Abbas Nasrollah Nejad (Institute for Advanced Studies in Basic Sciences)

• • Title: Quasihomogeneous Singularities and the Milnor–Tjurina Difference

  • Abstract: This talk, based on joint work with Aron Simis and Hamid Hassanzadeh, is about quasihomogeneous isolated hypersurface singularities and two of their fundamental invariants, the Milnor and Tjurina numbers. We present a syzygy-theoretic approach to Saito’s classical criterion for quasihomogeneity, which leads to an effective way of testing whether a singularity is quasihomogeneous. We then discuss the difference between the Milnor and Tjurina numbers, explain how it can be captured by algebraic methods, and show inequalities that extend known results to higher dimensions.

Aline Vilela Andrade (Universidade Federal de Minas Gerais)

• • Title: From the Tea Theorem to the Cheese Bread Theorem: a Taste of Lefschetz Properties

  • Abstract: Let A be a standard graded Artinian K-algebra over a field of characteristic zero. We prove that the failure of strong Lefschetz property (SLP) for A is equivalent to the osculating defect of a certain rational variety. Our results extend previous works by eliminating the need for the defining ideal of A to be equigenerated and by extending the study of WLP/SLP in a fixed degree to the study of the SLP in any degree and any range. As a consequence we provide a geometric interpretation of the vanishing of a higher Hessian, extending the classical Gordan-Noether criterion. Moreover, we reobtain some foundational results, including the presence of SLP for codimension 2 Artinian algebras, and the SLP for Artinian Gorenstein algebras with Hilbert function (1,3,6,6,3,1). Joint work with: Charles Almeida (UFMG/Brazil) and Rodrigo Gondim (UFRPE/Brazil)

Bernd Ulrich (Purdue University)

• • Title: Degrees of Vector Fields

  • Abstract: Motivated by the problem of finding algebraic curves that are left invariant by a given vector field in the plane, Poincaré asked whether the degree of such curves can be bounded above in terms of the degree of the vector field. Although the question has a negative answer in general, it has inspired a great deal of work for over a century. A broader goal of this research is to relate the degree of vector fields in projective n-space to properties of curves or even varieties that they leave invariant. We will survey some of the numerous earlier results and report on more recent joint work with Marc Chardin, Hamid Hassanzadeh, Claudia Polini, and Aron Simis, where the question is approached from a more algebraic point of view. We provide lower bounds for the degree of vector fields in terms of local and global invariants of the curves they leave invariant. Higher-dimensional varieties are considered as well, and the sharpness of the bounds will be discussed.

Claudia Polini (University of Notre Dame)

• • Title: The Behrend function and Rees valuations

  • Abstract: I will survey preliminary results of joint work with Alessio Sammartano and Bernd Ulrich. Given a scheme X of finite type over the complex numbers, the Behrend function is a constructible function that allows one to compute the degree of the virtual fundamental class of X under suitable assumptions, leading to the solution of numerous problems in enumerative geometry. Even in simple cases, though, the Behrend function is very difficult to compute.  In this talk, I will explain how we compute the Behrend function of arbitrary zero-dimensional monomial ideals in any number of variables and its connections to Rees rings and Rees valuations.

David Eisenbud (University of California, Berkeley)

• • Title: Syzygies in some infinite resolutions

  • Abstract: Much attention has been given to the ranks of free modules in infinite minimal free resolutions, but, oddly, much less to the actual syzygy modules. In work with Hai Long Dao, Claudia Polini and Bernd Ulrich we have uncovered unexpectedly simple structure in a number of cases. I will report on what we have learned and on the many open problems that remain.

Haydee Lindo (Harvey Mudd College)

• • Title: Arf rings and applications of trace ideals

  • Abstract: Following the work of Vasconcelos and Herzog, much recent attention has been given to applications of trace ideals in commutative algebra and algebraic geometry.  In joint work with Hai Long Dao we explore the intersection of the theories of trace ideals and stable ideals, that is, ideals that are stable under homomorphisms to the ring and ideals that are isomorphic to their endomorphism rings, respectively. We apply our results to the study of Arf rings making precise the relationship between various notions of closure that coincide in the Arf setting: reflexive, integrally closed, trace, and stable. 

I-Chiau Huang (Academia Sinica)

• • Title: Solution Module and Linear Closure

  • Abstract: Explicit description of injective modules are required for concrete realization of Grothendieck duality. We introduce the notion of an ``initial condition'' for a module M over a commutative Noetherian local ring (A,m), allowing for a recursive construction of its ``solution modules''. If M has zero-dimensional support, such as the residue field of A, we demonstrate that the solution module E(M) is an injective hull of M. The construction of E(M) for finitely generated M is explicit and computable, devoid of the need for Zorn's lemma. As an application, we improve Baer's criterion for a module N with zero-dimensional support to be injective: If any A-homomorphism from m to N lifts to A, then N is injective.

Jacqueline Rojas (Universidade Federal da Paraíba)

• • Title: Enumerative Geometry and Hilbert Schemes of Curves in Projective Space

  • Abstract: In this talk, we will address some enumerative questions that can be solved by providing an appropriate description of a suitable component of the Hilbert scheme of curves in complex projective space, using Bott’s localization formula.

Kuei-Nuan Lin (Penn State University)

• • Title: Rees Algebra of Determinantal Modules

  • Abstract: The Rees algebra is a central object in commutative algebra and algebraic geometry. Describing the defining equations of Rees algebras is important in elimination theory, geometric modeling, chemical reaction networks, and algebraic statistics. These questions remain largely open. In this talk, I will review recent results on determining the defining ideals of Rees algebras of determinantal ideals and modules.

Lorenzo Robbiano (Università di Genova)

• • Title: Re-embeddings of affine algebras

  • Abstract: PDF

Maral Mostafazadehfard (Universidade Federal do Rio de Janeiro)

• • Title: Open Loci of Ideals with Applications to Birational Maps

  • Abstract: We study families of ideals in a fixed parameter space and show that several natural conditions define Zariski open loci. Specifically, we prove that the sets of ideals in principal class, ideals of grade at least two, and ideals of maximal analytic spread are Zariski open sets in the parameter space. As an application, we show that the set of birational maps of clear polynomial degree d over an arbitrary projective variety X, denoted by Bir(X)_d is a constructible set. This extends a previous result by Blanc and Furter. This talk is based on joint work with Hamid Hassanzadeh.

Marc Chardin (Université Pierre et Marie Curie)

• • Title: TBA

  • Abstract: TBA

Marcos Jardim (IMECC - Universidade Estadual de Campinas)

• • Title: Ideals with 3 generators

  • Abstract: In joint work with Nejad and Simis, we introduced the Bourbaki degree of a plane curve as a measure of how far it is from being free. We now observe that all the arguments can be generalized to graded ideals with three generators, and we introduce the Bourbaki degree and related concepts in this context. This is joint work with Felipe Monteiro and Zaqueu Ramos.

Martin Kreuzer (Universität Passau)

• • Title: Applications of Separating Re-Embeddings

  • Abstract: Based on the concept of "separating re-embeddings" introduced and explained in Lorenzo Robbiano's talk, we present some applications of this method. In particular, the following topics are addressed: 

1. Applications to border basis schemes, especially studying the question of when these moduli spaces for 0-dimensional schemes are affine cells.

2. Extensions of the theory to Boolean polynomials and applications to algebraic attacks in cryptography.

The talk is based on joint work with Bernhard Andraschko, Julian Danner, Le Ngoc Long, and Lorenzo Robbiano.

Philippe Gimenez (Universidad de Valladolid)

• • Title: Syzygies of toric ideals through additive combinatorics, a sumset approach.

  • Abstract: Motivated by several recent papers, in this talk we will focus on the interaction between commutative algebra and additive combinatorics. Associated with a finite subset A of tuples of non-negative integers (and an infinite field), we have a projective variety and a (standard) homogeneous toric ideal. On the other hand, for any non-negative integer s, we can consider the s-fold iterated sumset of A, sA, which is the set formed by all sums of s elements in A. We will show in some specific cases (monomial curves, simplicial varieties either smooth or with a single singular point) how the syzygies of the toric ideal associated with A, and in particular its Castelnuovo-Mumford regularity, are related to some properties of the sumsets. This bridge between commutative algebra and additive combinatorics can be used in both directions to solve problems in one field using results from the other. This talk is based on joint work with Mario González-Sánchez and Ignacio García-Marco.

Rosa Maria Miró Roig  (Universitat de Barcelona)

• • Title: The weak Lefschetz property for Artinian Gorenstein algebras of small Sperner number.

  • Abstract: For artinian Gorenstein algebras in codimension four and higher, it is well known that the Weak Lefschetz Property (WLP) does not need to hold. For Gorenstein algebras in codimension three, it is still open whether all artinian Gorenstein algebras satisfy the WLP when the socle degree and the Sperner number are both higher than six. We here show that all artinian Gorenstein algebras with socle degree d and Sperner number at most d + 1 satisfy the WLP, independent of the codimension. This is a sharp bound in general since there are examples of artinian Gorenstein algebras with socle degree d and Sperner number d + 2 that do not satisfy the WLP for all d ≥ 3. Join work with M. Boij, J. Migliore and U. Nagel.

Stefan Tohaneanu (University of Idaho)

• Title: On the Rees algebra of the Jacobian ideal of a central hyperplane arrangement

• Abstact: PDF

Ugo Bruzzo (SISSA)

• Title: Higgs Grassmannians

• Abstact: Given a Higgs bundle (E,phi), its Higgs Grassmiannans are subschemes of the usual  Grassmannian bundles of E that parameterise Higgs quotients of (E,phi). I will recall how to define them, will show some results about their structure, and will show how they can be used to prove some results about Higgs bundles satisfying a strong semistability condition. 

• Aiury Azerêdo - On a Question of Christensen-Foxby-Holm about Gorenstein Dimensions

• Cesar Hilario - Regular but non-smooth curves of genus 3

• Ernesto Carlo Mistretta - Semiample Vector bundles, parallelizable complex manifolds, fundamental groups

• Liliana Gabriela Gheorghe - A Fair Barter: Two Apollonian Circles for Three Poncelet Pairs

• Lucas Mioranci - Normal and tangent bundles of rational curves in projective hypersurfaces

• Mario González-Sánchez - On Computing the Short Resolution of a Homogeneous Ideal

• Michele Graffeo - The geometry of Hilbert schemes of points and 2-step ideals

• Muhammad Imran Qureshi - Canonical 3-folds in P2xP2 format and K3 Transitions

• Ruben Edwin Lizarbe Monje - Foliations on Quadrics

• Thiago Henrique de Freitas - Homological conjectures over fiber product rings

• Thiago Holleben - From points to complexes: a notion of unexpectedness for simplicial complexes

• Vinicius Bouça - Koszul-Fitting Ideals and Residual Intersections

• Wodson Mendson - Non-algebraicity of foliations via reduction mod 2