The Local Structure of Algebraic K-Theory: 18 (Algebra and Applications)

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Summary "Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate.

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Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of 'nearby' calculations in K-theory.

For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds.

Mike Hopkins: Equivariant dual of Morava E-theory

In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. Subject K-theory. Algebraic topology. Category Theory, Homological Algebra. Bibliographic information. Publication date Series Algebra and applications, ; v.

Arithmetic and algebraic geometry.

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The beautiful utility of any duality is that it allows for the surprisingly meaningful flow of ideas between two disparate areas. Indeed, through Homological Mirror Symmetry, symplectic considerations have yielded concrete results in the theory of derived categories of coherent sheaves on algebraic varieties. These results are unexpected: the apparent flexibility of symplectic topology contrasts with the rigidity found in solutions of polynomial equations.

That difference has been resolved by the surprising abundance found in the group of autoequivalences of the derived category. One particularly successful example was the paper of Seidel and Thomas on spherical twists, which inspired many successors and enjoyed further expansions. Workshop on New Geometry of Quantum Dynamics.

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There are a number unifying principles which establish connections among the proposed topics of the meeting. We aim to construct quantum metric geometries of crossed products and graph algebras relating the Lipschitz norm and Dirac-operator approaches. This links the metric and spectral features of noncommutative geometry. A key research focus of the proposed activities is to make precise the impact of metric and spectral noncommutative geometry on noncommutative topology. Representation Theory and Integrable Systems.

AIM Workshop: Zeros of random polynomials. This workshop, sponsored by AIM and the NSF, will be devoted to the zero distribution of random polynomials spanned by various deterministic bases. Higher homotopical structures such as L-infinity algebras, gerbes, and generalised differential cohomologies are at the heart of string and M-theory.

This has been recognised in special cases, and there is much evidence for higher structures to play a key role in understanding these theories.

Bibliographic Information

Li, Duke Math. MATH Y. Let X be a regular scheme over spec Z. Review available group times 3. Ian Hambleton Depto. Galatius, I. Announcements, and other items that pertain to all sections, will be posted there.

In addition, higher structures have been receiving much attention in mathematics and physics independently. Therefore, it is time to bring together experts in those areas to identify the pertinent open problems and discuss solution strategies. This symposium will combine lectures on the foundations, talks on recent advances, and open discussion sessions.

High Energy Physics, Particles and Fields. Perspectives in Linear Algebraic Groups. Group theory is a fundamental and highly active field of research within the mathematical sciences and has ramifications throughout all of mathematics and the sciences. There are large communities of group theory researchers in all continents, and they keep in touch through international meetings and conferences. However, local conditions and cultural circumstances under which mathematical research is done differ quite a bit between different regions of the world.

William G. Dwyer

The focus of this conference is on research in group theory in the USA and in China. One major goal of the conference is to establish and strengthen ties between group theorists in China and the US. It is the first conference in group theory with this focus. We will have about 15 invited talks and a limited number of contributed talks. The speakers from both countries will cover a large variety of the subjects in the area, which include representations of solvable groups, representations of simple groups, character theory, classes of groups, groups and combinatorics, recognizing simple groups from group invariants, p-groups, and fusion systems.

Representations of solvable groups, representations of simple groups, character theory, classes of groups, groups and combinatorics, recognizing simple groups from group invariants, p-groups, and fusion systems. Quantum structures in Algebra and Geometry.

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Quantum Mechanics and Quantum Information Theory. Immeubles et grassmaniennes affines — Buildings and Affine Grassmannians.

The aim of the summer school is to gather young researchers arising from a quite wide mathematical area in order to build knowledge in two complementary fundamental theories. The first one is Bruhat-Tits theory which deals with the Euclidean buildings and integral models of reductive algebraic groups defined over a local field, the second one is the theory of Affine Grassmannians which are geometric objects arising from Representation Theory. The purpose of the CONCUR conferences is to bring together researchers, developers, and students in order to advance the theory of concurrency, and promote its applications.

All papers must be original, unpublished, and not submitted for publication elsewhere. The aim of the conference is to bring together leading scientists of the pure and applied mathematics and related areas to present their researches, to exchange new ideas, to discuss challenging issues, to foster future collaborations and to interact with each other.

\(K\)-Theory of Non-Archimedean Rings. I

Integrability, Combinatorics, and Representations. LMFDB as a microscope and a telescope. The LMFDB contains a wealth of information on L-functions, modular forms of several types, elliptic curves and genus 2 curves, number fields, and much more. In addition to detailed information about individual objects, the LMFDB also includes information about connections between objects, including the connections described by the Langlands Program. The workshop will involve a mixture of demonstrations, explorations, discussions about mathematical content, and discussions about the future of the LMFDB.

Illustrating Mathematics Semester Program. The Illustrating Mathematics program brings together mathematicians, makers, and artists who share a common interest in illustrating mathematical ideas via computational tools. The goals of the program are to: introduce mathematicians to new computational illustration tools to guide and inform their research; spark collaborations among and between mathematicians, makers and artists; and find ways to communicate research mathematics to as wide an audience as possible.

MAT TRIAD provides an opportunity to bring together researchers sharing an interest in a variety of aspects of matrix analysis and its applications in other area of science.

Mathematical Sciences Research Institute

Researchers and graduate students interested in recent developments in matrix and operator theory and computation, spectral problems, applications of linear algebra in statistics, statistical models, matrices and graphs as well as combinatorial matrix theory are particularly encouraged to attend. Workshop on Banach spaces and Banach lattices. The purpose of this workshop is to bring together researchers interested in Banach spaces and Banach lattices, and stimulate collaboration. The program consists of four mini-courses about recent developments of the theory, as well as plenary lectures by internationally renowned specialists.

The Geometry of Derived Categories. Perfectoid spaces. This will consist of a one-day meeting on zeta functions and a 2-day workshop on elliptic curves. The aim of the conference is to allow PhD students and young researchers to share their knowledge and discuss topics in Algebra.

The number of admissible participants is limited. Requests for a financial support to cover accommodation expenses must be indicated in the application form and will be evaluated by the organizers.

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Representation Theory In Venice. Kontsevich's original conjecture dealt solely with Calabi-Yau varieties. Later, it was realized that to incorporate Fano varieties into the statement one needed something a bit more general than varieties: the objects called Landau--Ginzburg models in physics. Landau--Ginzburg models provide theories where the interesting behavior localizes to the singular locus of w. As such, they are well-known in a number of mathematical fields. This workshop is meant to gather experts in a broad range of areas around the theme of Landau-Ginzburg models -- all of which are touched by HMS.

Specific problems of interest are are the study of Fan--Jarvis--Ruan--Witten Theory and Homological Projective Duality through matrix factorizations in Algebraic Geometry and their dual realization in Fukaya Categories.