Seminari di Geometria Algebrica a.a. 2015-2016
- 10 Ottobre 2016, ore 14.30, sala conferenze Tricerri, DiMaI
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Jan Draisma
(Università di Berna)
(Uniform) determinantal representations of polynomials
ABSTRACT. Given a polynomial p of degree d in n variables x_1,...,x_n, there always exists a square matrix M whose entries are affine-linear
in the x_i and whose determinant equals p. Probably the best-known instance of this is the case where n=1 and M is the companion matrix of
p. Across several areas of mathematics and theoretical computer science the challenge arises to minimise the number N of rows among all such
determinantal representations of p. I will discuss some of this existing literature, and then zoom in on recent, joint work with
Boralevi, van Doornmalen, Hochstenbach, and Plestenjak, in which we require that the entries of M, in addition to being affine-linear in
the x_i, depend polynomially (or even affine-linearly) on the coefficients of p. Then M is a uniform determinantal representation of
all polynomials of degree at most d in x_1,...,x_n. In this setting, which is motivated by Hochstenbach-Plestenjak's work in numerics, we
show that the minimal N grows like C(n) * d^(n/2) for n fixed and d tending to infinity.
- 31 Maggio 2016, ore 12.00, sala conferenze Tricerri, DiMaI
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Jerzy Weyman
(U. Connecticut)
Finite free resolutions and Kac-Moody Lie algebras
ABSTRACT. Let us recall that a format $(r_{n},\ldots ,r_{1})$ of a free complex $0 \to F_{n} \to F_{n-1} \to \ldots \to F_0$
over a commutative Noetherian ring is the sequence of ranks $r_{i}$ of the $i$-th differential $d_{i}$. We will assume that rank $F_{i} = r_{i}+r_{i+1}$.
We say that an acyclic complex $F_{gen}$ of a given format over a given ring $R_{gen}$ is generic if for every complex $G$ of this format over a Noetherian
ring $S$ there exists a homomorphism $f:R_{gen} \to S$ such that $G=F_{gen}\otimes_{R_{gen}} S$. For complexes of length $2$ the existence of the generic
acyclic complex was established by Hochster and Huneke in the 1980's. It is a normalization of the ring giving a generic complex (two matrices with composition
zero and rank conditions). I will discuss the ideas going into the proof of the following result: Associate to a triple of ranks $(r_{3}, r_{2}, r_{1})$
a triple $(p,q,r)=(r_{3}+1,r_{2}-1, r_{1}+1)$. Associate to $(p,q,r)$ the graph $T_{p,q,r}$ (three arms of lengths $p-1, q-1, r-1$ attached to the central vertex).
Then there exists a Noetherian generic ring for this format if and only if $T_{p,q,r}$ is a Dynkin graph. In other cases one can construct in a uniform way a
non-Noetherian generic ring, which deforms to a ring carrying an action of the Kac-Moody Lie algebra corresponding to the graph $T_{p,q,r}$.
- 18 Maggio 2016, ore 14.30, sala conferenze Tricerri, DiMaI
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Nick Vannieuwenhoven
(TU Leuven)
A condition number for the tensor rank decomposition
ABSTRACT. The tensor rank decomposition problem consists of recovering the unique parameters of the decomposition from a robustly identifiable low-rank tensor.
These parameters are subsequently analyzed and interpreted in many applications. As tensors are often perturbed by measurement errors in practice, one must investigate
insofar the unique parameters change in order to preserve the validity of the analysis. The magnitude of this change can be bounded asymptotically by the product of the
condition number and the magnitude of the perturbation to the tensor. This paper introduces such a condition number for the tensor rank decomposition problem. It admits
a closed expression as the inverse of a particular singular value of Terracini's matrix(a matrix representing the tangent space to the semi-algebraic set of tensors of
fixed rank). A practical algorithm for computing the condition number is presented. The latter's elementary properties such as scaling and orthogonal invariance are established.
The condition number of rank-1 tensors of order d equals d1/2d1/2; they are always well-conditioned. The class of weak 3-orthogonal tensors, which includes orthogonally
decomposable tensors, contains both well-conditioned and ill-conditioned problems. The numerical experiments confirm that the condition number yields a good upper bound
on the magnitude of the change of the parameters when the tensor is perturbed. They also suggest that the condition number may be inversely related to the distance to ill-posed
tensor rank decomposition problems, where the ill-posedness arises either from the nonclosedness of the set of tensors of fixed rank or from the existence of infinitely many
decompositions.
- 21 Aprile 2016, ore 12.30, sala conferenze Tricerri, DiMaI
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Alessandro Massarenti
(Fluminense, Rio de Janeiro)
On the biregular geometry of Fulton-MacPherson configuration spaces
ABSTRACT. allegato
- 23 Febbraio 2016, ore 11.30, sala conferenze Tricerri, DiMaI
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Ada Boralevi
(SISSA Trieste)
Un nuovo approccio allo studio dei tensori ortogonalmente decomponibili
ABSTRACT. Dare una decomposizione ortogonale di una matrice corrisponde a trovare la sua decomposizione a valori singolari (SVD),
ed è ben noto che ogni matrice ammette una fattorizzazione di questo tipo, dove i termini sono a due a due ortogonali.
Nel caso di tensori di ordine strettamente maggiore di due tuttavia solo un sottoinsieme relativamente
piccolo ammette tale decomposizione; i suoi elementi sono detti tensori odeco (sui reali) o udeco (sui complessi),
e formano varietà algebriche reali. In collaborazione con J. Draisma, E. Horobet e E. Robeva abbiamo affrontato lo studio dei tensori
odecoudeco da un punto di vista algebrico, e abbiamo trovato un collegamento con specifiche proprietà di algebre semisemplici
che fornisce una descrizione completa in tutti i casi di tensori ordinari, simmetrici e alternanti; e in entrambe le versioni reale (odeco) e complessa (udeco).
- 5 Febbraio 2016, ore 11.30, sala conferenze Tricerri, DiMaI
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Alexander Dimca
(Nice)
On free plane curves
ABSTRACT. First we recall the definition of free divisors going back to Kyoji Saito around 1980.
Next we explain a new method for constructing free curve arrangements by using pencils of curves.
Finally the relation between irreducible free curves and rational cuspidal curves will be explored.
- 26 Gennaio 2016, ore 11.30, sala conferenze Tricerri, DiMaI
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Simone Naldi
(Tolosa / Dortmund)
Computer algebra algorithms for semidefinite programming
ABSTRACT. Semidefinite programming (SDP) is the natural extension of linear programming to the convex cone of symmetric positive semidefinite matrices.
It consists in minimizing a linear function over the convex set, called spectrahedron, defined by a linear matrix inequality (i.e. over the set of real vectors x
such that a symmetric linear matrix A(x) is positive semidefinite, A(x)>=0). SDP finds numerous applications especially in combinatorial and polynomial
optimization, systems control theory, real algebra. While one can solve "numerically" a SDP problem with interior-point algorithms in polynomial time at
a given accuracy, neither efficient exact algorithms for SDP are available, nor a complete understanding of its theoretical complexity has been achieved.
In this talk I would like to present some new results in this direction, based on joint work with D. Henrion and M. Safey El Din.