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A homotopy theoretic description is given for trivial unit conjecture in the group ring ZG.

McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the…

In the paper we complete a case by case proof of Reeder's Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type $D$ and for the exceptional cases.

In \cite{Mil}, Milnor posed the {\em Monotonicity Conjecture} that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected. This conjecture was proved…

We give a short proof of Weintraub's conjecture by constructing explicit highest weight vectors in the symmetric power of an even exterior power.

We present a novel method for checking the Hurwitz stability of a polytope of matrices. First we prove that the polytope matrix is stable if and only if two homogenous polynomials are positive on a simplex, then through a newly proposed…

We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.

In this note we give a p-adic proof of Hodge symmetry for smooth, projective threefolds over complex numbers.

We formulate Vojta's conjecture for smooth weighted projective varieties, weighted multiplier ideal sheaves, and weighted log pairs and prove that all three versions of the conjecture are equivalent. In the process, we introduce generalized…

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous…

We give a geometric proof of a conjecture of W. Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL(r).

We prove Simon's conjecture for 3-manifolds.

We prove that the standard conjecture of Hodge type holds for powers of abelian threefolds. Along the way, we also prove the conjecture for powers of simple abelian variety of prime dimension over finite fields, and in other related cases…

In this paper we give a complete proof of the Brumer-Stark conjecture over $\mathbf{Z}$.

This paper gives a necessary and sufficient condition for robust D-stability of Polytopic Polynomial Matrices. Edge theorem is extended to multi-input-multi-output case.

In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula involves a new class of generalized Littlewood-Richardson coefficients, all of which surprisingly seem to be…

We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.

The conjectures of Manin and Peyre are confirmed for a certain threefold.

We prove that if the Morrison cone conjecture holds for a smooth Calabi-Yau threefold $Y$, it holds for any smooth Calabi-Yau threefold deformation-equivalent to $Y$. We use this result to prove a new case of the Morrison cone conjecture.

We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of a such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with…