In this paper we discuss different versions of Bertini’s theorem and prove a variant of a theorem of Bertini. We also prove the following theorem. Suppose A is a semilocal regular reduced affine algebra over an algebraically closed field of characteristic zero such that every maximal ideal in A[T ] is complete intersection, with dim(A) = n > 0. Suppose I is an ideal of A[T ], with height(I) = n...

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f : M → R is bijective if and only if Hn−1(M) = 0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological re...

Given an element w of a Coxeter group, let ai(w) be the number of elements less than w in Bruhat order. A theorem of Björner and Ekedahl states that if W is crystallographic, then ai(w) ≤ aj(w) for all 0 ≤ i < j ≤ `(w) − i. Their proof uses the hard Lefschetz property in intersection cohomology. In this note we extend Björner and Ekedahl’s theorem to all Coxeter groups using the hard Lefschetz ...

Susumu TANABÉ Abstract. This is a review article on the Gauss-Manin system associated to the complete intersection singularities of projection. We show how the logarithmic vector fields appear as coefficients to the Gauss-Manin system (Theorem 2.7). We examine further how the multiplication table on the Jacobian quotient module calculates the logarithmic vector fields tangent to the discriminan...

The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank's discrete separation theorem for submodular/supermodular set functions and Edmon...

Let d and t be fixed positive integers, and let K t,...,t denote the complete d-partite hypergraph with t vertices in each of its parts, whose hyperedges are the d-tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erdős [7], the number of hyperedges of a d-uniform hypergraph on n vertices that does not co...

We give an exposition and generalization of Orlov’s theorem on graded Gorenstein rings. We show the theorem holds for non-negatively graded rings which are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary non-commutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this ...

Susumu TANABÉ Abstract. This is a review article on the Gauss-Manin system associated to the complete intersection singularities of projection. We show how the logarithmic vector fields appear as coefficients to the Gauss-Manin system (Theorem 2.7). We examine further how the multiplication table on the Jacobian quotient module calculates the logarithmic vector fields tangent to the discriminan...

1. Classification problems in differential topology: 1/18/18 1 2. Review of the algebraic topology of manifolds: 1/23/18 4 3. Unimodular forms: 1/25/18 8 4. The intersection form and characteristic classes: 1/30/18 12 5. Tangent bundles of 4-manifolds: 2/1/18 16 6. Rokhlin’s theorem and the homotopy theorem: 2/6/18 19 7. The homotopy theorem: 2/8/18 22 8. The Hodge star: 2/13/18 22 9. Hodge the...