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Seminari de Geometria Algebraica 2020/2021 |
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Conferenciant
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Títol |
Data i hora |
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Barcelona Mathematical Days
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23 i 24 d'octubre, Virtual
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Carlos d'Andrea
Universitat de Barcelona
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The Canny-Emiris conjecture for the sparse resultant
Compact and easy-to-evaluate formulae for computing resultants is a kind of "Philosopher's Stone" in computer algebra.
In this search, determinantal formulae are important pearls although hard to find.
Macaulay showed in 1902 that resultants of homogeneous polynomials can be computed as the quotient of two determinants 'a la Sylvester'.
In 1993, Canny and Emiris extended his methods to sparse resultants, and conjectured that also these can be computed as the quotient of two determinants a la Sylvester.
In this talk, we will review the history of this problem and show how in a joint work with Gabriela Jeronimo and Martin Sombra, we managed to solve this conjecture.
slides
video
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Divendres 30 d'octubre, 15h00, virtual.
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Josep Àlvarez Montaner
Universitat Politècnica de Catalunya
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D-modules on singular varieties
A line of research that has attracted some attention lately is to extend the theory of D-modules to singular rings.
In this talk we will go to the basics of the theory and we will develop the notion of holonomic D-modules for certain singular rings.
This is a joint work with D.Hernández, J.Jeffries, L.Núñez-Betancourt, P.Teixeira and E.Witt.
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Divendres 27 de novembre, 15h00, virtual.
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Marina Garrote-López
Universitat Politècnica de Catalunya
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Semi-algebraic conditions for phylogenetic reconstruction
The main goal of phylogenetic reconstruction is to recover the evolutionary history of a group of living species by using solely the
information of their genome. To model evolution, one usually assumes that DNA sequences evolve according to a Markov process on a phylogenetic
tree ruled by a model of nucleotide substitutions. This allows to define a distribution at the leaves of the trees and one might be able to obtain
polynomial relationships among the probabilities of different characters. The study of these polynomials and the geometry of the algebraic varieties
defined by them can be used to reconstruct phylogenetic trees.
However, not all points in these algebraic varieties have biological sense. In this talk, we would like to discuss the importance of studying the
subset of these varieties that has biological sense, the stochastic region, and explore the extent to which restricting to these subsets can provide
insight into existent methods of phylogenetic reconstruction. Moreover, we will prove that, in some cases, considering the stochastic phylogenetic
region seems to be fundamental for the phylogenetic reconstruction problem and we will present the phylogenetic quartet reconstruction method SAQ
(Semi-algebraic quartet reconstruction) which is based on the algebraic and semi-algebraic description of distributions that arise from the general
Markov model on a quartet tree.
slides
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Divendres 11 de desembre, 15h00, virtual.
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Alessio Caminata
Università di Genova, Itàlia
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A Pascal's Theorem for rational normal curves
Pascal's Theorem gives a synthetic geometric condition for six points \(A,\ldots,F\) in the projective plane to lie on a conic:
Namely, that the intersection points of the lines \(AB\) and \(DE\), \(AF\) and \(CD\), \(EF\) and \(BC\) are aligned.
One could ask an analogous question in higher dimension:
Is there a linear coordinate-free condition for \(d + 4\) points in the \(d\)-dimensional projective space to lie on a degree \(d\) rational normal curve?
In this talk, we will discuss and give an answer to this problem by studying the parameter space of \(d+4\) ordered points that lie on a rational normal
curve of degree \(d\). This is a joint work with Luca Schaffler.
slides
video
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Divendres 15 de gener, 15h00, virtual.
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Roser Homs Pons
Technische Universität München, Alemanya
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Inverting catalecticants of ternary quartics
We study the reciprocal variety to the LSSM (linear space of symmetric matrices) of catalecticant matrices associated with ternary quartics.
We prove that only its rank-1 locus, namely the Veronese surface \(v_4(\mathbb{P}^2)\), contributes to the degree of the reciprocal variety.
As opposed to catalecticants of binary forms, we show that the ML-degree of the linear concentration model represented by the LSSM does not coincide
with the degree of its reciprocal variety.
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Divendres 12 de febrer, 15h00, virtual.
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José I. Burgos Gil
ICMAT-CSIC, Madrid
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Higher height pairing and extensions of mixed Hodge structures
The height pairing between algebraic cycles over global fields is an important arithmetic invariant.
It can be written as a sum of local contributions, one for each place of the ground field.
Following Hain, the Archimedean components of the height pairing can be interpreted in terms of biextensions of mixed Hodge structures.
In this talk we will explore how to extend the Archimedean contribution of the height pairing to higher cycles in the Bloch complex
and interpret it as an invariant associated to a mixed Hodge structure.
This is joint work with S. Goswami and G. Pearlstein.
video
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Divendres 19 de febrer, 15h00, virtual.
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Jordi Roca-Lacostena
Universitat Politècnica de Catalunya
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The embedding problem for Markov processes
The embedding problem for Markov matrices consists on deciding whether the substitution process ruled by a given stochastic
matrix is a homogeneous time-continuous process or not. The problem had been solved only for 2x2 and 3x3 matrices so far.
In this talk, we give a characterization of the embeddability of 4x4 matrices, which are used in phylogenetics to model nucleotide
substitution in DNA strands. We also provide a generic solution to the embedding for matrices of larger sizes.
slides
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Divendres 5 de març, 15h00, virtual.
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Yairon Cid-Ruiz
Ghent University
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Primary decomposition with differential operators
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. This is joint work with Bernd Sturmfels
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Divendres 16 d'abril, 15h00, virtual.
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Miguel Ángel Barja
Universitat Politècnica de Catalunya
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Higher dimensional slope inequalities for irregular fibrations
We will consider fibrations of higher dimensional irregular varieties over curves and introduce a new (continuous) invariant, the continuous degree.
Using this invariant we will study the slope of such fibrations, give sharp lower bounds of different classes of such fibrations and study the limit cases.
When the varieties are of maximal Albanese dimension we will establish a perfect correspondence between Clifford-Severi inequalities and Slope inequalities.
This correspondence allows to obtain both Slope and Clifford-Severi inequalites in any dimension by considering an inequality in dimension 1, by an induction argument.
We will also study the cases of non-maximal Albanese dimension where this correspondence does not hold in general.
The main technical tool is a continuous version of the Xiao's method.
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Divendres 30 d'abril, 15h00, virtual.
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Joan Carles Naranjo
Universitat de Barcelona
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Orbits and Voisin sets on abelian varieties
Several geometric problems on abelian varieties are related to families of rationally equivalent zero-cycles, called orbits.
This is the case of the study of the degree of irrationality or the minimum gonality of a curve contained in an abelian variety.
In a recent work, Voisin has given a bound for the dimension of zero-cycle orbits in very general abelian varieties by studying the following sets
(which we call "Voisin sets"):
\(V_k (A) = \{a \in A \mid ({a} - {0})^{*k} = 0 \} \subset CH_0 (A) _ {\mathbb Q}, \)
where the asterisk denotes the Pontryagin product.
The first part of the talk will be devoted to an introduction to the theory of zero-cycles in abelian varieties and we will review
the basic results on orbits.
Next we will describe the properties of Voisin sets and present a bound for the dimensions of the locus in the moduli space of abelian varieties whose sets
\(V_2 (A) \) have positive dimension. These results have been obtained in collaboration with E. Colombo and G.P. Pirola.
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Divendres 14 de maig, 15h00, virtual.
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Víctor González-Alonso
Humboldt-Universität zu Berlin, Alemanya
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Flat subbundles of PVHS and second order Kodaira-Spencer classes
Given a family of smooth complex projective curves, its associated polarized variation of Hodge structure contains a natural maximal flat unitary subbundle \(U\).
It plays an important role in the study of fibred surfaces and totally geodesic subvarieties of the Jacobian locus,
directly related to the Coleman-Oort conjecture on the non-existence of Shimura subvarieties of positive dimension.
The main obstacle to study \(U\) is that it is defined by conditions on open subsets of the base, but there is no useful characterization of its fibre at any (general) point.
In this talk I will present a recent joint work with S. Torelli,
where we provide an alternative characterization of \(U\) that evidences the relation with totally geodesic curves and moreover can be evaluated pointwise.
I will also show how this new characterization suggests a "second order Kodaira-Spencer class", depending on the second-order neighbourhoods of the fibres.
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Divendres 4 de juny, 15h00, virtual.
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Andrés Rojas
Universitat de Barcelona
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Chern degree functions
Given a smooth polarized surface, we will introduce Chern degree functions associated to any object of its derived category.
These functions classify the behaviour of the object along the boundary of a certain region of Bridgeland stability conditions.
We will discuss their extension to continuous real functions and the meaning of their (non-)differentiability at certain points.
In the final part we will see applications to abelian surfaces, where Chern degree functions recover the cohomological rank functions
defined by Jiang-Pareschi. This is joint work with Martí Lahoz.
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Divendres 18 de juny, 15h00, Sessió doble PRESENCIAL, Aula B7, FMI-UB.
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Eduard Casas-Alvero
Universitat de Barcelona
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Polar germs, Jacobian ideal and analytic classification of irreducible plane curve singularities
I will examine the relationship between the analytic type of an irreducible plane curve singularity with a single characteristic exponent and the germs of curve defined by the elements of its Jacobian ideal, in particular its polar germs.
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Divendres 18 de juny, 16h30, Sessió doble PRESENCIAL, Aula B7, FMI-UB.
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