Algebra & Geometry Seminar
Algebra & Geometry Seminar
Date: 18/12/2025 Time: 14.00 Room: 713
Title: Connectivity and linear codes
Abstract: Connectivity is a notion in graph theory that was later extended to matroid theory. Several notions of matroid connectivity exist, most notably Tutte connectivity and vertical connectivity for classical matroids, which we extend to q-matroids. After a brief introduction to coding theory and matroid theory, in this talk we will discuss some properties of the vertical connectivity of a matroid associated to a linear code and the q-matroid associated to a rank-metric code. In both cases, we will show a relation with the support structure of the code.
Date: 09/10/2025 Time: 14.00 Room: 713
Title: On the maximality of natural involutions of Hilbert schemes of points on regular surfaces
Abstract: The Smith inequality states that the total F2-Betti number of the fixed locus of an involution is no greater than the total F2-Betti number of the ambient manifold; the involution is called maximal when equality is achieved. In the context of real geometry, maximal anti-holomorphic involutions on complex manifolds determine particularly interesting real structures. In hyper-Kähler geometry, (anti-)holomorphic involutions are called branes and play a role in modern physical theories. It is a difficult task to construct maximal examples of real structures and branes in high dimensions, the aim of this talk is to give some partial answers to this problem.
We give a criterion for a (anti-)holomorphic involution of a surface S with H^1(S,F2)=0 to induce a maximal involution of the Hilbert scheme of n points S^[n].
When S is a K3 surface, then S^[n] and all its deformations admit a hyper-Kähler structure. We prove that branes in such manifolds can never be maximal.
Date: 16/10/2025 Time: 14.00 Room: 713
Title: Arithmetic complexes for stable sheaf cohomology
Abstract: A classical problem connecting algebraic geometry and representation theory is the computation of sheaf cohomology for line bundles on flag varieties. In positive characteristic, these cohomology groups can be expressed using Schur functors and the cotangent sheaf of the projective space P(V). Raicu and VandeBogert proved that these groups stabilize when the dimension of V is sufficiently large and, under certain conditions, can be explicitly computed using "arithmetic complexes". In joint work with E. Reed, S. Roshan-Zamir, and H. Yu, we prove a uniform identification formula for these complexes. This provides an alternative proof of Raicu and VandeBogert’s results, leading to a more general outcome in the projective space over the integers.
Date: 23/10/2025 Time: 14.00 Room: 713
Title: Lattices and locally symmetric spaces from an arithmetic point of view
Abstract: Lattices in Lie groups are naturally associated to complete, finite-volume, locally symmetric spaces, and for this reason are of natural interest to geometric topologists (and possibly also to algebraic geometers 😄). We will review their construction via arithmetic groups, the dichotomy between arithmetic and non-arithmetic lattices implied by Margulis' Superrigidity Theorem, and the relation between their arithmetic properties and the geometric/topological properties of the corresponding locally symmetric spaces. If time allows, we will review some recent developments in this field.
Date: 6/11/2025 Time: 14.00 Room: 713
Title: Universal Groebner bases for universal multiview ideals
Abstract: Huang and Larson have recently given a matroid based criterion for when a collection of squarefree polynomials form a universal Groebner basis. Using this method, we find universal Groebner bases for two families of ideals that arise in computer vision. To apply the Huang-Larson method to (infinite) families of ideals we rely on symmetry reduction and induction.
Date: 13/11/2025 Time: 14.00 Room: 713
Title: A survey on binomial edge ideals
Abstract: In this seminar I want to present several results about binomial edge ideals, covering part of the existing literature on Cohen-Macaulayness, Castelnuovo-Mumford regularity and other topics. Time permitting, I will provide a list of conjectures and open problems.
Date: 20/11/2025 Time: 14.00 Room: 713
Title: Graphical scattering equations
Abstract: We study the scattering equations associated with a graph G on n vertices, defined as the rational functions arising from the derivatives of the logarithmic potential of the corresponding graphical arrangement. These equations depend on certain parameters and determine a multiprojective variety, the scattering correspondence V_G. In the case of the complete graph K_n, the equations coincide with the classical CHY scattering equations on the moduli space M_{0,n}, which play a central role in particle physics and algebraic statistics. We focus on graphs G that contain sufficiently many edges for the scattering correspondence V_G to be non-empty and for its natural projection to exhibit desirable properties. We refer to such graphs as copious . We provide a geometric, algebraic, and combinatorial characterization of copious graphs. Furthermore, we give a partial topological description via a partial compactification of M_{0,n} that depends on G. This is a join work with Viktoriia Borovik, Bella Finkel, Bernd Sturmfels and Bailee Zacovic.
Date: 27/11/2025 Time: 13.00 Room: 713
Title: Positivity of tangent sheaves and the study of Frobenius pushforwards
Abstract: If X is a smooth projective variety, the sheaf-theoretic properties of the Frobenius pushforwards F_*^e \O_X reflect a great deal of information about X. In particular, there are close connections between the properties of the tangent and cotangent sheaves, and certain properties of F_*^e \O_X; in particular, the direct sum decompositions and (anti)positivity of F_*^e \O_X. From a commutative algebra point of view, this provides a way of understanding some of the Frobenius summands of graded rings with isolated singularities.
In this talk, we will discuss two main results regarding the connection between F_*^e \O_X and positivity of the tangent sheaf (or related sheaves): the first is a necessary criteria for F_*^e \O_X to have only finitely many indecomposable summands as e varies (i.e., for X to have global finite F-representation type) in terms of positivity of the tangent sheaf of X, and the second is a set of criteria for the cokernel of the Frobenius morphism to be ample or antiample, thus partially answering questions of Carvajal-Rojas and Patakfalvi..
Date: 27/11/2025 Time: 14.00 Room: 713
Title: Deformations and the homotopy Lie algebra
Abstract: A deformation of a local ring R is another local ring S such that R = S/f for some nonzero divisor f, and the deformation is called embedded if f is in the square of the maximal ideal of R. Detecting the presence of embedded deformations is a tricky problem. Avramov observed that embedded deformations give rise to nontrivial central elements in the "homotopy Lie algebra" associated to R, and he asked in 1989 whether all central elements arise in this way. I'll explain what all this means and then give a counterexample to the problem, which has been open until now. This is all joint work with Eloísa Grifo, Josh Pollitz, and Mark Walker.
Date: 4/12/2025 Time: 14.00 Room: 713
Title: Symmetry counts: an introduction to equivariant Hilbert and Ehrhart series
Abstract: The aim of this talk is to introduce equivariant versions of Ehrhart series of lattice polytopes and of Hilbert series of Stanley-Reisner rings of simplicial complexes. More precisely, we want to record how the action of a finite group affects the collections of lattice points or monomials that one usually "just" counts. Inspired by previous results by Betke-McMullen, Stembridge, Stapledon and Adams-Reiner, we will investigate which extra combinatorial features of the group action give rise to "nice" rational expressions of the equivariant Hilbert and Ehrhart series, and how the two are sometimes related. This is joint work with Emanuele Delucchi. A particularly well-behaved class of lattice polytopes is the one given by order polytopes of posets; if time permits, I will also mention a recent result about the gamma-effectiveness of order polytopes of graded posets. This is joint work with Akihiro Higashitani.
Date: 4/12/2025 Time: 15.00 Room: 713
Title: Nearly Gorenstein semigroup rings
Abstract: Among the Cohen-Macaulay rings, Gorenstein rings play a prominent role in many respects. Motivated by the search for rings exhibiting similar properties, several generalizations have been introduced; among them, the notion of nearly Gorenstein ring has attracted considerable attention in recent years. In this talk, after a brief introduction to these rings, I will focus on some nearly Gorenstein semigroup rings. In particular, I will explain characterizations in terms of properties of the associated semigroup and I will present some results concerning their Cohen-Macaulay type. This is based on joint works with Alessio Moscariello, Raheleh Jafari and Santiago Zarzuela Armengou.
Date: 11/12/2025 Time: 14.00 Room: 713
Title: Degenerations of Küchle c5 Fourfold
Abstract: Küchle c5 fourfold is a rather mysterious Fano fourfold of K3 type. Several questions naturally arise: Where does its K3 structure come from? Are there any hyperkähler varieties associated with it? Is it rational? In joint work with M. Bernardara, we investigate the geometry of the Küchle c5 fourfold and exhibit three rational degenerations of it.
Date: 11/12/2025 Time: 15.00 Room: 713
Title: The first Hilbert coefficient of stretched ideals
Abstract: In this talk we explore the almost Cohen-Macaulayness of the associated graded ring of stretched m-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring (A,m). In particular, we explore the structure of stretched m-primary ideals satisfying the inequality e_1(I) <= e_0(I)-l_A(A/I)+4 where e_0(I) and e_1(I) denote the multiplicity and the first Hilbert coefficient respectively.