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We present STT4SG-350 (Speech-to-Text for Swiss German), a corpus of Swiss German speech, annotated with Standard German text at the sentence level. The data is collected using a web app in which the speakers are shown Standard German sentences, which they translate to Swiss German and record. We make the corpus publicly available. It contains 343 hours of speech from all dialect regions and is the largest public speech corpus for Swiss German to date. Application areas include automatic speech recognition (ASR), text-to-speech, dialect identification, and speaker recognition. Dialect information, age group, and gender of the 316 speakers are provided. Genders are equally represented and the corpus includes speakers of all ages. Roughly the same amount of speech is provided per dialect region, which makes the corpus ideally suited for experiments with speech technology for different dialects. We provide training, validation, and test splits of the data. The test set consists of the same spoken sentences for each dialect region and allows a fair evaluation of the quality of speech technologies in different dialects. We train an ASR model on the training set and achieve an average BLEU score of 74.7 on the test set. The model beats the best published BLEU scores on 2 other Swiss German ASR test sets, demonstrating the quality of the corpus.

We present SDS-200, a corpus of Swiss German dialectal speech with Standard German text translations, annotated with dialect, age, and gender information of the speakers. The dataset allows for training speech translation, dialect recognition, and speech synthesis systems, among others. The data was collected using a web recording tool that is open to the public. Each participant was given a text in Standard German and asked to translate it to their Swiss German dialect before recording it. To increase the corpus quality, recordings were validated by other participants. The data consists of 200 hours of speech by around 4000 different speakers and covers a large part of the Swiss-German dialect landscape. We release SDS-200 alongside a baseline speech translation model, which achieves a word error rate (WER) of 30.3 and a BLEU score of 53.1 on the SDS-200 test set. Furthermore, we use SDS-200 to fine-tune a pre-trained XLS-R model, achieving 21.6 WER and 64.0 BLEU.

We provide a uniform bound for the index of cohomology classes in $H^i(F, \mu_\ell^{\otimes i-1})$ when $F$ is a semiglobal field (i.e., a one-variable function field over a complete discretely valued field $K$). The bound is given in terms of the analogous data for the residue field of $K$ and its finitely generated extensions of transcendence degree at most one. We also obtain analogous bounds for collections of cohomology classes. Our results provide recursive formulas for function fields over higher rank complete discretely valued fields, and explicit bounds in some cases when the information on the residue field is known. In the process, we prove a splitting result for cohomology classes of degree 3 in the context of surfaces over finite fields., Comment: 23 pages. Lemmas 3.1 and 3.3 have been generalized. Lemma 4.2 has been dropped. Section 7 has been reorganized. Sections 7 and 8 now focus on the function field case, and consider the number field case only for the prime 2

We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the semiglobal field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence., Comment: 59 pages. Some improved phrasing to clarify the presentation

We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat. For the proof we develop a new characterization of free proalgebraic groups in terms of split embedding problems, and we use patching techniques in order to solve a very general class of differential embedding problems. Our result about $\mathbb{C}(x)$ also applies to rational function fields over more general fields of coefficients., Comment: 20 pages. Version of Jan. 12, 2021. Expanded the third paragraph of Section 4.2; updated bibliography

In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution of a torus., Comment: 27 pages; final pre-publication version; correction to Theorem 6.5, rewording at Proposition 6.3 and beginning of Section 8.3

We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the absolute differential Galois group is free as a proalgebraic group. Building on this, we prove Matzat's freeness conjecture in the case that the field of constants is algebraically closed of countably infinite transcendence degree over the rationals. This is the first known case of the twenty year old conjecture., Comment: 16 pages. Version of Oct. 20, 2020. Expanded the discussion after Definition 2.4; added an assumption on not being finite type after the proof of Lemma 3.2

We consider local-global principles for rational points on varieties, in particular torsors, over one-variable function fields over complete discretely valued fields. There are several notions of such principles, arising either from the valuation theory of the function field, or from the geometry of a regular model of the function field. Our results compare the corresponding obstructions, proving in particular that a local-global principle with respect to valuations implies a local-global principle with respect to a sufficiently fine regular model., Comment: 10 pages; published version

We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a function field. In this context we also study algebraic versions of van Kampen's theorem, describing the global absolute Galois group as a pushout of local absolute Galois groups., Comment: 26 pages. Some reorganization of Section 3.2, with a new remark; notation in Section 3.3 for absolute Galois groupoids has been modified and made more consistent; slightly expanded introduction

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over ${\mathbb Q}$. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over ${\mathbb Q}_p(x)$., Comment: 16 pages. Shortened Sections 1 and 2; added Examples 3.2, 3.4, 4.3, 4.5; renumbering of the results in all sections due to these changes

We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case., Comment: 23 pages. Some explanations were expanded for greater clarity throughout the paper. Some results in Section 2 in the earlier version in the complete case were generalized here to the henselian case. Subsection 3.3 in the earlier version was broken into two subsections, and there was some renumbering of results in the new Subsection 3.4 due to an added remark

We solve the inverse differential Galois problem over the fraction field of $k[[t,x]]$ and use this to solve split differential embedding problems over $k((t))(x)$ that are induced from $k(x)$. The proofs use patching as well as prior results on inverse problems and embedding problems., Comment: 13 pages; simplified Sections 1-3 (resulting in a renumbering of the statements in Section 1-2), added Proposition 1.4, added Remark 3.4, streamlined proof of Theorem 4.2, added Remark 4.4

We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching techniques in differential Galois theory over fields of characteristic zero. We show that patching holds over function fields over the complex numbers. As the main application, we prove the solvability of all differential embedding problems over complex function fields, thereby providing new insight on the structure of the absolute differential Galois group, i.e., the fundamental group of the underlying Tannakian category., Comment: 42 pages. Added Rem. 1.13, added Thm. 2.14 and simplified proof of Prop. 3.5 using Thm. 2.14, added Ex. 3.8

In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic group (i.e. affine group scheme of finite type) over such a Laurent series field does occur as the differential Galois group of a linear differential equation with coefficients in any such function field (of one or several variables)., Comment: 25 pages

We introduce a notion of refinements in the context of patching, in order to obtain new results about local-global principles and field invariants in the context of quadratic forms and central simple algebras. The fields we consider are finite extensions of the fraction fields of two-dimensional complete domains that need not be local. Our results in particular give the u-invariant and period-index bound for these fields, as consequences of more general abstract results., Comment: 44 pages. We renumbered results to be consistent with the numbering in the published version. We added a needed hypothesis to Corollary 4.13 that was missing from the published version

This paper proves local-global principles for Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for $H^n(F, Z/mZ(n-1))$, for all $n>1$. This is motivated by work of Kato and others, where such principles were shown in related cases for $n=3$. Using our results in combination with cohomological invariants, we obtain local-global principles for torsors and related algebraic structures over $F$. Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture., Comment: 32 pages. Some changes of notation. Statement of Lemma 2.4.4 corrected. Lemma 3.3.2 strengthened and made a proposition. Some proofs modified to fix or clarify specific points or to streamline the presentation

We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period-index problem for division algebras, over fraction fields of complete two-dimensional rings. Our approach uses patching methods and matrix factorization results that can be viewed as analogs of Cartan's Lemma., Comment: 22 pages; more detail given in some arguments

We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for groups with rational components, we compute it explicitly and show that it is finite. This yields necessary and sufficient conditions for local-global principles to hold. Our results rely on first obtaining a Mayer-Vietoris sequence for Galois cohomology and then showing that torsors can be patched. We also give new applications to quadratic forms and central simple algebras., Comment: 50 pages. Clarified some explanations, updated references, fixed typographical errors

Given a field F, one may ask which finite groups are Galois groups of field extensions E/F such that E is a maximal subfield of a division algebra with center F. This question was originally posed by Schacher, who gave partial results over the field of rational numbers. Using patching, we give a complete characterization of such groups in the case that F is the function field of a curve over a complete discretely valued field with algebraically closed residue field of characteristic zero, as well as results in related cases., Comment: 20 pages. In Section 3 some statements were strengthened and proofs simplified. At the end of Section 4 the definition of the rank two valuation was fixed