Although Alain Lascoux and I never collaborated on a mathematical paper, his ideas have been a constant influence on my students’, my collaborators’ and my work ever since the appearance of the first draft of his doctoral thesis [12] His systematic use of characteristic-zero representation theory to find the resolutions of ideals generated by the minors, of any order, of generic matrices, impel...

We investigate the relations between the cut number, c(M), and the first Betti number, b1(M), of 3-manifolds M. We prove that the cut number of a “generic” 3-manifold M is at most 2. This is a rather unexpected result since specific examples of 3-manifolds with large b1(M) and c(M) ≤ 2 are hard to construct. We also prove that for any complex semisimple Lie algebra g there exists a 3-manifold M...

Abstract. The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard grad...

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over S1 with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over S1. In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold i...

We study the persistent homology of random Čech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we define the kth persistent Betti number of a random Čech complex and determine its asymptotic order in the subcritical regime. This extends a result of...

This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded R-module, where R = k[x1, . . . , xm] is the polynomial ring over a field k in m variables. The bound is given in terms of the rank and the first two Betti numbers of the module. An example is given which achieves these bounds simultaneously in each homological degree. Using Alexander duality, a bound...

We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group C∗ associated to a compact connected Riemann surface.