pi:X\rightarrow Z$ be a Fano type fibration with $\dim X-\dim Z=d$ and let $(X,B)$ be an $\epsilon$-lc pair with $K_X+B\sim_{\RR} 0/Z$. The canonical bundle formula gives $(Z,B_Z+M_Z)$ where $B_Z$ is the discriminant divisor and $M_Z$…

The finitistic dimension of a triangulated category is introduced. For the category of perfect complexes over a ring it is shown that this dimension is finite if and only if the small finitistic dimension of the ring is finite.

We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena…

A new upper bound is given for the dimension of the space of holomorphic cusp forms of weight one and prime level $q$: $$ \hbox{dim}\, S_1(q) << q^{11/12} \log^4{q} $$ with an absolute implied constant.

In this paper we are concerned with the following question: if the tensor product of finitely generated modules $M$ and $N$ over a local complete intersection domain is maximal Cohen-Macaulay, then must $M$ or $N$ be a maximal…

We show there is an upper bound on the diameter of a closed, hyperbolic 3-manifold in terms of the length of any presentation of its fundamental group.

In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for…

We obtain an optimal upper bound for the normalised volume of a hyperplane section of an origin-symmetric d-dimensional cube. This confirms a conjecture posed by Imre Barany and Peter Frankl.

We prove that singularities of area minimizing hypersurfaces can be perturbed away in ambient dimensions 9 and 10.

The paper introduces a number of new techniques to handle minimal hyersurface singularities. In particular, they allow to extend the obstruction theory for postive scalr curvature to any dimension.

Let $\textbf{k}$ be an algebraically closed field. We classify all maximal $\textbf{k}$-subalgebras of any one-dimensional finitely generated $\textbf{k}$-domain. In dimension two, we classify all maximal $\textbf{k}$-subalgebras of…

We study the partial regularity problem of the three-dimensional incompressible Navier--Stokes equations. We present a new boundary regularity criterion for boundary suitable weak solutions. As an application, a bound for the parabolic…

This note is concerned with the Rouquier dimension of the bounded derived category of coherent complexes on a Noetherian algebraic stack. Specifically, we study the diagonal dimension of a morphism, which can be used to produce upper bounds…

We establish upper bounds for the distance to finite-dimensional subspaces in inner product spaces and improve some generalisations of Bessel's inequality obtained by Boas, Bellman and Bombieri. Refinements of the Hadamard inequality for…

We consider a definition of the Fukaya category of a singular hypersurface proposed by Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel's natural transformation, and show that this possesses several desirable…

New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for…

For a commutative local ring $R$, consider (noncommutative) $R$-algebras $\Lambda$ of the form $\Lambda = End_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $\Lambda$ of finite global dimension can…