1 - Oberwolfach Preprints (OWP)
http://publications.mfo.de/handle/mfo/19
The Oberwolfach Preprints (OWP) mainly contain research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs program and the Oberwolfach Leibniz Fellows, but this can also include an Oberwolfach Lecture, for example.Tue, 28 May 2024 18:22:44 GMT2024-05-28T18:22:44ZOn 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].Ground State of Bose Gases Interacting through Singular Potentials
http://publications.mfo.de/handle/mfo/4087
Ground State of Bose Gases Interacting through Singular Potentials
Boßmann, Lea; Leopold, Nikolai; Petrat, Sören; Rademacher, Simone
We consider a system of $N$ bosons on the three-dimensional unit torus. The particles interact through repulsive pair interactions of the form $N^{3β-1} v (N^βx)$ for $β\in (0,1)$. We prove the next order correction to Bogoliubov theory for the ground state and the ground state energy.
Mon, 27 Nov 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40872023-11-27T00:00:00ZBoßmann, LeaLeopold, NikolaiPetrat, SörenRademacher, SimoneWe consider a system of $N$ bosons on the three-dimensional unit torus. The particles interact through repulsive pair interactions of the form $N^{3β-1} v (N^βx)$ for $β\in (0,1)$. We prove the next order correction to Bogoliubov theory for the ground state and the ground state energy.A Note on Endpoint Bochner-Riesz Estimates
http://publications.mfo.de/handle/mfo/4086
A Note on Endpoint Bochner-Riesz Estimates
Beltran, David; Roos, Joris; Seeger, Andreas
We revisit an $\varepsilon$-removal argument of Tao to obtain sharp $L^p \to L^r(L^p)$ estimates
for sums of Bochner-Riesz bumps which are conditional on non-endpoint bounds for single scale bumps. These can be used to obtain sharp conditional sparse bounds for Bochner-Riesz multipliers at the critical index, refining the conditional weak-type $(p,p)$ estimates of Tao.
Mon, 27 Nov 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40862023-11-27T00:00:00ZBeltran, DavidRoos, JorisSeeger, AndreasWe revisit an $\varepsilon$-removal argument of Tao to obtain sharp $L^p \to L^r(L^p)$ estimates
for sums of Bochner-Riesz bumps which are conditional on non-endpoint bounds for single scale bumps. These can be used to obtain sharp conditional sparse bounds for Bochner-Riesz multipliers at the critical index, refining the conditional weak-type $(p,p)$ estimates of Tao.Bochner-Riesz Means at the Critical Index: Weighted and Sparse Bounds
http://publications.mfo.de/handle/mfo/4085
Bochner-Riesz Means at the Critical Index: Weighted and Sparse Bounds
Beltran, David; Roos, Joris; Seeger, Andreas
We consider Bochner-Riesz means on weighted $L^p$ spaces, at the critical index $\lambda(p)=d(\frac 1p-\frac 12)-\frac 12$. For every $A_1$-weight we obtain an extension of Vargas' weak type $(1,1)$ inequality in some range of $p>1$. To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension $d= 2$; partial results as well as conditional results are proved in higher dimensions. For the means of index $\lambda_*= \frac{d-1}{2d+2}$ we prove fully optimal sparse bounds.
Mon, 27 Nov 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40852023-11-27T00:00:00ZBeltran, DavidRoos, JorisSeeger, AndreasWe consider Bochner-Riesz means on weighted $L^p$ spaces, at the critical index $\lambda(p)=d(\frac 1p-\frac 12)-\frac 12$. For every $A_1$-weight we obtain an extension of Vargas' weak type $(1,1)$ inequality in some range of $p>1$. To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension $d= 2$; partial results as well as conditional results are proved in higher dimensions. For the means of index $\lambda_*= \frac{d-1}{2d+2}$ we prove fully optimal sparse bounds.The Character Triple Conjecture for Maximal Defect Characters and the Prime 2
http://publications.mfo.de/handle/mfo/4060
The Character Triple Conjecture for Maximal Defect Characters and the Prime 2
Rossi, Damiano
We prove that Späth’s Character Triple Conjecture holds for every finite group with respect to maximal defect characters at the prime 2. This is done by reducing the maximal defect case of the conjecture to the so-called inductive Alperin–McKay condition whose verification has recently been completed by Ruhstorfer for the prime 2. As a consequence we obtain the Character Triple Conjecture for all 2-blocks with abelian defect groups by applying Brauer’s Height Zero Conjecture, a proof of which is now available. We also obtain similar results for the block-free version of the Character Triple Conjecture at the prime 3.
Tue, 22 Aug 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40602023-08-22T00:00:00ZRossi, DamianoWe prove that Späth’s Character Triple Conjecture holds for every finite group with respect to maximal defect characters at the prime 2. This is done by reducing the maximal defect case of the conjecture to the so-called inductive Alperin–McKay condition whose verification has recently been completed by Ruhstorfer for the prime 2. As a consequence we obtain the Character Triple Conjecture for all 2-blocks with abelian defect groups by applying Brauer’s Height Zero Conjecture, a proof of which is now available. We also obtain similar results for the block-free version of the Character Triple Conjecture at the prime 3.The Brown Complex in Non-Defining Characteristic and Applications
http://publications.mfo.de/handle/mfo/4059
The Brown Complex in Non-Defining Characteristic and Applications
Rossi, Damiano
We study the Brown complex associated to the poset of $\ell$-subgroups in the case of a finite reductive group defined over a field $\mathbb{F}_q$ of characteristic prime to $\ell$. First, under suitable hypotheses, we show that its homotopy type is determined by the generic Sylow theory developed by Broué and Malle and, in particular, only depends on the multiplicative order of $q$ modulo $\ell$. This result leads to several interesting applications to generic Sylow theory, mod $\ell$ homology decompositions, and $\ell$-modular representation theory. Then, we conduct a more detailed study of the Brown complex in order to establish an explicit connection between the local-global conjectures in representation theory of finite groups and the generic Sylow theory. This is done by isolating a family of $\ell$-subgroups of finite reductive groups that corresponds bijectively to the structures controlled by the generic Sylow theory.
Tue, 25 Jul 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40592023-07-25T00:00:00ZRossi, DamianoWe study the Brown complex associated to the poset of $\ell$-subgroups in the case of a finite reductive group defined over a field $\mathbb{F}_q$ of characteristic prime to $\ell$. First, under suitable hypotheses, we show that its homotopy type is determined by the generic Sylow theory developed by Broué and Malle and, in particular, only depends on the multiplicative order of $q$ modulo $\ell$. This result leads to several interesting applications to generic Sylow theory, mod $\ell$ homology decompositions, and $\ell$-modular representation theory. Then, we conduct a more detailed study of the Brown complex in order to establish an explicit connection between the local-global conjectures in representation theory of finite groups and the generic Sylow theory. This is done by isolating a family of $\ell$-subgroups of finite reductive groups that corresponds bijectively to the structures controlled by the generic Sylow theory.Multi-Dimensional Summation-by-Parts Operators for General Function Spaces: Theory and Construction
http://publications.mfo.de/handle/mfo/4058
Multi-Dimensional Summation-by-Parts Operators for General Function Spaces: Theory and Construction
Glaubitz, Jan; Klein, Simon-Christian; Nordström, Jan; Öffner, Philipp
Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.
Tue, 25 Jul 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40582023-07-25T00:00:00ZGlaubitz, JanKlein, Simon-ChristianNordström, JanÖffner, PhilippSummation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.