Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Rank Deviations for Overpartitions
No full textWe 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 denotes the number of overpartitions of with rank congruent to modulo , is the number of overpartitions of with -rank congruent to modulo and is the number of overpartitions of . 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 and .The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for their
support as this work began during their stay from January 9-22, 2022 as part of the Research
in Pairs program. The second author was partially funded by a SSHN 2022 grant from the
Embassy of France in Ireland during his visit to the Université Paris Cité from December 4-17,
2022. Finally, the authors are grateful to the Max-Planck-Institut für Mathematik for their
hospitality and support during their joint stay from May 1-31, 2023
Logical Relations for Partial Features and Automatic Differentiation Correctness
No full textWe 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.This project has received funding via NWO Veni grant number VI.Veni.202.124 as well as the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 895827.
This research was supported through the programme “Oberwolfach Leibniz Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2022. It was also partially supported by the CMUC, Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES
Recent Trends in Algebraic Geometry
No full textAlgebraic geometry has grown into a broad subject, with many different
streams often advancing quite independently of each
other. Nonetheless, important advances have often come from visionary
applications of ideas in one part of the subject to another. This
workshop brought together leaders and future leaders in
different areas of the subject, centered on geometric methods or
geometric problems.
It also brought together groups from different regions of the globe,
in order to bridge communities of different sorts, and help new ideas
quickly spread throughout algebraic geometry. Some of the best
freshly-minted algebraic geometers were deliberately invited, so
that they could meet their peers from around the world and learn about different perspectives on the subject
Arbeitsgemeinschaft: Cluster Algebras
No full textCluster algebras, invented by Sergey Fomin
and Andrei Zelevinsky around the year 2000, are commutative algebras
endowed with a rich combinatorial structure. Fomin-Zelevinsky's original motivations
came from Lie theory but in the past two decades, cluster algebras
have had strikingly fruitful interactions with a large array of other
subjects including Poisson geometry, discrete dynamical systems,
(higher) Teichmüller spaces, commutative and non-commutative
algebraic geometry, representation theory,.... In this Arbeitsgemeinschaft,
we have focused on 1) basic definitions and theorems, 2) cluster structures on
algebraic varieties and 3) the recent connection between cluster algebras and
symplectic topology, with its recent application to the construction of cluster
structures on braid varieties
A Note on Endpoint Bochner-Riesz Estimates
No full textWe revisit an -removal argument of Tao to obtain sharp 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 estimates of Tao.This research was supported through the program Oberwolfach Research Fellows by Mathematisches Forschungsinstitut Oberwolfach in 2023.
The authors were supported in part by National Science Foundation grants DMS-1954479 (D.B.), DMS-2154835 (J.R.), DMS-2054220 (A.S.), and by the AEI grants RYC2020-029151-I and PID2022-140977NA-I00 (D.B.)
Differentialgeometrie im Grossen
No full textOver the past several decades, classical differential geometry has undergone a remarkable expansion, helped by the integration of tools and insights from neighboring fields like partial differential equations, complex analysis, and geometric topology. In keeping with the spirit of previous gatherings, this meeting aimed to bridge the gaps between researchers working in seemingly disparate subfields of differential geometry, illuminating the connections that unite them.
Amongst other things, this meeting was centered around the theme of scalar curvature, which has recently emerged as a fundamental element across various fields, including differential geometry, metric geometry, topology, and complex geometry. This shared topic presented an ideal opportunity for scholars from these distinct areas to convene, discuss their individual progress, and foster a vibrant exchange of ideas
Mini-Workshop: Interpolation and Over-parameterization in Statistics and Machine Learning
No full textIn recent years it has become clear that, contrary to traditional statistical beliefs, methods that interpolate (fit exactly) the noisy training data, can still be statistically optimal. In particular, this phenomenon of "benign overfitting'' or "harmless interpolation'' seems to be close to the practical regimes of modern deep learning systems, and, arguably, underlies many of their behaviors. This workshop brought together experts on the emerging theory of interpolation in statistical methods, its theoretical foundations and applications to machine learning and deep learning
Algebraische Zahlentheorie
No full textAlgebraic Number Theory is an area of Mathematics that has a legendary history and lies at the interface of
Algebra and Number Theory. The last four decades of the last century witnessed rapid developments that led to connections with
other areas such as Algebraic Geometry, Representation Theory, Harmonic Analysis, Iwasawa theory, to mention a few.
In the last two decades, emergent areas such as -adic Analysis, -adic Geometry ( is a prime number) led
to additional new facets. More recent developments in Arithmetic Geometry via Perfectoid Spaces and other emerging areas
have added newer facets. The lectures in this workshop present current developments in these diverse areas
Mini-Workshop: Felix Klein's Foreign Students: Opening Up the Way for Transnational Mathematics
No full textExtending existing analyses of the topic, the workshop aimed to investigate the influence of Felix Klein on the development of mathematics (especially number theory, algebra, geometry, analysis, applications of mathematics in scientific and technical fields as well as in mathematics education) in countries other than Germany. The goal of the workshop was to take a look at mathematicians of foreign origin who studied with Klein that have received little attention so far (including Czech, Greek, Hungarian, Japanese, Polish, Russian, and Ukrainian mathematicians) and uncover how Klein guided them through his lectures and seminars. The protocols of the lectures held in Klein's seminars (from 1872 to 1912 in Göttingen, Erlangen, and Leipzig), which are a unique and so far largely unexplored source, were the basis for the workshop
Transport and Scale Interactions in Geophysical Flows
No full textThis interdisciplinary workshop brought together researchers working on different aspects of transport and scale interactions across the spectrum of geophysical fluid dynamics: geometry and computation of transport and exchange processes in geophysical flows, Lagrangian coherent structures, (geostrophic) turbulence, nonlinear waves and coherent structures in the Eulerian description of fluids, and stochastic methods in multiscale systems. Each of these topics have their own vibrant communities as well as well-established and emerging connections. This meeting aimed to bridge across the entire span of topics from a dynamical systems perspective, and to connect classical approaches with new developments in data-driven modeling and stochastic modeling