Social Dinner

A social dinner will take place on Thursday June 19, 2025 at the restaurant "I Tre Merli" at 19:30.

Titles and Abstracts

Monday - Room 509

Joan Elias - 30 years of collaboration with M.E. Rossi

Abstract: In this talk, I will present some of the main results that Marilina and I have found over the last 30 years.

To do this, I will review all the nine  papers that we have written.  I illustrate the presentation with some old pictures. 

Holger Brenner - Module Schemes in Invariant Theory. Or: What is a Module?

Abstract: Let G be a finite group acting linearly on the polynomial ring  with invariant ring R. We assign, to a  linear representation of G, a corresponding quotient scheme over Spec R, and we show how to reconstruct the action from the quotient scheme. This works in particular in the case of a reflection group, where Spec R itself is an affine space, in contrast to the Auslander correspondence, where one has to assume that the basic action is small, i.e. contains no pseudo reflection.  These quotient schemes exhibit rich geometric features which mirror properties of the representation. In order to understand the image of this construction, we encounter module schemes (a forgotten notion of Grothendieck), module schemes up to modification and fiberflat bundles. These notions give rise to the question: What is (the best geometric realization of) a module?

Eloísa Grifo - Embedded deformations

Abstract: We will discuss an old question of Avramov about when a local ring R has an embedded deformation, meaning R is nontrivially a quotient of another ring by a regular element. We will give a complete positive answer when R is defined by monomials, and some negative answers in the general case. This is joint work with subsets of Ben Briggs, Josh Pollitz, and Mark Walker.

Bruno Benedetti - Handle decompositions discretized

Abstract: Exactly 100  years ago, Morse discovered that any manifold M can be studied by looking at the critical points of generic functions from M to R. Any such function induces a very convenient way to chop up the manifold M into balls, called a ‘handle decomposition of M’.  Our main new result is that for any manifold M with boundary, the following are equivalent:

(1) M admits a handle decomposition into handles of index at most k;

(2) M is "k-stacked"; that is, M admits a triangulation in which all (d-k-1)-faces are on the boundary.

This proves a conjecture by Ed Swartz. Time permitting, we’ll show how  this solves an open problem by Gil Kalai on the g-vector of homology spheres.

This is joint work with Karim Adiprasito.

Tuesday - Room 509

Srikanth Iyengar - Unstable elements in cohomology and a question of Lescot

Abstract: This talk concerns the Bass numbers of syzygy modules of modules over a noetherian local ring. This topic was investigated by Lescot in a paper published in 1986. I will discuss some results from an ongoing project with Maitra and Tribone  that  grew out of our reading of Lescot's work.

Hai Long Dao - Reflexive modules over commutative rings: old and new

Abstract: Let R be commutative ring. An R-module M is called reflexive if the natural map M -> M** (where M* denotes Hom_R(M,R)) is an isomorphism. The category of reflexive modules is well-studied and well-behaved when R is normal, or more generally, (G_1) and (S_2). When R is not, for instance, if it represents a non-Gorenstein curve singularity or the homogenous coordinate ring of a non-arithmetically Cohen-Macaulay projective curve, they become much more mysterious. Some new tools are needed, such as trace ideals and generalized Ulrich modules. I will describe the past, present and future of this topic.

Rosa Maria Miró-Roig - 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.

Wednesday - Room 509

Claudia Polini - The dream for Licci ideals

Abstract: Linkage has been used for over a century to study and classify curves in projective three-space, as well as varieties in projective space and homogeneous ideals in polynomial rings. A key focus in this field is on licci ideals, which are ideals that can be linked to a complete intersection in a finite number of steps. 

It is known that the Castelnuovo-Mumford regularity of a licci ideal imposes a strict upper bound on the initial degree of the ideal. Now, in joint work with Craig Huneke and Bernd Ulrich, we have envisioned an algorithm that uses only the Betti table of an ideal to determine its licci property. So far, we have successfully proven this DREAM for ideals containing a maximal regular sequence of quadrics. 

Bernd Ulrich - Linkage and Applications

Abstract: This talk on joint work with Craig Huneke and Claudia Polini is a continuation of Claudia's lecture. I will report on a surprising conjecture relating the number of generators and the Castelnuovo-Mumford regularity of ideals in the linkage class of a complete intersection (licci ideals). I will also give a new sufficient condition for an ideal to be licci and a characterization for when a zero-dimensional monomial ideal is licci. I will explain the connection with recent work by Jelisiejew, Ramkumar, and Sammartano on the Hilbert scheme of points on smooth threefolds.

Lisa Seccia - F-splittings of Geometrically Vertex Decomposable ideals

Abstract: In this talk, we discuss the connection between Frobenius splittings and geometrically vertex decomposable ideals. Motivated by earlier work of Knutson, who showed that certain Frobenius splittings descend to degenerations obtained in a way compatible with the splitting, we show a partial converse to this result. After a brief overview of F-splittings and geometric vertex decomposition, we present an iterative method for constructing Frobenius splittings of geometrically vertex decomposable ideals. The talk is based on joint work with Emanuela De Negri, Elisa Gorla, Patricia Klein, Jenna Rajchgot, and Mayada Shahada.

Kazuho Ozeki - The first Euler characteristic and almost Cohen-Macaulayness of associated graded rings

Abstract: The homological property of the associated graded ring of an ideal is an important problem in commutative algebra.

In this talk, we explore the almost Cohen-Macaulayness of the associated graded ring in terms of the first Euler characteristic of it.

As an application, we present the structure of associated graded ring of stretched ideals with small reduction number.

Claudiu Raicu - Cohomology on the incidence correspondence and related questions

Abstract: A fundamental problem at the confluence of algebraic geometry, commutative algebra and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on (partial) flag varieties. I will describe an answer in the case of the incidence correspondence (the partial flag variety consisting of pairs of a point in projective space and a hyperplane containing it), and highlight surprising connections to other questions of interest in commutative algebra. This is based on joint work with Annet Kyomuhangi, Emanuela Marangone, and Ethan Reed.

Thursday - Room 509

Aldo Conca - From resolution of points to Koszul algebras

Abstract: We will explore how the investigation of the Minimal Resolution Conjecture for points naturally led  to a  study of Koszul algebras initially focused on point configurations but  soon also on more  general classes of algebras. In particular, we will revisit  concepts such as Koszul filtrations and flags, examining how these structural tools have been employed to establish results—and formulate   conjectures—across a variety of algebraic contexts.

Emanuela De Negri - Algebras with minimal rate

Abstract: The rate of a standard graded K-algebra A is a measure of the growth of the shifts in a minimal free resolution of K as an A-module. In particular, A has rate one if and only if it is Koszul. If I is the defining ideal of A, then the rate of A is bigger than or equal to  m(I)-1, where m(I) denotes the highest degree of a generator of I.

In this talk we present three classes of algebras with minimal rate: the coordinate rings of certain sets of points of the projective space, some algebras defined by a space of forms of a fixed degree and of small codimension, and generic Artinian Gorenstein algebras of socle degree bigger than or equal to 3.

These  results are part of two joint works: one with A.Conca and M.E. Rossi, and the other with M.Boij, A. De Stefani and M.E. Rossi. 

Giulio Caviglia - Three kinds of upper bounds for Betti numbers

Abstract: In this talk I will overview three kinds of upper bounds for Betti numbers with a particular focus on the ones of prime ideals: first, the double exponential types, whose existence often relies on computational considerations and whose most notable example is the estimation of the size of Groebner bases;  second, the (sometimes overly optimist) linear or polynomial ones, which arise from geometric consideration; and finally, the unconceivably large ones, which are the offsprings of the solution of Stillman's conjecture. This is based on joint works with A. De Stefani and with C. Meng and Y. Liang.

Winfried Bruns - Fusion rings from lattice points

Abstract: Fusion rings are abstract versions of Grothendieck rings of certain tensor categories, i.e., categories that are endowed with a bifunctor called tensor product. The prototypical example is rhe category of finite-dimensional representations of a finite group. The corresponding fusion ring is a finite rank free algebra over the integers whose base elements correspond to isomorphism classes of irreducible representations and whose relations are defined by the decomposition of the tensor product of two irreducibles into a direct sum of irreducibles. Another source is conformal field theory, which is undoubtedly a driving force in the theory of fusion rings.

The compoutation of fusion rings amounts to finding their multiplication matrices for given basic data. The multiplicatioin matrices must satisfy linear equations that result from the Frobenius-Perron theorem. Therefore they are given by lattice points in polytopes of extremely high dimension, often > 200. The computation is only possible since the points must additionally satisfy quadratic equations that represent the associativity of the algebra. Normaliz contains an interface for fusion rings and is an efficient solver for them.

Part of the results so far are reported in the paper "Classification of modular data of integral modular fusion categories up to rank 13" with Max A. Alekseyev, Sébastien Palcoux and Fedor V. Petrov (arXiv:2302.14345).

Patricia Klein - Polarization and Gorenstein liaison

Abstract: A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen-Macaulay subscheme of P^n can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of P^{n+1}, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme.

Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley-Reisner complexes. Given a monomial ideal I and a vertex decomposition of the Stanley-Reisner complex of its polarization P(I), we give conditions that allow for the lifting of an associated basic double G-link of P(I) to a basic double G-link of I itself.  We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage.

The work described in this talk is joint with Sara Faridi, Jenna Rajchgot and Alexandra Seceleanu. 

Friday - Room 509

Anurag Singh - Local cohomology of modular invariant rings

Abstract: Consider a finite subgroup of the general linear group, with its natural action on a polynomial ring.  We discuss how the local cohomology module of the invariant ring compares with the invariant part of the local cohomology of the polynomial ring.  This has various consequences, such as for the a-invariant and for the Hilbert series.  This is joint work with Kriti Goel and Jack Jeffries.

Alessandro De Stefani - Connectedness of the associated graded ring

Abstract: We will discuss how certain connectedness properties of a complete local ring descend to its associated graded ring G. Our main tool is a homogenization technique which produces a complete “Gröbner-like” deformation with special fibre the completion of G. Time permitting, we will also discuss some consequences of these results on certain numerical invariants of the ring. This talk is based on joint work with Maria Evelina Rossi and Matteo Varbaro.

Jugal Verma - A survey of the Hoskin-Deligne formula for complete ideals and modules

Abstract: The Hoskin-Deligne formula for the colength of a complete m-primary ideal of a two-dimensional regular local ring (R,m)  was discovered independently by M. A. Hoskin, P. Deligne and D. Rees. It has been generalised to finitely supported complete ideals   by Clare D'Cruz and for complete submodules of finite co-length of free modules by Vijay Kodiyalam and Radha Mohan. The formula helps in recovering results of O. Zariski and J. Lipman-B. Teissier about the reduction numbers and the  Hilbert-Samuel polynomial of complete ideals. The module theoretic analogue yields the  Buchsbaum-Rim polynomial of complete modules over 2-dimensional regular local rings. We also report on some old and recent results about  the adjoint and core of product of complete ideals and modules using  their joint reductions.

Davide Bolognini - O-Sequences and Bruhat intervals

Abstract: Hilbert functions of standard graded Artinian algebras (i.e. finite O-sequences) coincide with h-vectors of pure shellable simplicial complexes. Rephrasing a construction by Bjorner-Frankl-Stanley, we improve this result by showing that they coincide with h-vectors of pure vertex-decomposable simplicial complexes.

In the second part of the talk, this construction combined with the notion of Lehmer code lead us to prove results concerning the shape of lower Bruhat intervals in symmetric groups, which encode the cohomology of Schubert varieties in flag varieties.

This is a joint work with Paolo Sentinelli.

Ngo Viet Trung - An effective formula for the regularity function

Abstract: To compute the function reg I^n of a graded ideal I is usually very hard. We will present an explicit formula for reg I^n in terms of an initial ideal of the defining ideal of the Rees algebra of I. This formula is inspired by Bayer-Stilman' method for the computation of reg I in terms of Gin(I).