Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Mathematical Logic: Proof Theory, Constructive Mathematics
No full textThe workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of core mathematics and theoretical computer science as well as homotopy type theory and logical aspects of computational complexity
Counting Curves on Toric Surfaces Tropical Geometry & the Fock Space
No full textResearch in Pairs 2015We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams, and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are shown to coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttche, Cooper–Pandharipande, and Block–Gathmann–Markwig
Inductive Freeness of Ziegler’s Canonical Multiderivations for Reflection Arrangements
No full textMSC: 20F55; 51F15; 52C35; 14N20; 32S22; 51D20Research in Pairs 2017Let be a free hyperplane arrangement. In 1989, Ziegler showed that the
restriction of to any hyperplane endowed with the natural
multiplicity is then a free multiarrangement. We initiate a study of the
stronger freeness property of inductive freeness for these canonical free
multiarrangements and investigate them for the underlying class of reflection
arrangements.
More precisely, let be the reflection arrangement of a complex
reflection group . By work of Terao, each such reflection arrangement is
free. Thus so is Ziegler's canonical multiplicity on the restriction of
to a hyperplane. We show that the latter is inductively free as a
multiarrangement if and only if itself is inductively free
Geometric Structures in Group Theory
No full textGeometric group theory has natural connections and rich interfaces with many of the other major fields of modern mathematics. The basic motif of the field is the construction and exploration of actions by infinite groups on spaces that admit further structure, with an emphasis on geometric structures of different sorts: one usually seeks actions in order to illuminate the structure of groups of particular interest, but one also explores actions in order to understand the underlying spaces. The dramatic growth of the field in the late twentieth century was closely associated with the study of generalized forms of non-positive and negative curvature, and classically the spaces at hand were cell complexes with some additional structure. But the scope of the field, the range of groups embraced by its techniques, and the nature of the spaces studied, have expanded enormously in recent years, and they continue to do so. This meeting provided an exciting snapshot of some of the main strands in the recent development of the subject
Partial Differential Equations
No full textThe workshop dealt with nonlinear partial differential equations and some applications in geometry, touching several different topics such as minimal surfaces and geometric measure theory, conformal geometry, geometric flows, metric geometry and structure of Riemannian manifolds
Algebraic Geometry: Birational Classification, Derived Categories, and Moduli Spaces
No full textThe workshop covered a number of active areas of research in algebraic geometry with a focus on derived categories, moduli spaces (of varieties and sheaves) and birational geometry (often in positive characteristic) and their interactions. Special emphasis was put on hyperkähler manifolds and singularity theory
Algebraic Statistics
No full textAlgebraic Statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in statistics and data science. This workshop was the first at Oberwolfach dedicated to this emerging subject area. The participants highlighted recent achievements in this field, explored exciting new applications, and mapped out future directions for research
Algebraic Groups
No full textLinear algebraic groups is an active research area in contemporary mathematics. It has rich connections to algebraic geometry, representation theory, algebraic combinatorics, number theory, algebraic topology, and differential equations. The foundations of this theory were laid by A. Borel, C. Chevalley, J.-P. Serre, T. A. Springer and J. Tits in the second half of the 20th century. The Oberwolfach workshops on algebraic groups, led by Springer and Tits, played an important role in this effort as a forum for researchers, meeting at approximately 3 year intervals since the 1960s. The present workshop continued this tradition, covering a range of topics, with an emphasis on recent developments in the subject
Mathematische Modellierung von Krebswachstum
No full textKrebs ist eine der größten Herausforderungen der modernen
Medizin. Der WHO zufolge starben 2012 weltweit
8,2 Millionen Menschen an Krebs. Bis heute sind
dessen molekulare Mechanismen nur in Teilen verstanden,
was eine erfolgreiche Behandlung erschwert.
Mathematische Modellierung und Computersimulationen
können helfen, die Mechanismen des Tumorwachstums
besser zu verstehen. Sie eröffnen somit
neue Chancen für zukünftige Behandlungsmethoden.
In diesem Schnappschuss steht die mathematische
Modellierung von Glioblastomen im Fokus, einer Klasse
sehr agressiver Tumore im menschlichen Gehirn
Matrix Elements of Irreducible Representations of SU(n+1) x SU(n+1) and Multivariable Matrix-Valued Orthogonal Polynomials
No full textResearch in Pairs 2017In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary
rank under a suitable multiplicity freeness assumption and additional conditions
on the branching rules. The spherical functions are taking values in the spaces of linear
operators of a finite dimensional representation of the subgroup, so the spherical functions
are matrix-valued. Under these assumptions these functions can be described in terms of
matrix-valued orthogonal polynomials in several variables, where the number of variables is
the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined
as simultaneous eigenfunctions of a commutative algebra of differential operators.
In Part 2 we verify that the group case SU(+ 1) meets all the conditions that we impose
in Part 1. For any we obtain families of orthogonal polynomials in n variables with
values in the -matrices, where
The case leads to the classical
Heckman-Opdam polynomials of type with geometric parameter. For we obtain
the most complete results. In this case we give an explicit expression of the matrix weight,
which we show to be irreducible whenever . We also give explicit expressions of the
spherical functions that determine the matrix weight for . These expressions are used
to calculate the spherical functions that determine the matrix weight for general up to
invertible upper-triangular matrices. This generalizes and gives a new proof of a formula
originally obtained by Koornwinder for the case . The commuting family of differential
operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains
an element of order one. We give explicit formulas for differential operators of order one
and two for equal to and