Related papers: Statement of the Alexandru Conjecture
We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat…
A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.
Using translation from the regular block, we construct and analyze properties of BGG complexes in singular blocks of BGG category ${\mathcal{O}}$. We provide criteria, in terms of the Kazhdan-Lusztig-Vogan polynomials, for such complexes to…
In this paper we prove two results pertaining to the (unramified and global) geometric Langlands program. The first result is an analogue of the Ramanujan conjecture: any cuspidal D-module on Bun_G is tempered. We actually prove a more…
We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases…
The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix,…
We give a proof of Alexandrov's conjecture on a formula connecting the Kontsevich-Witten and Hodge tau-functions using only the Virasoro operators. This formula has been confirmed up to an unknown constant factor. In this paper, we show…
The Hodge conjecture is a major open problem in complex algebraic geometry. In this survey, we discuss the main cases where the conjecture is known, and also explain an approach by Griffiths-Green to solve the problem.
We review what is known about the Hodge conjecture for abelian varieties, with some emphasis on how Mumford-Tate groups have been applied to this problem.
In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…
The Hodge conjecture is shown to be equivalent to a question about the homology of very ample divisors with ordinary double point singularities. The infinitesimal version of the result is also discussed.
We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we…
Many algorithms in verification and automated reasoning leverage some form of duality between proofs and refutations or counterexamples. In most cases, duality is only used as an intuition that helps in understanding the algorithms and is…
We give a simple proof of a Theorem of Cline, Parshall and Scott about the category O of BGG and suggest an analog for Harish-Chandra modules.
In this work we classify the thick subcategories of the bounded derived category of dg modules over a Koszul complex on any list of elements in a regular ring. This simultaneously recovers a theorem of Stevenson when the list of elements is…
Despite the failure of the integral Hodge conjecture, we show that the rational Hodge conjecture implies an integral version (modulo torsion) of the absolute Hodge conjecture.
A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory duality for affine Lie algebras. The functor sends objects in the DG category of G(O)-equivariant positive level affine Lie algebra modules to objects in the DG…
We show that the structure of blocks outside the critical hyperplanes of category O over any symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules by…
Using a characterization of a generalized $\tau$-invariant for intermediate parameter Hecke algebras in type $B_n$, we verify a conjectural description of Kazhdan-Lusztig cells in this setting due to C. Bonnaf\'e, L. Iancu, M. Geck, and T.…
The article reviews different definitions for a convolutional code which can be found in the literature. The algebraic differences between the definitions are worked out in detail. It is shown that bi-infinite support systems are dual to…