Related papers: On the Griffiths numbers for higher dimensional si…

There are many generalizations of the McKay correspondence for higher dimensional Gorenstein quotient singularities and there are some applications to compute the topological invariants today. But some of the invariants are completely…

Landau's work on the singularities of Feynman diagrams suggests that they can only be of three types: either poles, logarithmic divergences, or the roots of quadratic polynomials. On the other hand, many Feynman integrals exist whose…

This is the announcement of an alternative approach to the 3-dimensional Poincar\'e Conjecture, different from Perelman's big and spectacular breakthrough. No claim concerning the other parts of the Thurston Geometrization Conjecture, come…

In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over Suzuki algebra $A_{N\,2n+1}^{\mu\lambda}$ and investigates the Nichols algebras over those irreducible Yetter-Drinfeld modules. The finite dimensional…

We provide a fine classification of rigid three-dimensional torus quotients with isolated canonical singularities, up to biholomorphism and diffeomorphism. This complements the classification of Calabi-Yau 3-folds of type $\rm{III}_0$,…

In 2018, Kalck and Yang showed that the singularity categories associated with $3$-dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with…

In general the multiplicity one theorem fails for Fourier-Jacobi models over finite fields. In this paper we prove that there is an upper bound for the multiplicities of Fourier-Jacobi models which is independent of $q$. As a consequence,…

A celebrated result by Keller--Reiten says that $2$-Calabi--Yau tilted algebras are Gorenstein and stably $3$-Calabi--Yau. This note shows that the converse holds in the monomial case: a $1$-Gorenstein monomial algebra with a…

We prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal…

In this paper we investigate three unsolved conjectures in geometric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We formulate natural $\delta$-discretized…

In this paper we prove the validity of Gibbons' conjecture for a coupled competing Gross-Pitaevskii system. We also provide sharp a priori bounds, regularity results and additional Liouville-type theorems.

K. Harada conjectured for any finite group $G$, the product of sizes of all conjugacy classes is divisible by the product of degrees of all irreducible characters. We study this conjecture when $G$ is the general linear group over a finite…

A simple example is given to show that the gauge equivalence classes of physical states in Chern Simons theory are not in one-to-one correspondence with those of Einstein gravity in three spacetime dimensions. The two theories are therefore…

We establish that finite-time singularities do not occur in four-dimensional Yang-Mills flow, confirming the conjecture of Schlatter, Struwe, and Tahvildar-Zadeh. The proof relies on a weighted energy identity and sharp decay estimates in…

We prove that compact Calabi--Yau varieties with certain isolated singularities are projective. In dimension 3 we do this by analysis, supposing given conifold metrics. In higher dimensions it follows more readily from Ohsawa's degenerate…

We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being…

This is a survey of recent studies of singularity formation in solutions of spherically symmetric Yang-Mills equations in higher dimensions. The main attention is focused on five space dimensions because this case exhibits interesting…

In this paper, we use Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity to establish a sharp upper bound for the quotient of the generalized Milnor numbers and the Tjurina numbers for isolated hypersurface…

The well-known Nakai Conjecture concerns a very natural question: For an algebra of finite type over a characteristic zero field, if the ring of its differential operators is generated by the first order derivations, is the algebra regular?…

In Special Geometry there are two inequivalent notions of "Kodaira type" for a singular fiber: one associated with its local monodromy and one with its Hwang-Oguiso characteristic cycle. When the two Kodaira types are not equal the geometry…