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I then study the relationship between the algebraic invariants of these spaces and the combinatorial invariants of the input data. The flow of information goes in both directions: sometimes I use combinatorics to compute objects of intrinsic geometric interest categories of sheaves, cohomology rings, etc. I work on probability and statistics as applied to understanding ecology and evolution, in particular developing new stochastic models of biological evolution and using statistical inference and visualization methods to find out what genomes can tell us about biology.

I am interested in algebraic geometry and mathematical physics.

My recent work focus on curve counting theories, such as Gromov-Witten theory and Fan-Jarvis-Ruan-Witten theory, and the mirror symmetry beyond them. These theories have deep connections to complex geometry, number theory, and representation theory. I am interested in the statistics of eigenvalues of random matrices and the roots of random polynomials. I am also interested in measures of complexity of polynomials heights and the distribution of roots of polynomials with low height.

I work in a range of topics in topology, mostly in algebraic topology but some in geometric topology as well. I like to see the geometry which underlies homotopical structures. My most recent projects are in cohomology of alternating and symmetric groups, in rational homotopy theory, and in knot theory. The first two topics are an interesting juxtaposition since the cohomology of symmetric groups is all torsion while rational homotopy theory systematically ignores torsion. Configuration spaces are a part of much of what I do, and I like to study topology, geometry, algebra and combinatorics related to them.

I study algebra and geometry motivated by physics. My current interests involve algebraic geometry, in particular orbifolds. My past interests have included knot theory and representation theory. My current research interest is in the area of measure-valued processes or superprocesses that come out as limits in distribution of a sequence of branching particle systems. My most recent work has involved fourth order elliptic minimal surface equations, and other recent work has been on constructions of Ricci curvature with applications to machine learning.

My main research is in approximation theory, Fourier analysis, orthogonal polynomials and special functions, which are really all connected.

Most of my work focuses on multidimensional problems. I work in enumerative, bijective and algebraic combinatorics. Most of what I am working on at the moment is related to the dimer model, or to Schubert calculus and the combinatorics of reduced words. I use computers heavily in my work.

Persistent sheaf cohomology

University of Oregon. UO Home Dept Index. Faculty Research Interests Nicolas Addington : algebraic geometry I work in algebraic geometry, mainly using derived categories of coherent sheaves. Yashar Ahmadian : theoretical neuroscience, statistical physics, random matrix theory My research is in the field of theoretical neuroscience. Arkady Berenstein : quantum groups, representation theory, algebraic combinatorics My research interests include Representation Theory of Lie algebras, Quantum Groups, and Coxeter Groups, Hopf Algebras, Algebraic Combinatorics Cluster algebras, Noncommutative Algebra and related aspects.

Group representations Coxeter groups Lie algebra representations Hopf algebras Quantum Group Cluster algebra Noncommutative algebraic geometry Boris Botvinnik : differential topology, positive scalar curvature, Morse theory I study algebraic topology and differential geometry, with a focus on conformal geometry and the space of metrics of positive scalar curvature. Marcin Bownik : harmonic analysis, wavelets, approximation theory I work in the area of harmonic analysis and wavelets. If M and N are not assumed to be simply connected, then an h -cobordism need not be a cylinder.

This generalizes the h -cobordism theorem because the simple connectedness hypotheses imply that the relevant Whitehead group is trivial. In fact the s -cobordism theorem implies that there is a bijective correspondence between isomorphism classes of h -cobordisms and elements of the Whitehead group. An obvious question associated with the existence of h -cobordisms is their uniqueness. The natural notion of equivalence is isotopy. Jean Cerf proved that for simply connected smooth manifolds M of dimension at least 5, isotopy of h -cobordisms is the same as a weaker notion called pseudo-isotopy.

The proper context for the s -cobordism theorem is the classifying space of h -cobordisms. Consideration of these questions led Waldhausen to introduced his algebraic K -theory of spaces. In order to fully develop A -theory, Waldhausen made significant technical advances in the foundations of K -theory.

Quillen suggested to his student Kenneth Brown that it might be possible to create a theory of sheaves of spectra of which K -theory would provide an example. The sheaf of K -theory spectra would, to each open subset of a variety, associate the K -theory of that open subset. Brown developed such a theory for his thesis. Simultaneously, Gersten had the same idea.

This is now called the Brown—Gersten spectral sequence. This is known as Bloch's formula. While progress has been made on Gersten's conjecture since then, the general case remains open. Lichtenbaum conjectured that special values of the zeta function of a number field could be expressed in terms of the K -groups of the ring of integers of the field. Quillen therefore generalized Lichtenbaum's conjecture, predicting the existence of a spectral sequence like the Atiyah—Hirzebruch spectral sequence in topological K -theory. In the case studied by Lichtenbaum, the spectral sequence would degenerate, yielding Lichtenbaum's conjecture.

William G. Throughout the s and early s, K -theory on singular varieties still lacked adequate foundations. While it was believed that Quillen's K -theory gave the correct groups, it was not known that these groups had all of the envisaged properties. For this, algebraic K -theory had to be reformulated. This was done by Thomason in a lengthy monograph which he co-credited to his dead friend Thomas Trobaugh, who he said gave him a key idea in a dream.

There, K 0 was described in terms of complexes of sheaves on algebraic varieties. Thomason discovered that if one worked with in derived category of sheaves, there was a simple description of when a complex of sheaves could be extended from an open subset of a variety to the whole variety. By applying Waldhausen's construction of K -theory to derived categories, Thomason was able to prove that algebraic K -theory had all the expected properties of a cohomology theory.

In , Keith Dennis discovered an entirely novel technique for computing K -theory based on Hochschild homology.

Math 256A-B — Algebraic Geometry — 2018-19

While the Dennis trace map seemed to be successful for calculations of K -theory with finite coefficients, it was less successful for rational calculations. Goodwillie, motivated by his "calculus of functors", conjectured the existence of a theory intermediate to K -theory and Hochschild homology. He called this theory topological Hochschild homology because its ground ring should be the sphere spectrum considered as a ring whose operations are defined only up to homotopy.

In the mids, Bokstedt gave a definition of topological Hochschild homology that satisfied nearly all of Goodwillie's conjectural properties, and this made possible further computations of K -groups. This transformation factored through the fixed points of a circle action on THH , which suggested a relationship with cyclic homology. In the course of proving an algebraic K -theory analog of the Novikov conjecture , Bokstedt, Hsiang, and Madsen introduced topological cyclic homology, which bore the same relationship to topological Hochschild homology as cyclic homology did to Hochschild homology.

In , Dundas, Goodwillie, and McCarthy proved that topological cyclic homology has in a precise sense the same local structure as algebraic K -theory, so that if a calculation in K -theory or topological cyclic homology is possible, then many other "nearby" calculations follow. The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring. The functor K 0 takes a ring A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modules , regarded as a monoid under direct sum.

If the ring A is commutative, we can define a subgroup of K 0 A as the set. If B is a ring without an identity element , we can extend the definition of K 0 as follows. An algebro-geometric variant of this construction is applied to the category of algebraic varieties ; it associates with a given algebraic variety X the Grothendieck's K-group of the category of locally free sheaves or coherent sheaves on X. Given a compact topological space X , the topological K-theory K top X of real vector bundles over X coincides with K 0 of the ring of continuous real-valued functions on X.

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The relative K-group is defined in terms of the "double" [50]. The independence from A is an analogue of the Excision theorem in homology. If A is a commutative ring, then the tensor product of projective modules is again projective, and so tensor product induces a multiplication turning K 0 into a commutative ring with the class [ A ] as identity. Hyman Bass provided this definition, which generalizes the group of units of a ring: K 1 A is the abelianization of the infinite general linear group :.

Define an elementary matrix to be one which is the sum of an identity matrix and a single off-diagonal element this is a subset of the elementary matrices used in linear algebra. The relative K-group is defined in terms of the "double" [53]. There is a natural exact sequence [54]. When A is a Euclidean domain e. When this fails, one can ask whether K 1 is generated by the image of GL 2. For Dedekind domains with all quotients by maximal ideals finite, SK 1 is a torsion group.

Wang's theorem states that if A has prime degree then SK 1 A is trivial, [59] and this may be extended to square-free degree. It can also be defined as the kernel of the map. For a field, K 2 is determined by Steinberg symbols : this leads to Matsumoto's theorem. One can compute that K 2 is zero for any finite field. For non-Archimedean local fields, the group K 2 F is the direct sum of a finite cyclic group of order m , say, and a divisible group K 2 F m. Matsumoto's theorem states that for a field k , the second K -group is given by [71] [72].

Matsumoto's original theorem is even more general: For any root system , it gives a presentation for the unstable K-theory. This presentation is different from the one given here only for symplectic root systems. For non-symplectic root systems, the unstable second K-group with respect to the root system is exactly the stable K-group for GL A. Unstable second K-groups in this context are defined by taking the kernel of the universal central extension of the Chevalley group of universal type for a given root system.

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If A is a Dedekind domain with field of fractions F then there is a long exact sequence. There is also an extension of the exact sequence for relative K 1 and K 0 : [74].

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There is a pairing on K 1 with values in K 2. The above expression for K 2 of a field k led Milnor to the following definition of "higher" K -groups by. For integer m invertible in k there is a map. Convert currency. Add to Basket. Book Description Springer-Verlag, Condition: Good. Shows some signs of wear, and may have some markings on the inside. Seller Inventory GRP More information about this seller Contact this seller. Book Description Springer, Bumped corner, light wear to covers, otherwise text clean and solid; Lecture Notes in Mathematics ; 0. Seller Inventory