## ANA BRAVO ZARZA

 Algebra Seminar

Monday, December 3rd, 2018, 11:30

Room  420  module 17

Angus J. Macintyre (Queen Mary University of London)

Title: " Analogues for exponential fields of algebraic-geometric notions  "

Abstract: Tarski raised in the 1930's some logical questions about the real exponential field. Answers to these questions did not come till the 1990's in the work of Wilkie and Wilkie-Macintyre (based on work of Hovanski, and insights of Schanuel). Wilkie's work initiated numerous very important uses of the notion of o-minimality.
Work of Zilber about twenty years ago initiated serious study of the complex exponential, from a logical point of view concerning exponential algebraic sets. More dramatically, it revealed the existence of other exponential fields (the Zilber fields) with highly structured notions of exponential dependence and exponential dimension (including a general Steinitz theory). It was conjectured by Zilber that the complex exponential field is a Zilber field. This would imply Schanuel's Conjecture and a very deep Hilbert Nullstellensatz for exponential-algebraic sets.
I will discuss the analysis, algebra and model theory that has gone in to establishing

a rich theory of exponential dimension.

Thursday, October 11th, 2018, 11:00

Room  420  module 17

Título: " Sobre el invariante Mather-Jacobian mld y el espacio de arcos según S. Ishii

Thursday, October 4th, 2018, 10:30

Room  420  module 17

Ricardo Podestá (FaMAF, Universidad Nacional de Córdoba)

Título: "Formas cuadráticas, sumas exponenciales, grafos de Ramanujan y códigos cíclicos asociados"

Resumen: Utilizando polinomios q-linealizados es posible definir formas cuadráticas sobre el cuerpo finito Fq. Introduciremos ciertos grafos de Cayley asociados a estas formas cuadráticas y estudiaremos sus propiedades. El espectro de dichos grafos depende del cómputo de sumas exponenciales asociadas. El cálculo explícito de dicho espectro permitirá dar condiciones para que estos grafos resulten ser enteros, no bipartitos, fuertemente regulares y de Ramanujan. Para una forma cuadrática particular, se pueden describir en qué casos los grafos son Ramanujan y, además, dar explícitamente el espectro del código cíclico asociado.
La charla se basa en trabajos conjuntos en curso con Denis Videla (FaMAF, UNC).

Monday, September  24th, 2018, 11:00

Room 520, module 17

Title: " Differential Galois Theory for some Spectral Problems "

Abstract: In this talk we will introduce the Picard-Vessiot Theory for integrable systems and the Darboux transformations. First, we will present our results on the differential Galois groups for Ablowitz-Kaup-Newell-Segur systems, which are an important kind of integrable systems depending on a spectral parameter $\lambda$.

Next we will focus on  the Schrödinger equation $(-\partial^2+u)\psi=-\lambda^2 \psi$ associated to the Korteweg de Vries hierarchy (KdV hierarchy for short). We will show the algebraic structure of the  fundamental matrices for the Schrödinger equation with potential $u$ in a fixed family of KdV rational potentials. As a by product, we will obtain the differential Galois groups  associated with the mentioned spectral problem. We will also compute  non trivial examples in the $1+1$ dimensional case using SAGE.

Moreover, we will establish the deep relationship between the singularities of the spectral curves, the Darboux transformations and the fundamental matrices for the KdV hierarchy.

Secondly, we will present a family of rational complex potentials $u$ depending on a parameter. We will show that these functions are KdV potentials and compute fundamental matrices for the corresponding Schrödinger equation.

Finally, we will use Darboux transformations for studying orthogonal differential systems from a galoisian point of view. Here the techniques of tensor products of  differential systems are essential tools. Explicit formulas for these matrix Darboux transformations are computed using Maple.

Monday, June  25th, 2018, 12:00

Room 420, module 17

Olivier Piltant (Université Rennes 1)

Title: " A weak problema of local uniformization"

Abstract: Dado un cuerpo de funciones $K$ sobre un cuerpo base $k$, la uniformización local de una $k$-valoración $v$ de $K$  trata de encontrar un modelo afín y regular de $K$ en el que esté $v$ centrada.
Motivados por cuestiones de irreductibilidad en el espacio de arcos de los modelos de $K$, introduciremos un problema análogo cambiando la palabra "regular" por "cuyo espacio de arcos es irreducible". Sorprendentemente, se trata de un problema abierto en general para cuerpos de característica positiva.
Trabajo conjunto con A. Benito y A. Reguera.

Monday, June  25th, 2018, 11:00

Room 420, module 17

Mario Morán (Université Rennes 1)

Title: "On local rings of arc spaces of toric varieties”

Monday June 4th 2018, 12:00

Room 320  module 17

Pooneh Afsharijoo (Paris Diderot -Paris 7)

Title: Looking for a new version of Gordon's Identitites and differential ideals

Monday February 5th,   2018, 11:30

Room 101-6  module 16

Hanna Melánová (University of Vienna)

Title: Resolution of singular plane curves via geometric invariants

Wednesday January  17th,   2018, 11:30h

Room 420 module 17

Hussein Mourtada (Université Paris Diderot, Paris 7)

Title: A geometric approach to resolution of singularities

Abstract: We will explain an approach via jet schemes to a conjecture of Teissier on resolution of singularities with toric morphisms.

Wednesday December 20th, 2017, 10:00h

Room 420 module 17

Pre-thesis defense: Beatriz Pascual Escudero

Title: Algorithmic resolution of singularities and Nash multiplicity sequences

Monday December 18th, 2017, 11:00h

Room  420 module  17

Eleonore Faber (University of Leeds)

Title: Endomorphism rings and rings of differential operators of finite global dimension

Abstract: In this talk we consider a normal toric algebra R over a field k of arbitrary characteristic. The module M of p^e-th roots of R, where p and e are positive integers, is then the direct sum of so-called conic modules. With a combinatorial method we construct certain complexes of conic modules over R and explain how these yield projective resolutions of simple modules over the endomorphism ring End_R(M). Thus we obtain a bound on the global dimension of End_R(M), which shows that this endomorphism ring is a so-called noncommutative resolution of singularities (NCR) of R (or Spec(R)). If the characteristic of k is p>0, then this fact allows us to bound the global dimension of the ring of differential operators D(R). This is joint work with Greg Muller and Karen E. Smith.

Tuesday November  7th,  2017: “Arcs and Singularities”

Title: “Explicit computations of local rings of the space of arcs at stable points”.

Title:Integrable derivations and base change”

Title: “Nash multiplicity sequences and their relation to invariants from constructive resolution of singularities”.

Title: “p-bases and differential operators on varieties over non-perfect fields”

Tuesday July 18th,  12:00

Fuensanta Aroca, Instituto de Matemáticas, Universidad Nacional Autónoma de México

Title: The algebraic closure of the power series field  in several variables.

Tuesday July 4th,   11:30

Ana Zumalacárregui (University of New South Wales, Australia)

Title: "Strategies to solve problems in congruencies”

Monday June 19th,  2017, 12:00

Teresa Krick (universidad de Buenos Aires-IMAS-CONICET)

Title:  Subresultantes en raíces múltiples

Friday June 16th, 2017, 10:30

Luis Núñez Betancourt (CIMAT, Guanajuato, Méjico)

Title: F-thresholds of local and graduated  rings.

Friday June 9th,  2017,  10:00

Juan de Vicente Guijarro

Pre-thesis defense

Title: "Locally Nash Groups"

Room  420, module 17

Wednesday June  7th,  2017, 14:30

Hema Srinivasan (University of Missouri)

Title: Generating graded Cohen Macaulay algebras

Room 420. Module 17

Monday June 5th, 2017, 12:30

S. Dale Cutkosky (University of  Missoury)

Title: Extension under projection of associated graded rings along a valuation

Abstract: A central method in resolution of singularities is to take a finite projection to a regular variety, and then to make a local analysis of the ramification of this projection to understand which blow ups are required to improve the singularity. In local uniformization, this analysis is made along a fixed, arbitrary  valuation, so it can be very complicated (the value group may not be finitely generated).

The relevant information about this projection, and the effect of the possible blow ups along the valuation, is captured in the extension of associated graded rings along the valuation. The associated graded ring along a valuation was introduced by Teissier; it is central in his work on local uniformization in positive characteristic.

In this talk we define the associated graded ring along the valuation, and consider the structure of the extension of associated graded rings along a  projection, and stable forms of the extension after sufficient blowing up along the valuation.

Room 420, module 17

Firday April 21st,  2017, 11:30h

Title: Terminal valuations and the Nash problema according to de Fernex - Docampo.

Room 420, module 17

Tuesday March 21st,  2017, 14:30

André Belotto (Université Toulouse-III-Paul-Sabatier)

Title: Resolution of singularities of the cotangent sheaf of a singular variety

Room 420, module 17

Friday March 3rd, 2017

Roberto Miatello (Universidad Nacional de Córdoba, Argentina)

Title: Hilbert cusp forms with prescribed Casimir and Hecke eigenvalues

Abstract: The main goal is to describe joint work with R Bruggeman (Utrecht) on the distribution of cusp forms for the Hilbert-Blumenthal group in terms of their Laplace and Hecke eigenvalues. A main tool will be a version of the Kuznetsov trace formula that will be first introduced  in the case of the modular group.

Friday February 24th,   2017, 12:00

Thesis defense


Title: "Multiplicity along embedded schemes and differential operators"

Carlos Abad Reigadas
Advisor: Orlando Villamayor Uriburu

Room 307, module 0

Thursday February  23rd,  2017

Metting on Singularities (organized by  A. Benito)

11:30 - 12:30

"Derivaciones asociadas a una derivación de Hasse-Schmidt"

12:40 - 13:40

"Cuerpos de coeficientes y derivaciones de Hasse-Schmidt"

16:00 - 17:00

"D-módulos, polinomios de Bernstein-Sato y F-invariantes de sumandos

directos"

Josep Àlvarez-Montaner (Universitat Politècnica de Catalunya)

17:10 - 18:10

"Sobre la clausura entera de un ideal plano"

Guillem Blanco (Universitat Politècnica de Catalunya)