2024
http://publications.mfo.de/handle/mfo/4095
Wed, 11 Sep 2024 20:42:33 GMT2024-09-11T20:42:33ZThe 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.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.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.Ky Fan Theorem for Sphere Bundles
http://publications.mfo.de/handle/mfo/4131
Ky Fan Theorem for Sphere Bundles
Panina, Gaiane; Živaljević, Rade
The classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker’s lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of a triangulated sphere Sn. Here we describe generalizations of Ky Fan theorem for the case when the sphere is replaced by the total space of a triangulated sphere bundle.
Fri, 05 Apr 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41312024-04-05T00:00:00ZPanina, GaianeŽivaljević, RadeThe classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker’s lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of a triangulated sphere Sn. Here we describe generalizations of Ky Fan theorem for the case when the sphere is replaced by the total space of a triangulated sphere bundle.A Gentle Introduction to Interpolation on the Grassmann Manifold
http://publications.mfo.de/handle/mfo/4097
A Gentle Introduction to Interpolation on the Grassmann Manifold
Ciaramella, Gabriele; Gander, Martin J.; Vanzan, Tommaso
Wed, 10 Jan 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40972024-01-10T00:00:00ZCiaramella, GabrieleGander, Martin J.Vanzan, TommasoArm Exponent for the Gaussian Free Field on Metric Graphs in Intermediate Dimensions
http://publications.mfo.de/handle/mfo/4096
Arm Exponent for the Gaussian Free Field on Metric Graphs in Intermediate Dimensions
Drewitz, Alexander; Prévost, Alexis; Rodriguez, Pierre-François
We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $\alpha$ and that the Green's function for the random walk on ${G}$ exhibits a power law decay with exponent $\nu$, in the regime $1\leq \nu \leq \frac{\alpha}{2}$. In particular, this includes the cases of ${G}=\mathbb Z^3$, for which $\nu=1$, and ${G}= \mathbb Z^4$, for which $\nu=\frac{\alpha}{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$ like $R^{-\frac{\nu}{2}+o(1)}$. Our results are in fact more precise and yield logarithmic corrections when $\nu > 1$ as well as corrections of order $\log \log R$ when $\nu=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $\nu > 1$ and essentially optimal when $\nu=1$. This extends previous results from [16].
Mon, 08 Jan 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40962024-01-08T00:00:00ZDrewitz, AlexanderPrévost, AlexisRodriguez, Pierre-FrançoisWe investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $\alpha$ and that the Green's function for the random walk on ${G}$ exhibits a power law decay with exponent $\nu$, in the regime $1\leq \nu \leq \frac{\alpha}{2}$. In particular, this includes the cases of ${G}=\mathbb Z^3$, for which $\nu=1$, and ${G}= \mathbb Z^4$, for which $\nu=\frac{\alpha}{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$ like $R^{-\frac{\nu}{2}+o(1)}$. Our results are in fact more precise and yield logarithmic corrections when $\nu > 1$ as well as corrections of order $\log \log R$ when $\nu=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $\nu > 1$ and essentially optimal when $\nu=1$. This extends previous results from [16].