## Spring 2019

**February 25**: Alessandra Costantini, Purdue University

**Title**: Rees algebras of modules and their defining ideal

**Abstract**: Rees algebras of ideals and modules arise in Commutative Algebra and Algebraic Geometry in connection with the study of singularities of algebraic varieties. In this talk, I will describe the problem of determining the defining ideal of the Rees algebra, i.e. a presentation in terms of generators and relations. This is a difficult problem in general, but becomes easier to treat in case the ideal or module has a nice presentation matrix.

I will provide a class of modules for which it is possible to determine the defining ideal of the corresponding Rees algebra. This is a generalization of a result by Jacob Boswell and Vivek Mukundan.

**February 18**: Jonathan Montaño, New Mexico State University

**Title**: Vanishing of Ext for Cohen-Macaulay local rings of small multiplicity II

**Abstract**: Let R be a Cohen-Macaulay local ring and let M,N be Noetherian R-modules. We find sufficient conditions on R so that whenever Ext_{R}^{i}(M,N) = 0 for i ≫ 0, then either M has finite projective dimension or N has finite injective dimension. As a consequence, we prove that the Auslander-Reiten conjecture holds for Cohen-Macaulay local rings of multiplicity at most 8. This is joint work with Justin Lyle.

**February 11**: Jonathan Montaño, New Mexico State University

**Title**: Vanishing of Ext for Cohen-Macaulay local rings of small multiplicity I

**Abstract**: Let R be a Cohen-Macaulay local ring and let M,N be Noetherian R-modules. We find sufficient conditions on R so that whenever Ext_{R}^{i}(M,N) = 0 for i ≫ 0, then either M has finite projective dimension or N has finite injective dimension. As a consequence, we prove that the Auslander-Reiten conjecture holds for Cohen-Macaulay local rings of multiplicity at most 8. This is joint work with Justin Lyle.

**February 1**: Cameron Calk, Université de Lyon

**Title**: Time-reversal and directed homotopy

**Abstract**: Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain “directed” paths in the topological space representing such a program. Algebraic invariants which respect this directedness have been introduced to classify directed spaces. We will discuss the properties of such invariants with respect to the reversal of the flow of time in directed spaces. A known invariant, natural homotopy, has been shown to be unchanged under time-reversal. We will see that it can be equipped with additional algebraic structure witnessing this reversal; when applied to a directed space and to its reversal, the refined invariant yields dual objects.

**January 23**: Giulio Peruginelli, University of Padova

**Title**: Valuation domains of the field of the rational functions associated to pseudo-monotone sequences

**Abstract**: Let V be a valuation domain with quotient field K. If V has rank one, Ostrowski introduced in 1935 the notion of pseudo-convergent sequence in order to describe all the possible rank one extensions of V to K(X). In 2010, Chabert generalized this notion and gave the definition of pseudo-monotone sequence, in order to describe the polynomial closure of a subset Sof V, which is the largest subset T of V with the property that if an element f in K[X] maps S into V then f(X) maps also T in V. In this talk, we will show how the notion of pseudo-monotone sequence can be used to generalize Ostrowski’s result for general valuation domains V in order to describe all the possible extensions of V to K(X).

## Past Semesters

By clicking on Past Seminars you will get a PDF listing the speakers, titles, and abstracts of the seminars for the past several years.