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Alexander Schmitt

Publications and source records attributed to Alexander Schmitt.

14 recordsLinked to original sources
arXiv2016

Analysis of Temporal Features for Interaction Quality Estimation

Many different approaches for estimating the Interaction Quality (IQ) of Spoken Dialogue Systems have been investigated. While dialogues clearly have a sequential nature, statistical classification approaches designed for sequential problems do not seem to work better on automatic IQ estimation than static approaches, i.e., regarding each turn as being independent of the corresponding dialogue. Hence, we analyse this effect by investigating the subset of temporal features used as input for statistical classification of IQ. We extend the set of temporal features to contain the system and the user view. We determine the contribution of each feature sub-group showing that temporal features contribute most to the classification performance. Furthermore, for the feature sub-group modeling the temporal effects with a window, we modify the window size increasing the overall performance significantly by +15.69%.

cs.HC↗
arXiv2011

On the motives of moduli of chains and Higgs bundles

We take another approach to Hitchin's strategy of computing the cohomology of moduli spaces of Higgs bundles by localization with respect to the circle-action. Our computation is done in the dimensional completion of the Grothendieck ring of varieties and starts by describing the classes of moduli stacks of chains rather than their coarse moduli spaces. As an application we show that the n-torsion of the Jacobian acts trivially on the middle dimensional cohomology of the moduli space of twisted SL_n-Higgs-bundles of degree coprime to n and we give an explicit formula for the motive of the moduli space of Higgs bundles of rank 4 and odd degree. This provides new evidence for a conjecture of Hausel and Rodríguez-Villegas. Along the way we find explicit recursion formulas for the motives of several types of moduli spaces of stable chains.

math.AG↗
arXiv2004

A universal construction for moduli spaces of decorated vector bundles over curves

Let $X$ be a smooth projective curve over the complex numbers. To every representation $ρ\colon \GL(r)\lra \GL(V)$ of the complex general linear group on the finite dimensional complex vector space $V$ which satisfies the assumption that there be an integer $α$ with $ρ(z \id_{\C^r})=z^α\id_V$ for all $z\in\C^*$ we associate the problem of classifying triples $(E,L,ϕ)$ where $E$ is a vector bundle of rank $r$ on $X$, $L$ is a line bundle on $X$, and $ϕ\colon E_ρ\lra L$ is a non trivial homomorphism. Here, $E_ρ$ is the vector bundle of rank $\dim V$ associated to $E$ via $ρ$. If we take, for example, the standard representation of $\GL(r)$ on $\C^r$ we have to classify triples $(E,L,ϕ)$ consisting of $E$ as before and a non-zero homomorphism $ϕ\colon E\lra L$ which includes the so-called Bradlow pairs. For the representation of $\GL(r)$ on $S^2\C^3$ we find the conic bundles of Gomez and Sols. In the present paper, we will formulate a general semistability concept for the above triples which depends on a rational parameter $δ$ and establish the existence of moduli spaces of $δ$-(semi)stable triples of fixed topological type. The notion of semistability mimics the Hilbert-Mumford criterion for $SL(r)$ which is the main reason that such a general approach becomes feasible. In the known examples (the above, Higgs bundles, extension pairs, oriented framed bundles) we show how to recover the "usual" semistability concept. This process of simplification can also be formalized. Altogether, our results provide a unifying construction for the moduli spaces of most decorated vector bundle problems together with an automatism for finding the right notion of semistability and should therefore be of some interest.

math.AG↗
arXiv2002

On the classification of certain piecewise linear and differentiable manifolds in dimension eight and automorphisms of $#_{i=1}^b(S^2\times S^5)$

In this paper, we will be concerned with the explicit classification of closed, oriented, simply-connected spin manifolds in dimension eight with vanishing cohomology in the odd dimensions. The study of such manifolds was begun by Stefan Müller. In order to understand the structure of these manifolds, we will analyze their minimal handle presentations and describe explicitly to what extent these handle presentations are determined by the cohomology ring and the characteristic classes. It turns out that the cohomology ring and the characteristic classes do not suffice to reconstruct a manifold of the above type completely. In fact, the group ${\rm Aut_0}\bigl(#_{i=1}^b(S^2\times S^5)\bigr)/{\rm Aut}_0\bigl(#_{i=1}^b (S^2\times D^6)\bigr)$ of automorphisms of $#_{i=1}^b(S^2\times S^5)$ which induce the identity on cohomology modulo those which extend to $#_{i=1}^b(S^2\times D^6)$ acts on the set of oriented homeomorphy classes of manifolds with fixed cohomology ring and characteristic classes, and we will be also concerned with describing this group and some facts about the above action.

math.GT↗
arXiv2002

Topological rigidity of algebraic $¶_3$-bundles over curves

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $g\ge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case $g\ge 2$.

math.AG↗
arXiv2002

Singular principal bundles over higher dimensional manifolds and their moduli spaces

In this note, we introduce the notion of a singular principal G-bundle, associated to a reductive algebraic group G over the complex numbers by means of a faithful representation $\varrho^\p\colon G\lra \SL(V)$. This concept is meant to provide an analogon to the notion of a torsion free sheaf as a generalization of the notion of a vector bundle. We will construct moduli spaces for these singular principal bundles which compactify the moduli spaces of stable principal bundles.

math.AG↗
arXiv2001

Framed Hitchin Pairs

We provide a construction of the moduli spaces of framed Hitchin pairs and their master spaces. These objects have come to interest as algebraic versions of solutions of certain coupled vortex equations by work of Lin and Stupariu. Our method unifies and generalizes constructions of several similar moduli spaces. Here are some points which are also of interest in other similar situations: - Our construction does not require the symmetricity condition that the map ^2G x E -> E be zero, usually appearing in the context of Higgs bundles. - We carry out a detailed analysis of the polynomial stability parameter without referring to GIT. This sheds some light on intrinsic properties of such parameter dependent stability concepts. - The construction corrects an inaccuracy in our previous construction of the compactification of the Hitchin space and generalizes it.

math.AG↗
arXiv2001

Moduli problems of sheaves associated with oriented trees

To every oriented tree, we associate a moduli problem for sheaves over a projective manifold $X$. We define the corresponding notion of semistability and establish the existence of moduli spaces. Applying the results to the tree *->*, we obtain a GIT construction for the moduli space of holomorphic triples of Bradlow and Garcia-Prada.

math.AG↗
arXiv1999

The Equivalence of Hilbert and Mumford Stability for Vector Bundles

In this paper, we prove that the notions of Hilbert stability and Mumford stability agree for vector bundles of arbitrary rank over smooth curves. The notion of Hilbert stability was introduced by Gieseker and Morrison in 1984, and they showed that for smooth curves and vector bundles of rank two it agrees with Mumford stability. A different proof for the rank two case was given by M. Teixidor i Bigas. Our proof uses a new approach and avoids complicated computations. Our results might serve as a first step in the construction of the Hilbert stable compactification of the universal moduli space of stable vector bundles over the moduli space of smooth curves as suggested by Teixidor.

math.AG↗
arXiv1999

Walls for Gieseker semistability and the Mumford-Thaddeus principle for moduli spaces of sheaves over higher dimensional bases

Let $X$ be a complex projective manifold. Fix two ample line bundles $H_0$ and $H_1$ on $X$. It is the aim of this note to study the variation of the moduli spaces of Gieseker semistable sheaves for polarizations lying in the cone spanned by $H_0$ and $H_1$. We attempt a new definition of walls which naturally describes the behaviour of Gieseker semistability. By means of an example, we establish the possibility of non-rational walls which is a substantially new phenomenon compared to the surface case. Using the approach of Ellingsrud and Goettsche via parabolic sheaves, we were able to show that the moduli spaces undergo a sequence of GIT flips while passing a rational wall.

math.AG↗
arXiv1997

Master Spaces for stable pairs

We construct master spaces for oriented torsion free sheaves coupled with morphisms into a fixed reference sheaf. These spaces are projective varieties endowed with a natural $\C^*$-action. The fixed point set of this action contains the moduli space of semistable oriented torsion free sheaves and the quot scheme associated with the given data. In the case of curves with trivial reference sheaf, our master spaces compactify the moduli spaces constructed by Bertram, Daskalopoulos and Wentworth. In the 2-dimensional case with trivial rank 1 reference sheaf, master spaces provide algebraic analoga of compactified moduli spaces of twisted quaternionic monopoles.

alg-geom↗
arXiv1996

Projective Moduli for Hitchin Pairs

We give an algebraic geometric compactification of certain moduli spaces of semistable E-pairs in the sense of Yokogawa. In particular, we obtain a compactification of the moduli spaces of semistable Higgs pairs on a curve which were constructed by N. Hitchin.

alg-geom↗