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The proalgebraic fundamental group of a connected topological space $X$, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces on $X$. In this article, we further develop the theory of the proalgebraic fundamental group, in particular, we establish homotopy invariance and a Seifert-van Kampen theorem. To facilitate the latter, we study amalgamated free product of affine group schemes. We also compute the proalgebraic fundamental group of the arithmetically relevant Kucharcyzk-Scholze spaces and compare it to the motivic Galois group., Comment: 41 pages

Dose escalation radiotherapy allows increased control of prostate cancer (PCa) but requires segmentation of dominant index lesions (DIL), motivating the development of automated methods for fast, accurate, and consistent segmentation of PCa DIL. We evaluated five deep-learning networks on apparent diffusion coefficient (ADC) MRI from 500 lesions in 365 patients arising from internal training Dataset 1 (1.5Tesla GE MR with endorectal coil), external ProstateX Dataset 2 (3Tesla Siemens MR), and internal inter-rater Dataset 3 (3Tesla Philips MR). The networks include: multiple resolution residually connected network (MRRN) and MRRN regularized in training with deep supervision (MRRN-DS), Unet, Unet++, ResUnet, and fast panoptic segmentation (FPSnet) as well as fast panoptic segmentation with smoothed labels (FPSnet-SL). Models were evaluated by volumetric DIL segmentation accuracy using Dice similarity coefficient (DSC) and detection accuracy, as a function of lesion aggressiveness, size, and location (Dataset 1 and 2), and accuracy with respect to two-raters (on Dataset 3). In general MRRN-DS more accurately segmented tumors than other methods on the testing datasets. MRRN-DS significantly outperformed ResUnet in Dataset2 (DSC of 0.54 vs. 0.44, p<0.001) and the Unet++ in Dataset3 (DSC of 0.45 vs. p=0.04). FPSnet-SL was similarly accurate as MRRN-DS in Dataset2 (p = 0.30), but MRRN-DS significantly outperformed FPSnet and FPSnet-SL in both Dataset1 (0.60 vs 0.51 [p=0.01] and 0.54 [p=0.049] respectively) and Dataset3 (0.45 vs 0.06 [p=0.002] and 0.24 [p=0.004] respectively). Finally, MRRN-DS produced slightly higher agreement with experienced radiologist than two radiologists in Dataset 3 (DSC of 0.45 vs. 0.41)., Comment: https://pubmed.ncbi.nlm.nih.gov/36856092/

This paper derives a free analog of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking fSDEs are stochastic differential equations in the context of non-commutative random variables (e.g. large random matrices). By applying the theory of multiple operator integrals we derive a free It\^{o} formula from Taylor expansion of operator valued functions. Iterating the free It\^{o} formula allows to motivate and define fEMM. Then we consider weak and strong convergence in the fSDE setting and prove strong convergence order of $\frac{1}{2}$ and weak convergence order of ${1}$. Numerical examples support the theoretical results and show solutions for equations where no analytical solution is known.

We determine the differential Galois group of the family of all regular singular differential equations on the Riemann sphere. It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$., Comment: 13 pages

We study families of linear differential equations parametrized by an algebraic variety $\mathcal{X}$ and show that the set of all points $x\in \mathcal{X}$, such that the differential Galois group at the generic fibre specializes to the differential Galois group at the fibre over $x$, is Zariski dense in $\mathcal{X}$. As an application, we prove Matzat's conjecture in full generality: The absolute differential Galois group of a one-variable function field over an algebraically closed field of characteristic zero is a free proalgebraic group., Comment: 88 pages, this is a major revision and reorganization of the first version, to appear in Memoirs of the AMS

We present an elementary proof of the fact that every torsor for an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of fibre functors on neutral tannakian categories., Comment: 6 pages

\'{E}tale difference algebraic groups are a difference analog of \'{e}tale algebraic groups. Our main result is a Jordan-H\"{o}lder type decomposition theorem for these groups. Roughly speaking, it shows that any \'{e}tale difference algebraic group can be build up from simple \'{e}tale algebraic groups and two finite \'{e}tale difference algebraic groups. The simple \'{e}tale algebraic groups occurring in this decomposition satisfy a certain uniqueness property., Comment: 45 pages

We develop a Galois theory for difference ring extensions, inspired by Magid's separable Galois theory for ring extensions and by Janelidze's categorical Galois theory. Our difference Galois theorem states that the category of difference ring extensions split by a chosen Galois difference ring extension is classified by actions of the associated difference profinite Galois groupoid. In particular, difference locally \'etale extensions of a difference ring are classified by its difference profinite fundamental groupoid. The emergence of difference profinite spaces, viewed as discrete dynamical systems in the realm of topological dynamics, leads us to investigate the interaction of difference algebra and symbolic dynamics. As an application of this interaction, we prove the near-rationality of a certain difference zeta function counting solutions of systems of difference algebraic equations over algebraic closures of finite fields with Frobenius.

We show that a closed finite index subgroup of a free proalgebraic group is itself a free proalgebraic group. Our main motivation for this result is an application in differential Galois theory: The absolute differential Galois group of a one-variable function field is a free proalgebraic group., Comment: 14 pages

We consider pairs of automorphisms $(\phi,\sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(\phi\colon x\mapsto x+h_1, \sigma\colon x\mapsto x+h_2)$, of $q$-difference operators $(\phi\colon x\mapsto q_1x,\ \sigma\colon x\mapsto q_2x)$, and of Mahler operators $(\phi\colon x\mapsto x^{p_1},\ \sigma\colon x\mapsto x^{p_2})$. Given a solution $f$ to a linear $\phi$-equation and a solution $g$ to a linear $\sigma$-equation, both transcendental, we show that $f$ and $g$ are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the $\sigma$-Galois theory of linear $\phi$-equations.

A profinite group equipped with an expansive endomorphism is equivalent to a one-sided group shift. We show that these groups have a very restricted structure. More precisely, we show that any such group can be decomposed into a finite sequence of full one-sided group shifts and two finite groups., Comment: 30 pages

Affine difference algebraic groups are a generalization of affine algebraic groups obtained by replacing algebraic equations with algebraic difference equations. We show that the isomorphism theorems from abstract group theory have meaningful analogs for these groups and we establish a Jordan-H\"{o}lder type theorem that allows us to decompose any affine difference algebraic group into almost-simple affine difference algebraic groups. We also characterize almost-simple affine difference algebraic groups via almost-simple affine algebraic groups., Comment: 49 pages. This submission contains the results of Sections 7, 8, 10, 11.2, 12 of arXiv:1405.6603v1, which has been replaced by arXiv:1405.6603v2

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

We introduce a notion of dimension for the solution set of a system of algebraic difference equations that measures the degrees of freedom when determining a solution in the ring of sequences. This number need not be an integer, but, as we show, it satisfies properties suitable for a notion of dimension. We also show that the dimension of a difference monomial is given by the covering density of its set of exponents., Comment: 28 pages, some corrections and more details added, mainly in Section 1

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$., Comment: 29 pages, new Section 6.2

$ $We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (e.g., standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assuption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable: $\bullet$ testing radical difference ideal membership or, equivalently, determining whether a given difference polynomial vanishes on the solution set of a given system of difference polynomials; $\bullet$ determining consistency of a system of difference equations in the ring of real-valued sequences; $\bullet$ determining consistency of a system of equations with action of $\mathbb{Z}^2$, $\mathbb{N}^2$, or the free monoid with two generators in the corresponding ring of sequences over any field of characteristic zero.

We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomatization is based on the tannakian philosophy. Through a tensor analog of skeletal categories we are able to consider neutral tannakian categories with a fibre functor as many-sorted first order structures. The class of diagonalizable proalgebraic groups is analyzed in detail. We show that the theory of a diagonalizable proalgebraic group $G$ is determined by the theory of the base field and the theory of the character group of $G$. Some initial steps towards a comprehensive study of types are also made., Comment: 39 pages

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

Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main motivation for this endeavor is a differential analog of a conjecture of Shafarevic., Comment: 36 pages, final accepted version

Let k be an algebraically closed field of characteristic 0, let X=P^1\times A^N and let f be a rational endomorphism of X given by (x,y)--->(g(x), A(x)y), where g is a rational function, while A is an N-by-N matrix with entries in k(x). We prove that if g is of the form x--->ax+b, then each irreducible curve C of X which intersects some orbit of f in infinitely many points must be periodic under the action of f. In the case g is an endomorphism of degree greater than 1, then we prove that each irreducible subvariety Y of X intersecting an orbit of f in a Zariski dense set of points must be periodic as well. Our results provide the desired conclusion in the Dynamical Mordell-Lang Conjecture in a couple new instances. Also, our results have interesting consequences towards a conjecture of Rubel and towards a generalized Skolem-Mahler-Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard-Vessiot extensions in the ring of sequences.