TIME TABLE: TBA

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

• • Title: Milnor to Tjurina Difference

  • Abstract: TBA

Aline Vilela Andrade (Universidade Federal de Minas Gerais)

• • Title: TBA

  • Abstract: TBA

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: TBA

  • Abstract: TBA

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: On the Hilbert Scheme of Curves in the Projective Space

  • Abstract: TBA

Joseph Brennan (University of Central Florida)

• • Title: TBA

  • Abstract: TBA

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: TBA

  • Abstract: TBA

Marc Chardin (Université Pierre et Marie Curie)

• • Title: TBA

  • Abstract: TBA

Marcos Jardim (IMECC - Universidade Estadual de Campinas)

• • Title: Generalized Saito criterion

  • Abstract: We establish generalizations of Saito's criterion for the freeness of divisors in projective spaces that apply both to sequences of several homogeneous polynomials and to divisors on other complete varieties. As an application, the new criterion is applied to several examples, including sequences whose polynomials depend on disjoint sets of variables, some sequences that are equivariant for the action of a linear group, blow-ups of divisors, and certain sequences of polynomials in positive characteristics. Joint work with Daniele Faenzi and Jean Vallès.

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: TBA

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. 

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• Title: TBA

• Abstact: TBA

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