ENG 
ESP Session S07 - Differential operators in algebraic geometry and commutative algebra
Talks
No date set.
On the Partial derivatives of determinant of square Hankel matrix and its degenerations
Maral Mostafazadehfard
Federal University of Rio de Janeiro, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
The generic Hankel matrix is the "super"-symmetric matrix of the following form. This type of matrix is the main member of the family of $1$-generic matrices $$ \left( \begin{matrix} x_1&x_2&\ldots &x_{m-1} & x_{m}\\ x_2&x_3&\ldots &x_{m}& x_{m+1}\\ \vdots &\vdots &\ldots &\vdots \\ x_{m-1}&x_{m}&\ldots &x_{2m-3} &x_{2m-2}\\ x_{m}&x_{m+1}&\ldots &x_{2m-2}&x_{2m-1} \\ \end{matrix} \right). \; $$ By degeneration of the Hankel matrix, we mean to set all the last $r$ variables zero, whenever $r$ is varying from $1$ to $m-2$.
Suppose that $f$ is the determinant of the Hankel matrix or its degenerations. We consider the polar map defined by $f$ and study the properties of this map through the Hessian matrix and ideal of sub-maximal minors. Homaloidalness is our target.
Throughout one deals with the effect of the degenerateness on the numerical invariants and ideal theoretic properties of the gradient ideal of $f$. Among others are Reduction Number, minimal reduction, codimension, Cohen-Macaulayness, and Normality.
Joint work with Rainelly Cunha (Federal institute of education, science, and technology of Rio Grande do Norte), Zaqueu Ramos (Universidade Federal de Sergipe) and Aron Simis (Universidade Federal de Pernambuco).
Friday, July 16, 12:00 ~ 12:30 UTC-3
A survey on $d$-simplicity
Daniel Levcovitz
University of Sao Paulo (USP), Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $k$ be a field of characteristic zero and $d$ a $k$-derivation of a commutative $k$-algebra $R$. We say that $d$ is a simple derivation of $R$ (or just that $R$ is $d$-simple) if $R$ does not have any proper non-zero ideal $I$ such that $d(I)\subseteq I$. Such an ideal is called a $d$-invariant ideal, a $d$-stable ideal or simply a $d$-ideal. Research on simple derivations of commutative $k$-algebras has increased significantly in the past years. This was motivated by several connections of $d$-simple algebras with other branches of mathematics such as noncommutative noetherian ring theory; D-modules; holomorphic foliations and also with the difficult question of algebraic independence of formal solutions of differential equations. In this talk we will present a survey on $d$-simplicity including some examples, main results and some conjectures about the isotropy group of simple derivations.
Friday, July 16, 12:30 ~ 13:00 UTC-3
Bernstein's inequality and holonomicity for certain singular rings
Josep Àlvarez Montaner
Universitat Politècnica de Catalunya, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
We prove the Bernstein inequality and develop the theory of holonomic D-modules for rings of invariants of finite groups in characteristic zero, and for strongly F-regular finitely generated graded algebras with FFRT in prime characteristic.
Joint work with Daniel J. Hernández (University of Kansas, USA), Jack Jeffries (University of Nebraska-Lincoln, USA), Luis Núñez-Betancourt (CIMAT , Mexico), Pedro Teixeira (Knox College, USA) and Emily E. Witt (University of Kansas, USA).
Friday, July 16, 13:00 ~ 13:30 UTC-3
Primary decomposition with differential operators
Yairon Cid Ruiz
Ghent University , Belgium - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis-Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed.
Joint work with Bernd Sturmfels (MPI-MiS Leipzig and UC Berkeley).
Friday, July 16, 13:30 ~ 14:00 UTC-3
D-Modules in (Algebraic) Statistics
Anna-Laura Sattelberger
Max Planck Institute for Mathematics in the Sciences, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
D-modules encode systems of linear partial differential equations with polynomial coefficients in algebraic terms. Among others, they help to solve problems arising in statistics. For an observed outcome of a discrete statistical experiment, the maximum likelihood estimation problem is to find the parameters that best explain these data. In this talk, I explain several ways how to tackle this problem in terms of D-modules.
Friday, July 16, 14:00 ~ 14:30 UTC-3
Differential powers in mixed characteristic
Eloísa Grifo
University of Nebraska – Lincoln, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.
The differential operators version of the Zariski--Nagata theorem says that over a polynomial ring over a perfect field, the differential powers of a radical ideal coincide with its symbolic powers. In mixed characteristic, differential powers are larger than symbolic powers, but there is a version of Zariski--Nagata that works when we mix in p-derivations.
In singular rings, differential powers can still be used as tools to prove results about symbolic powers, even though the two notions no longer coincide. We will discuss a uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic by introducing a new type of differential powers, which do not require the existence of a p-derivation on the direct summand.
Joint work with Alessandro De Stefani (University of Genova, Italy) and Jack Jeffries (University of Nebraska--Lincoln, United States of America).
Friday, July 16, 14:30 ~ 15:00 UTC-3
Valuative aspects of differential algebraic geometry
Cristhian Garay López
Center for Research in Mathematics (CIMAT), México - This email address is being protected from spambots. You need JavaScript enabled to view it.
Broadly speaking, the key ingredient for tropicalization is a generalized non-Archimedean absolute value; that is, is a function defined on a commutative ring with unit that satisfies properties analogous to those of a non-Archimedean absolute value (i.e., Krull valuation), but for which we allow the target to be a commutative idempotent semiring.
In this talk we will introduce non-Krull absolute values for rings of multivariate formal power series (and their fields of fractions) which are defined in terms of Newton polyhedra and polytopes associated to their supports in $\mathbb{N}^m$. We show how to use them to study systems of classical partial algebraic differential equations from a valuative (tropical) point of view.
Friday, July 16, 15:00 ~ 15:30 UTC-3
The discriminant locus of a vector bundle
Gregory G. Smith
Queen's University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
We analyze the locus of singular sections of a very ample vector bundle on a smooth complex variety. We characterize when this projective subscheme is a hypersurface. In the hypersurface case, we also compute its degree.
Joint work with Hirotachi Abo (University of Idaho, USA) and Robert Lazarsfeld (Stony Brook University, USA).
Monday, July 19, 16:00 ~ 16:30 UTC-3
On the isotropy of a polynomial derivation in two variables
Iván Pan
Universidad de la República, Uruguay - This email address is being protected from spambots. You need JavaScript enabled to view it.
If $D:k[x,y]\to k[x,y]$ is a derivation, where $k$ is an algebraically closed field of characteristic zero, we denote by $Aut(D)$ the group consisting of (plane) polynomial automorphisms which commute with $D$. We determine when $Aut(D)$ is an algebraic group.
Monday, July 19, 16:30 ~ 17:00 UTC-3
The length of $\mathcal{D}\frac{1}{f}.$
Thomas Bitoun
University of Calgary , Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $f$ be an absolutely irreducible multivariate polynomial with rational coefficients. We consider the $\mathcal{D}$-submodule of the complex local cohomology module $H^1_f(\mathcal{O})$ generated by the class of $\frac{1}{f}.$ Its $\mathcal{D}$-module length is closely related to that of the local cohomology of the reduction modulo $p$ of $f, H^1_{f_p}(\mathcal{O}_p),$ for large primes $p.$ We compute the lengths in the case of $f$ quasi-homogeneous with an isolated singularity and present a conjecture for the general isolated singularity case.
Partly joint work with Travis Schedler (Imperial College, United Kingdom).
Monday, July 19, 17:00 ~ 17:30 UTC-3
Bernstein-Sato theory for singular rings in positive characteristic
Eamon Quinlan-Gallego
University of Michigan, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Given an ideal $\mathfrak{a}$ in a smooth $\mathbb{C}$-algebra $R$, its Bernstein-Sato polynomial $b_{\mathfrak{a}}(s)$ is an invariant with origins in complex analysis that measures the singularities of the zero locus of $\mathfrak{a}$.
There are two recent generalizations of this construction. In one direction, Àlvarez-Montaner, Huneke, and Núñez-Betancourt have shown that ideals in certain non-smooth $\mathbb{C}$-algebras (direct summands) still admit Bernstein-Sato polynomials. In the other direction, work of Mustaţă, Bitoun and myself shows that we can also define Bernstein-Sato invariants in smooth algebras over fields of positive characteristic.
In this talk I present joint work with J. Jeffries and L. Núñez-Betancourt in which we that the combined generalization is possible. Namely, we show that Bernstein-Sato invariants exist in positive characteristic when the ambient ring has mild singularities (direct summand or graded F-finite representation type).
Joint work with Jack Jeffries (University of Nebraska-Lincoln) and Luis Núñez-Betancourt (Centro de Investigación en Matemáticas).
Monday, July 19, 17:30 ~ 18:00 UTC-3
High order differentials and Hasse-Schmidt derivations
Daniel Duarte
Universidad Autónoma de Zacatecas - CONACyT, México - This email address is being protected from spambots. You need JavaScript enabled to view it.
It was recently proved by T. de Fernex and R. Docampo that the module of differentials of the algebra of Hasse-Schmidt derivations of a ring can be described in terms of the module of differentials of the ring. This result was then applied to find a projectivization of induced maps on jets schemes. In this talk, we explore the analogous statements for the module of high order differentials.
Joint work with Paul Barajas (Universidad Autónoma de Zacatecas).
Monday, July 19, 18:00 ~ 18:30 UTC-3
Differential ideals and homological problems about derivation modules
Cleto B. Miranda-Neto
Universidade Federal da Paraíba, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We will present recent generalizations of results of A. K. Maloo concerning maximally differential ideals of local rings, and will propose some problems on the finiteness of the Gorenstein and the complete intersection dimensions of derivation modules of certain rings, inspired by classical long-standing conjectures stated in terms of projective dimension.
Monday, July 19, 18:30 ~ 19:00 UTC-3
Lines on Extremal Surfaces
Karen Smith
University of Michigan, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
What is the most singular possible singularity? What can we say about it's geometric and algebraic properties? This seemingly naive question has a sensible answer in characteristic p. The "F-pure threshold," which is an analog of the log canonical threshold, can be used to "measure" how bad a singularity is. The F-pure threshold is a numerical invariant of a point on (say) a hypersurface---a positive rational number that is 1 at any smooth point (or more generally, any F-pure point) but less than one in general, with "more singular" points having smaller F-pure thresholds. We explain a recently proved lower bound on the F-pure threshold in terms of the multiplicity of the singularity. We also show that there is a nice class of hypersurfaces--which we call "Extremal hypersurfaces"---for which this bound is achieved. These have very nice (extreme!) geometric properties. For example, the affine cone over a non Frobenius split cubic surface of characteristic two is one example of an "extremal singularity". Geometrically, these are the only cubic surfaces with the property that *every* triple of coplanar lines on the surface meets in a single point (rather than a "triangle" as expected)--a very extreme property indeed. In recent work with Anna Brosowski, Janet Page and Tim Ryan, we have unravelled the story of the lines on extremal surfaces of any degree.
Joint work with Anna Brosowksi (University of Michigan), Janet Page (University of Michigan) and Tim Ryan (University of Michigan).