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.Sun, 10 Dec 2023 16:19:39 GMT2023-12-10T16:19:39ZA 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.The Simplicial Complex of Brauer Pairs of a Finite Reductive Group
http://publications.mfo.de/handle/mfo/4056
The Simplicial Complex of Brauer Pairs of a Finite Reductive Group
Rossi, Damiano
In this paper we study the simplicial complex induced by the poset of Brauer pairs ordered by inclusion for the family of finite reductive groups. In the defining characteristic case the homotopy type of this simplicial complex coincides with that of the Tits building thanks to a well-known result of Quillen. On the other hand, in the non-defining characteristic case, we show that the simplicial complex of Brauer pairs is homotopy equivalen to a simplicial complex determined by generalised Harish-Chandra theory. This extends earlier results of the author on the Brown complex and makes use of the theory of connected subpairs and twisted block induction developed by Cabanes and Enguehard.
Mon, 17 Jul 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40562023-07-17T00:00:00ZRossi, DamianoIn this paper we study the simplicial complex induced by the poset of Brauer pairs ordered by inclusion for the family of finite reductive groups. In the defining characteristic case the homotopy type of this simplicial complex coincides with that of the Tits building thanks to a well-known result of Quillen. On the other hand, in the non-defining characteristic case, we show that the simplicial complex of Brauer pairs is homotopy equivalen to a simplicial complex determined by generalised Harish-Chandra theory. This extends earlier results of the author on the Brown complex and makes use of the theory of connected subpairs and twisted block induction developed by Cabanes and Enguehard.Rank Deviations for Overpartitions
http://publications.mfo.de/handle/mfo/4054
Rank Deviations for Overpartitions
Lovejoy, Jeremy; Osburn, Robert
We prove general fomulas for the deviations of two overpartition ranks from the average, namely \begin{equation*} \overline{D}(a, M) := \sum_{n \geq 0} \Bigl( \overline{N}(a, M, n) - \frac{\overline{p}(n)}{M} \Bigr) q^n \end{equation*} and \begin{equation*} \overline{D}_{2}(a,M) := \sum_{n \geq 0} \Bigl( \overline{N}_{2}(a, M, n) - \frac{\overline{p}(n)}{M} \Bigr) q^n \end{equation*} where $\overline{N}(a, M, n)$ denotes the number of overpartitions of $n$ with rank congruent to $a$ modulo $M$, $\overline{N}_{2}(a, M, n)$ is the number of overpartitions of $n$ with $M_2$-rank congruent to $a$ modulo $M$ and $\overline{p}(n)$ is the number of overpartitions of $n$. These formulas are in terms of Appell-Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the numerous overpartition rank difference identities in the literature. We give examples for $M=3$ and $6$.
Wed, 12 Jul 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40542023-07-12T00:00:00ZLovejoy, JeremyOsburn, RobertWe prove general fomulas for the deviations of two overpartition ranks from the average, namely \begin{equation*} \overline{D}(a, M) := \sum_{n \geq 0} \Bigl( \overline{N}(a, M, n) - \frac{\overline{p}(n)}{M} \Bigr) q^n \end{equation*} and \begin{equation*} \overline{D}_{2}(a,M) := \sum_{n \geq 0} \Bigl( \overline{N}_{2}(a, M, n) - \frac{\overline{p}(n)}{M} \Bigr) q^n \end{equation*} where $\overline{N}(a, M, n)$ denotes the number of overpartitions of $n$ with rank congruent to $a$ modulo $M$, $\overline{N}_{2}(a, M, n)$ is the number of overpartitions of $n$ with $M_2$-rank congruent to $a$ modulo $M$ and $\overline{p}(n)$ is the number of overpartitions of $n$. These formulas are in terms of Appell-Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the numerous overpartition rank difference identities in the literature. We give examples for $M=3$ and $6$.Hypergroups and Twin Buildings, I
http://publications.mfo.de/handle/mfo/4048
Hypergroups and Twin Buildings, I
French, Christopher; Zieschang, Paul-Hermann
We discuss a conjecture on thick twin buildings the verification of which is needed in order to show that thick twin buildings are mathematically equivalent to regular actions of certain twin Coxeter hypergroups. (A corresponding result for buildings is shown in [5; Sections 10.2, 10.3].) We prove that the conjecture holds in the case where the support of its sagittal has cardinality 2 and in the case where its sagittal has length at most 3. (Sagittals are defined in Section 1.) Our exposition is based on an earlier treatment of the subject; cf. [3].
Mon, 19 Jun 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40482023-06-19T00:00:00ZFrench, ChristopherZieschang, Paul-HermannWe discuss a conjecture on thick twin buildings the verification of which is needed in order to show that thick twin buildings are mathematically equivalent to regular actions of certain twin Coxeter hypergroups. (A corresponding result for buildings is shown in [5; Sections 10.2, 10.3].) We prove that the conjecture holds in the case where the support of its sagittal has cardinality 2 and in the case where its sagittal has length at most 3. (Sagittals are defined in Section 1.) Our exposition is based on an earlier treatment of the subject; cf. [3].Logical Relations for Partial Features and Automatic Differentiation Correctness
http://publications.mfo.de/handle/mfo/4046
Logical Relations for Partial Features and Automatic Differentiation Correctness
Lucatelli Nunes, Fernando; Vákár, Matthijs
We present a simple technique for semantic, open logical relations arguments about languages with recursive types, which, as we show, follows from a principled foundation in categorical semantics. We demonstrate how it can be used to give a very straightforward proof of correctness of practical forward- and reverse-mode dual numbers style automatic differentiation (AD) on ML-family languages. The key idea is to combine it with a suitable open logical relations technique for reasoning about differentiable partial functions (a suitable lifting of the partiality monad to logical relations), which we introduce.
Mon, 19 Jun 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40462023-06-19T00:00:00ZLucatelli Nunes, FernandoVákár, MatthijsWe present a simple technique for semantic, open logical relations arguments about languages with recursive types, which, as we show, follows from a principled foundation in categorical semantics. We demonstrate how it can be used to give a very straightforward proof of correctness of practical forward- and reverse-mode dual numbers style automatic differentiation (AD) on ML-family languages. The key idea is to combine it with a suitable open logical relations technique for reasoning about differentiable partial functions (a suitable lifting of the partiality monad to logical relations), which we introduce.Lax Comma Categories of Ordered Sets
http://publications.mfo.de/handle/mfo/4045
Lax Comma Categories of Ordered Sets
Clementino, Maria Manuel; Lucatelli Nunes, Fernando
Let $\mathsf{Ord} $ be the category of (pre)ordered sets. Unlike $\mathsf{Ord}/X$, whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category $\mathsf{Ord} //X$. In this paper we show that the forgetful functor $\mathsf{Ord} //X\to \mathsf{Ord} $ is topological if and only if $X$ is complete. Moreover, under suitable hypothesis, $\mathsf{Ord} // X$ is complete and cartesian closed if and only if $X$ is. We end by analysing descent in this category. Namely, when $X$ is complete and cartesian closed, we show that, for a morphism in $\mathsf{Ord} //X$, being pointwise effective for descent in $\mathsf{Ord} $ is sufficient, while being effective for descent in $\mathsf{Ord} $ is necessary, to be effective for descent in $\mathsf{Ord} //X$.
Mon, 19 Jun 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40452023-06-19T00:00:00ZClementino, Maria ManuelLucatelli Nunes, FernandoLet $\mathsf{Ord} $ be the category of (pre)ordered sets. Unlike $\mathsf{Ord}/X$, whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category $\mathsf{Ord} //X$. In this paper we show that the forgetful functor $\mathsf{Ord} //X\to \mathsf{Ord} $ is topological if and only if $X$ is complete. Moreover, under suitable hypothesis, $\mathsf{Ord} // X$ is complete and cartesian closed if and only if $X$ is. We end by analysing descent in this category. Namely, when $X$ is complete and cartesian closed, we show that, for a morphism in $\mathsf{Ord} //X$, being pointwise effective for descent in $\mathsf{Ord} $ is sufficient, while being effective for descent in $\mathsf{Ord} $ is necessary, to be effective for descent in $\mathsf{Ord} //X$.