Oberwolfach Preprints (OWP)
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- Item2-Minimal subgroups in classical groups: Linear and unitary groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Parker, Chris; Rowley, Peter[no abstract available]
- ItemA 3-Local characterization of Co2(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Parker, Christopher; Rowley, PeterConway's second largest simple group, Co2, is characterized by the centralizer of an element of order 3 and certain fusion data.
- ItemA 3-local identification of the alternating group of degree 8, the McLaughlin simple group and their automorphism groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Parker, Christopher; Rowley, PeterIn this article we give 3-local characterizations of the alternating and symmetric groups of degree 8 and use these characterizations to recognize the sporadic simple group discovered by McLaughlin from its 3-local subgroups.
- ItemA Cheeger Type Inequality in Finite Cayley Sum Graphs(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Biswas, Arindam; Saha, Jyoti PrakashLet G be a finite group and S be a symmetric generating set of G with |S|=d. We show that if the undirected Cayley sum graph CΣ(G,S) is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from −1. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval (−1+h(G)4η,1−h(G)22d2], where h(G) denotes the (vertex) Cheeger constant of the d-regular graph CΣ(G,S) and η=29d8. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph C(G,S).
- ItemA Deformed Quon Algebra(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Randriamaro, HeryThe quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators a_(i,k), (i,k)∈N^∗ × [m], on an infinite dimensional vector space satisfying the deformed q-mutator relations aj,a^(\dag)_(i,k) = q^(\dag)_(i,k)aj,l + q^(β−k,l)δ_(i,j). We prove the realizability of our model by showing that, for suitable values of q, the vector space generated by the particle states obtained by applying combinations of ai,k's and a^(\dag)_(i,k)'s to a vacuum state |0⟩ is a Hilbert space. The proof particularly needs the investigation of the new statistic cinv and representations of the colored permutation group.
- ItemA Function Algebra Providing New Mergelyan Type Theorems in Several Complex Variables(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Falcó, Javier; Gauthier, Paul Montpetit; Manolaki, Myrto; Nestoridis, VassiliFor compact sets K⊂Cd, we introduce a subalgebra AD(K) of A(K), which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.
- ItemA Gentle Introduction to Interpolation on the Grassmann Manifold(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Ciaramella, Gabriele; Gander, Martin J.; Vanzan, Tommaso[no abstract available]
- ItemA McKay Correspondence for Reflection Groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Buchweitz, Ragnar-Olaf; Faber, Eleonore; Ingalls, ColinWe construct a noncommutative desingularization of the discriminant of a finite reflection group G as a quotient of the skew group ring A=S∗G. If G is generated by order two reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement A(G) viewed as a module over the coordinate ring SG/(Δ) of the discriminant of G. This yields, in particular, a correspondence between the nontrivial irreducible representations of G to certain maximal Cohen--Macaulay modules over the coordinate ring SG/(Δ). These maximal Cohen--Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement A(G) viewed as a module over SG/(Δ). We identify some of the corresponding matrix factorizations, namely the so-called logarithmic co-residues of the discriminant.
- ItemA Note on Endpoint Bochner-Riesz Estimates(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Beltran, David; Roos, Joris; Seeger, 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.
- ItemA Quantitative Analysis of the “Lion-Man” Game(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Kohlenbach, Ulrich; López-Acedo, Genaro; Nicolae, AdrianaIn this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an ε-capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.
- ItemA Well-Posedness Result for Viscous Compressible Fluids with Only Bounded Density(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Danchin, Raphaël; Fanelli, Francesco; Paicu, MariusWe are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in th L^∞ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension d≥2 and for general pressure laws.
- ItemAbstract bivariant Cuntz semigroups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Antoine, Ramon; Perera, Francesc; Thiel, HannesWe show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups S and T, there is another Cuntz semigroup JS, TK playing the role of morphisms from S to T. Applied to C*-algebras A and B, the semigroup JCu(A),Cu(B)K should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We explore its behaviour under the tensor product with the Cuntz semigroup of strongly self-absorbing C*-algebras and the Jacelon-Razak algebra. We also show that order-zero maps between C*-algebras naturally define elements in the respective bivariant Cuntz semigroup.
- ItemAbundance of 3-planes on real projective hypersurfaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Finashin, S.; Kharlamov, V.We show that a generic real projective n-dimensional hypersurface of odd degree d , such that 4(n - 2) = (d + 3 3), contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, d3 log d, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.
- ItemAeppli-Bott-Chern-Massey Products, Bigraded Notions of Formality, and Non-Zero Degree Maps(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2022) Milivojević, Aleksandar; Stelzig, JonasWe introduce and study notions of bigraded formality for the algebra of forms on a complex manifold, along with their relation to higher Aeppli-Bott-Chern-Massey products which extend, in an augmented setting, the case of triple products studied by Angella-Tomassini. We show that these Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to the blow-up along a complex submanifold, as long their degree is less than the real codimension of the submanifold. We then consider the general question of under which conditions formality is preserved by non-zero degree maps.
- ItemAffine Space Fibrations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Gurjar, Rajendra V.; Masuda, Kayo; Miyanishi, MasayoshiWe discuss various aspects of affine space fibrations. Our interest will be focused in the singular fibers, the generic fiber and the propagation of properties of a given smooth special fiber to nearby fibers.
- ItemAlexander r-tuples and Bier complexes(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Jojic, Dusko; Nekrasov, Ilya; Panina, Gaiane; Zivaljevic, RadeWe introduce and study Alexander r-tuples K = Kiir i=1 of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [BFZ-1]. In the same vein, the Bier complexes, defined as the deleted joins K delta of Alexander r-tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases. Our main results are Theorem 4.3 saying that (1) the r-fold deleted join of Alexander r-tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the r-fold deleted join of a collective unavoidable r-tuple is (n - r - 1)-connected, and a classification theorem (Theorem 5.1 and Corollary 5.2) for Alexander r-tuples and Bier complexes.
- ItemThe algebra of differential operators for a Gegenbauer weight matrix(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Zurrián, Ignacio NahuelIn this work we study in detail the algebra of differential operators D(W) associated with a Gegenbauer matrix weight. We prove that two second order operators generate the algebra, indeed D(W) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W) is a finitely-generated torsion-free module over its center, but it is not at and therefore neither projective. After [Tir11], this is the second detailed study of an algebra D(W) and the first one coming from spherical functions and group representation theory.
- ItemThe algebraic combinatorial approach for low-rank matrix completion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Király, Franz J.; Theran, Louis; Tomioka, Ryota; Uno, TakeakiWe propose an algebraic combinatorial framework for the problem of completing partially observed low-rank matrices. We show that the intrinsic properties of the problem, including which entries can be reconstructed, and the degrees of freedom in the reconstruction, do not depend on the values of the observed entries, but only on their position. We associate combinatorial and algebraic objects, differentials and matroids, which are descriptors of the particular reconstruction task, to the set of observed entries, and apply them to obtain reconstruction bounds. We show how similar techniques can be used to obtain reconstruction bounds on general compressed sensing problems with algebraic compression constraints. Using the new theory, we develop several algorithms for low-rank matrix completion, which allow to determine which set of entries can be potentially reconstructed and which not, and how, and we present algorithms which apply algebraic combinatorial methods in order to reconstruct the missing entries.
- ItemAlgebraic geometric comparison of probability distributions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Király, Franz J.; von Bünau, Paul; Meinecke, Frank C.; Blythe, Duncan A.J.; Müller, Klaus-RobertWe propose a novel algebraic framework for treating probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of Algebraic Geometry, which we demonstrate in a compact proof for an identifiability criterion.
- ItemAlgebraic matroids with graph symmetry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Király, Franz J.; Rosen, Zvi; Theran, LouisThis paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal polynomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely.