Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 20 Sep 2024 23:19:02 GMT2024-09-20T23:19:02ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
Voronoi Cells: Or How to Find the Nearest Bakery
http://publications.mfo.de/handle/mfo/4163
Voronoi Cells: Or How to Find the Nearest Bakery
Hess, Sarah; Torres, Angélica; van der Eyden, Mirte
Deciding which mall, hospital or school is closest to us is a problem we face everyday. It even comes on holidays with us, when we optimize our plans to make sure that we have enough time to visit all the attractions we want to see. In this article, we show how concepts from metric algebraic geometry help us to rise to this task while planning a weekend trip to the Black Forest.
Thu, 05 Sep 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41632024-09-05T00:00:00ZHess, SarahTorres, Angélicavan der Eyden, MirteDeciding which mall, hospital or school is closest to us is a problem we face everyday. It even comes on holidays with us, when we optimize our plans to make sure that we have enough time to visit all the attractions we want to see. In this article, we show how concepts from metric algebraic geometry help us to rise to this task while planning a weekend trip to the Black Forest.The Subgroup Structure of Pseudo-Reductive Groups
http://publications.mfo.de/handle/mfo/4160
The Subgroup Structure of Pseudo-Reductive Groups
Bate, Michael; Martin, Benjamin; Röhrle, Gerhard; Sercombe, Damian
Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.
Wed, 31 Jul 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41602024-07-31T00:00:00ZBate, MichaelMartin, BenjaminRöhrle, GerhardSercombe, DamianLet $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.The Alternating Halpern-Mann Iteration for Families of Maps
http://publications.mfo.de/handle/mfo/4157
The Alternating Halpern-Mann Iteration for Families of Maps
Firmino, Paulo; Pinto, Pedro
We generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature.
Mon, 15 Jul 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41572024-07-15T00:00:00ZFirmino, PauloPinto, PedroWe generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature.Proof Mining and the Convex Feasibility Problem : the Curious Case of Dykstra's Algorithm
http://publications.mfo.de/handle/mfo/4156
Proof Mining and the Convex Feasibility Problem : the Curious Case of Dykstra's Algorithm
Pinto, Pedro
In a recent proof mining application, the proof-theoretical analysis of Dykstra's cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector’s bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining.
Mon, 15 Jul 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41562024-07-15T00:00:00ZPinto, PedroIn a recent proof mining application, the proof-theoretical analysis of Dykstra's cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector’s bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining.Free Boundary Problems in Fluid Dynamics
http://publications.mfo.de/handle/mfo/4155
Free Boundary Problems in Fluid Dynamics
Ai, Albert; Alazard, Thomas; Ifrim, Mihaela; Tataru, Daniel
This book, originating from a seminar held at Oberwolfach in 2022, introduces to state-of-the-art methods and results in the study of free boundary problems which are arising from compressible as well as from incompressible Euler’s equations in general. A particular set of such problems is given by gaseous stars considered in a vacuum (modeled via the compressible Euler equations) as well as water waves in their full generality (seen as recasts of incompressible irrotational Euler equations). This is a broad research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade.
Oberwolfach Seminar 2243a: Free Boundary Problems in Fluid Dynamics
Thu, 20 Jun 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41552024-06-20T00:00:00ZAi, AlbertAlazard, ThomasIfrim, MihaelaTataru, DanielThis book, originating from a seminar held at Oberwolfach in 2022, introduces to state-of-the-art methods and results in the study of free boundary problems which are arising from compressible as well as from incompressible Euler’s equations in general. A particular set of such problems is given by gaseous stars considered in a vacuum (modeled via the compressible Euler equations) as well as water waves in their full generality (seen as recasts of incompressible irrotational Euler equations). This is a broad research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade.On Overgroups of Distinguished Unipotent Elements in Reductive Groups and Finite Groups of Lie Type
http://publications.mfo.de/handle/mfo/4153
On Overgroups of Distinguished Unipotent Elements in Reductive Groups and Finite Groups of Lie Type
Bate, Michael; Böhm, Sören; Martin, Benjamin; Röhrle, Gerhard
Suppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good A1 subgroups of G, introduced by Seitz. We also formulate a counterpart of Korhonen’s theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen’s theorem for Lie algebras.
Tue, 18 Jun 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41532024-06-18T00:00:00ZBate, MichaelBöhm, SörenMartin, BenjaminRöhrle, GerhardSuppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good A1 subgroups of G, introduced by Seitz. We also formulate a counterpart of Korhonen’s theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen’s theorem for Lie algebras.Geometric, Algebraic, and Topological Combinatorics
http://publications.mfo.de/handle/mfo/4147
Geometric, Algebraic, and Topological Combinatorics
The 2023 Oberwolfach meeting "Geometric, Algebraic, and Topological
Combinatorics''
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference were (1) Federico Ardila and Tom Braden
discussed recent exciting developments in the intersection theory of matroids;
(2) Stavros Papadakis and Vasiliki Petrotou presented their proof of the
Lefschetz property for spheres, and, more generally, for pseudomanifolds and
cycles (this second part is joint with Karim Adiprasito); (3) Gaku Liu reported
on his joint work with Spencer Backman that establishes the existence of a
regular unimodular triangulation of an arbitrary matroid base polytope.
Sun, 01 Jan 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41472023-01-01T00:00:00ZThe 2023 Oberwolfach meeting "Geometric, Algebraic, and Topological
Combinatorics''
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference were (1) Federico Ardila and Tom Braden
discussed recent exciting developments in the intersection theory of matroids;
(2) Stavros Papadakis and Vasiliki Petrotou presented their proof of the
Lefschetz property for spheres, and, more generally, for pseudomanifolds and
cycles (this second part is joint with Karim Adiprasito); (3) Gaku Liu reported
on his joint work with Spencer Backman that establishes the existence of a
regular unimodular triangulation of an arbitrary matroid base polytope.Randomness is Natural - an Introduction to Regularisation by Noise
http://publications.mfo.de/handle/mfo/4146
Randomness is Natural - an Introduction to Regularisation by Noise
Djurdjevac, Ana; Elad Altman, Henri; Rosati, Tommaso
Differential equations make predictions on the future state of a system given the present. In order to get a sensible prediction, sometimes it is necessary to include randomness in differential equations, taking microscopic effects into account. Surprisingly, despite the presence of randomness, our probabilistic prediction of future states is stable with respect to changes in the surrounding environment, even if the original prediction was unstable. This snapshot will unveil the core mathematical mechanism underlying this "regularisation by noise" phenomenon.
Wed, 22 May 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41462024-05-22T00:00:00ZDjurdjevac, AnaElad Altman, HenriRosati, TommasoDifferential equations make predictions on the future state of a system given the present. In order to get a sensible prediction, sometimes it is necessary to include randomness in differential equations, taking microscopic effects into account. Surprisingly, despite the presence of randomness, our probabilistic prediction of future states is stable with respect to changes in the surrounding environment, even if the original prediction was unstable. This snapshot will unveil the core mathematical mechanism underlying this "regularisation by noise" phenomenon.Mini-Workshop: Homological Aspects for TDLC-Groups
http://publications.mfo.de/handle/mfo/4137
Mini-Workshop: Homological Aspects for TDLC-Groups
This mini-workshop aimed at bringing together experts and early career researchers on finiteness conditions for discrete groups, and experts on varying aspects of locally compact groups to find a common framework to develop a systematic theory of homological finiteness conditions for totally disconnected locally compact groups. Whereas the homological theory of
finiteness conditions of discrete groups is well developed and the structure theory of totally disconnected locally compact
groups has seen some important breakthroughs in the last decade, the homological theory for (non-compact) totally disconnected locally compact groups is an emerging research area. Specific
topics include finiteness conditions for locally compact groups, Mackey functors
and Bredon cohomology for topological groups, connections to condensed mathematics, connections to $\ell^2$-invariants and $\Sigma$-invariants.
Sun, 01 Jan 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41372023-01-01T00:00:00ZThis mini-workshop aimed at bringing together experts and early career researchers on finiteness conditions for discrete groups, and experts on varying aspects of locally compact groups to find a common framework to develop a systematic theory of homological finiteness conditions for totally disconnected locally compact groups. Whereas the homological theory of
finiteness conditions of discrete groups is well developed and the structure theory of totally disconnected locally compact
groups has seen some important breakthroughs in the last decade, the homological theory for (non-compact) totally disconnected locally compact groups is an emerging research area. Specific
topics include finiteness conditions for locally compact groups, Mackey functors
and Bredon cohomology for topological groups, connections to condensed mathematics, connections to $\ell^2$-invariants and $\Sigma$-invariants.On Dykstra’s Algorithm with Bregman Projections
http://publications.mfo.de/handle/mfo/4134
On Dykstra’s Algorithm with Bregman Projections
Pinto, Pedro; Pischke, Nicholas
We provide quantitative results on the asymptotic behavior of Dykstra’s algorithm with Bregman projections, a combination of the well-known Dykstra’s algorithm and the method of cyclic Bregman projections, designed to find best approximations and solve the convex feasibility problem in a non-Hilbertian setting. The result we provide arise through the lens of proof mining, a program in mathematical logic which extracts computational information from non-effective proofs. Concretely, we provide a highly uniform and computable rate of metastability of low complexity and, moreover, we also specify general circumstances in which one can obtain full and effective rates of convergence. As a byproduct of our quantitative analysis, we also for the first time establish the strong convergence of Dykstra’s method with Bregman projections in infinite dimensional (reflexive) Banach spaces.
Tue, 16 Apr 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41342024-04-16T00:00:00ZPinto, PedroPischke, NicholasWe provide quantitative results on the asymptotic behavior of Dykstra’s algorithm with Bregman projections, a combination of the well-known Dykstra’s algorithm and the method of cyclic Bregman projections, designed to find best approximations and solve the convex feasibility problem in a non-Hilbertian setting. The result we provide arise through the lens of proof mining, a program in mathematical logic which extracts computational information from non-effective proofs. Concretely, we provide a highly uniform and computable rate of metastability of low complexity and, moreover, we also specify general circumstances in which one can obtain full and effective rates of convergence. As a byproduct of our quantitative analysis, we also for the first time establish the strong convergence of Dykstra’s method with Bregman projections in infinite dimensional (reflexive) Banach spaces.