Algebra & Discrete Mathematics Seminar
Spring 2016 Schedule

Location: Minard 404 (Seminar Room)
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă
Previous Semesters

12 April 2016

Jessica Striker:
Combinatorial dynamics of monomial ideals

Abstract:
We introduce the notion of combinatorial dynamics on algebraic ideals by translating combinatorial results involving rowmotion and other toggle group actions on order ideals of posets to the setting of monomial ideals. This is joint work with David Cook.

23 February 2016

Trevor McGuire: Resolutions of $$k[M]$$-modules

Abstract: Resolutions of $$k[x_1,...x_n]$$-modules have been widely studied, and in particular, combinatorial methods have been applied to large classes of modules. If we replace the variables with more general objects, the problems get proportionally more difficult. Specifically, we will investigate $$k[M]$$-modules where $$M$$ is a monoid. There are two equally realistic avenues to take with $$M$$, and we will discuss both avenues. The presentation will begin with a review of aforementioned combinatorial methods for the traditional case.

16 February 2016

Cătălin Ciupercă: Reduction numbers of equimultiple ideals (Part II)

9 February 2016

Cătălin Ciupercă: Reduction numbers of equimultiple ideals

Abstract: Let $$(A,\mathfrak{m})$$ be an unmixed local ring containing a field. If $$J$$ is an $$\mathfrak{m}$$-primary ideal with Hilbert-Samuel multiplicity $$\operatorname{e}(J)$$, a recent result of Hickel shows that every element in the integral closure of $$J$$ satisfies an equation of integral dependence over $$J$$ of degree at most $$\operatorname{e}(J)$$. We extend this result to equimultiple ideals $$J$$ by showing that the degree of such an equation of integral dependence is at most $$c(J)$$, which is one of the elements of the so-called multiplicity sequence introduced by Achilles and Manaresi.  As a consequence, if the characteristic of the field contained in $$A$$ is zero, it follows that the reduction number of an equimultiple ideal $$J$$ with respect to any minimal reduction is at most $$c(J)-1$$.