相关论文: On a Proof of a Conjecture of Marino-Vafa on Hodge…

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We…

We complete the proof of the Howe duality conjecture in the theory of local theta correspondence by treating the remaining case of quaternionic dual pairs in arbitrary residual characteristic.

A constructible sheaf corresponding to Gel'fand Zelevinski hypergeometric functions on a torus is called hypergeometric sheaf. We consider Hodge and Tate conjectrue for hypergeomtric sheaves. Hodge conjecture is formulated in terms of…

Let $\nu_{f}(n)$ be the $n$-th nomalized Fourier coefficient of a Hecke--Maass cusp form $f$ for ${\rm SL}(2,\Z)$ and let $\alpha$ be a real number. We prove strong oscillations of the argument of $\nu_{f}(n)\mu (n) \exp (2\pi i n \alpha)$…

In this paper we extend the discussion on Homological Mirror Symmetry for Fano toric varieties presented by Hori and Vafa to more general case of monotone symplectic manifolds with real polarizations. We claim that the Hori -- Vafa…

Following an article of Dettweiler and Sabbah, this article studies the behaviour of various Hodge invariants by middle additive convolution with a Kummer module. The main result gives the behaviour of the nearby cycle local Hodge numerical…

Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of…

We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.

The Rota--Heron--Welsh conjecture (now a theorem of Adiprasito, Huh, and the author) asserts the log-concavity of the characteristic polynomial of matroids. We give an exposition of the Lorentzian polynomial proof following the work of…

We consider the conjectures from the paper by Katzarkov, Kontsevich, and Pantev about Landau-Ginzburg Hodge numbers associated to tamely compactifiable Landau-Ginzburg models. We test these conjectures in case of dimension two, verifying…

We study complexified elliptic Calogero-Moser integrable systems. We determine the value of the potential at isolated extrema, as a function of the modular parameter of the torus on which the integrable system lives. We calculate the…

We prove a conjecture of Schmid and the second named author that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof…

Khabibullin's conjecture for integral inequalities has two numeric parameters $n$ and $\alpha$ in its statement, $n$ being a positive integer and $\alpha$ being a positive real number. This conjecture is already proved in the case where…

Let $p(n)$ be the partition function. Ahlgren and Ono conjectured that every arithmetic progression contains infinitely many integers $N$ for which $p(N)$ is not congruent to $0\pmod{3}$. Radu proved this conjecture in 2010 using work of…

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0, \ldots, n$, is algebraic. We show that this holds for all degree $k$ away from the…

A family of congruences interpolating between those of Wilson and Giuga is constructed. Several elementary results are established, in order to present a possible approach to establishing Giuga's conjecture.

We show that the Hodge and pole order filtrations are globally different for sufficiently general singular projective hypersurfaces in case the degree is 3 or 4 assuming the dimension of the projective space is at least 5 or 3 respectively.…

We propose a new method to investigate the dimension of the space of integrals on a Hopf algebra $H$ and other properties of $H$-comodules.

Recently, Gaiotto, Moore and Neitzke \cite{GMN08} proposed a new construction of hyperk\"{a}hler metrics. In particular, they gave a new construction of the Ooguri-Vafa metric, in which they came across certain formulas. We interpret those…

In this note, we will give a partial answer for arithmetic analogues of Grothendieck's standard conjectures due to H. Gillet and C. Soule. (Remark : I changed the title of this note.)