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Analysis

Our group conducts research in modern analysis, an area of pure mathematics which combines analytic and algebraic methods to study complex systems, their symmetries and global characteristics. Examples of such objects include systems of operators entering the mathematical formulation of quantum physics, symmetries and invariants of such systems, as well as geometric and algebraic structures associated to infinite graphs, groups or matrices. Moreover, our research fits naturally into the general scheme known as non-commutative geometry, which provides a vast and powerful generalization of classical geometrical concepts with interesting applications in theoretical physics.
Moreover we do research in the philosophy of mathematics informed by case studies in, e.g., analysis. We strive to understand how different representations, including diagrams, contribute to mathematical knowledge. A recent interest concerns the writings on mathematics of Charles Sanders Peirce. The studies on the role of representations for the development of mathematical knowledge draw on Peirce’s semiotics.

Topics of research

·        C. S. Peirce’s writings on mathematics

·        Classical Groups and Quantum Groups

·        L²- Cohomology, K-Theory, KK-Theory

·        Noncommutative Dynamical Systems

·        Noncommutative Geometry

·        Operator Algebras

·        Structuralism

·        The role of diagrams in mathematics

 

Funding and external collaborations/partners

DFF-Research Project 2, Automorphisms and invariants of operator algebras, 2017-2021, DKK 5,5 million, W. Szymanski (PI), D. Kyed (AI), J. Kaad (AI), in collaboration with Copenhagen University and Aarhus University.

Villum project, Local and global structures of groups and their algebras, 2014-2018, DKK 4.6 million, W. Szymanski (PI), D. Kyed (AI).

Marie-Curie Fellowship, Operator algebras and single operators via dynamical properties of dual objects, 2014-2016, B. K. Kwasniewski (fellow) and W. Szymanski (scientist-in-charge). 

Members of group

Wojciech Szymanski, professor
David Kyed, associate professor
Jens Kaad, associate professor
Jessica Carter, associate professor
Thomas Gotfredsen, Ph.D.
Sophie Emma Mikkelsen, Ph.D.

Contact persons

David Kyed
Jessica Carter

Publications

T. Bates, D. Pask, I. Raeburn and W. Szymanski, The C*-algebras of row-finite graphs, New York J. Math. 6 (2000), 307-324.

A. Carey and J. Kaad, Topological invariance of the homological index, J. reine angew. Math. 729 (2017), 229-261.

J. H. Hong and W. Szymanski, Quantum spheres and projective spaces as graph algebras, Commun. Math. Phys. 232 (2002), 157-188.

D. Kyed and S. Raum, On the ℓ²- Betti numbers of universal quantum groups, Math. Ann. 369 (3), (2017), 957-975.

D. Kyed, H. D. Petersen and S. Vaes, L²- Betti numbers of locally compact groups and their cross section equivalence relations, Trans. Amer. Math. Soc. 367 (2015), 4917-4956.

J. Kaad and M. Lesch, Spectral flow and the unbounded Kasparov product, Adv. Math. 248 (2013), 495-530.

Jessica Carter, 2008: Structuralism as a Philosophy of Mathematical Practice. Synthese 163, 119-131.

Jessica Carter, 2010: Diagrams and proofs in analysis. International Studies in the Philosophy of Science 24 (1), 1-13.

Jessica Carter 2013: Handling Mathematical Objects: Representations and context. Synthese 190 (17), 3983-3999.

Jessica Carter 2014: Mathematical Objects as Hypothetical States of Things. Philosophia Mathematica 22, 209-230.

Jessica Carter 2017: Exploring the fruitfulness of diagrams in mathematics. Synthese. DOI 10.1007/s11229-017-1635-1.

Jessica Carter forthcoming: Logic of Relations and Diagrammatic Reasoning: Structural elements in the writings of C.S. Peirce (1839- 1914). To appear in the book ’The Prehistory of Structuralism’ edited by Erich Rech and Georg Schiemer. Oxford University Press.

URL to unofficial group site 

www.imada.sdu.dk/~analysis-group

Philosophy of Mathematics

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Operator Algebra

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