A Goodwillie-type Theorem for Milnor K-Theory

Goodwillie's rational isomorphism between relative algebraic K-theory and relative cyclic homology, together with the lambda decomposition of cyclic homology, illustrates the close relationships among algebraic K-theory, cyclic homology, and differential forms. In this paper, I prove a Goodwillie-type theorem for relative Milnor $K$-theory, working over a very general class of commutative rings, defined via the stability criterion of Van der Kallen. Early results of Van der Kallen and Bloch are special cases. The result likely generalizes in terms of de Rahm-Witt complexes, by weakening some invertibility assumptions, but the class of rings considered is already more than sufficiently general for the intended applications. The main motivation for this paper arises from applications to the infinitesimal theory of Chow groups, first pointed out by Bloch in the 1970's, and prominent in recent work of Green and Griffiths. Related results and geometric applications are discussed in the final section.Comment: 34 page

On the Infinitesimal Theory of Chow Groups

The Chow groups of codimension-p algebraic cycles modulo rational equivalence on a smooth algebraic variety X have steadfastly resisted the efforts of algebraic geometers to fathom their structure. This book explores a "linearization" approach to this problem, focusing on the infinitesimal structure of the Chow groups near their identity elements. This method was adumbrated in recent work of Mark Green and Phillip Griffiths. Similar topics have been explored by Bloch, Stienstra, Hesselholt, Van der Kallen, and others. A famous formula of Bloch expresses the Chow groups as Zariski sheaf cohomology groups of algebraic K-theory sheaves on X. "Linearization" of the Chow groups is thereby related to "linearization" of algebraic K-theory, which may be described in terms of negative cyclic homology. The "proper formal construction" arising from this approach is a "machine" involving the coniveau spectral sequences arising from four different generalized cohomology theories on X, with the last two sequences connected by the algebraic Chern character. Due to the critical role of the coniveau filtration, I refer to this construction as the coniveau machine. The main theorem in this book establishes the existence of the coniveau machine for algebraic K-theory on a smooth algebraic variety. An immediate corollary is a new formula expressing generalized tangent groups of Chow groups in terms of negative cyclic homology.Comment: 167 pages, 41 figures, preface update

Pathogenicity and Impact of HLA Class I Alleles in Aplastic Anemia Patients of Different Ethnicities

Acquired aplastic anemia (AA) is caused by autoreactive T cell-mediated destruction of early hematopoietic cells. Somatic loss of human leukocyte antigen (HLA) class I alleles was identified as a mechanism of immune escape in surviving hematopoietic cells of some patients with AA. However, pathogenicity, structural characteristics, and clinical impact of specific HLA alleles in AA remain poorly understood. Here, we evaluated somatic HLA loss in 505 patients with AA from 2 multi-institutional cohorts. Using a combination of HLA mutation frequencies, peptide-binding structures, and association with AA in an independent cohort of 6,323 patients from the National Marrow Donor Program, we identified 19 AA risk alleles and 12 non-risk alleles and established a potentially novel AA HLA pathogenicity stratification. Our results define pathogenicity for the majority of common HLA-A/B alleles across diverse populations. Our study demonstrates that HLA alleles confer different risks of developing AA, but once AA develops, specific alleles are not associated with response to immunosuppression or transplant outcomes. However, higher pathogenicity alleles, particularly HLA-B*14:02, are associated with higher rates of clonal evolution in adult patients with AA. Our study provides insights into the immune pathogenesis of AA, opening the door to future autoantigen identification and improved understanding of clonal evolution in AA

Discrete causal theory: emergent spacetime and the causal metric hypothesis

This book evaluates and suggests potentially critical improvements to causal set theory, one of the best-motivated approaches to the outstanding problems of fundamental physics. Spacetime structure is of central importance to physics beyond general relativity and the standard model. The causal metric hypothesis treats causal relations as the basis of this structure. The book develops the consequences of this hypothesis under the assumption of a fundamental scale, with smooth spacetime geometry viewed as emergent. This approach resembles causal set theory, but differs in important ways; for example, the relative viewpoint, emphasizing relations between pairs of events, and relationships between pairs of histories, is central. The book culminates in a dynamical law for quantum spacetime, derived via generalized path summation

Tangents to Chow groups: on a question of Green–Griffiths

We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We settle a question recently raised by Mark Green and Phillip Griffiths concerning the existence of Bloch–Gersten–Quillen-type resolutions of algebraic K-theory sheaves on infinitesimal thickenings of nonsingular varieties, and the relationships between these sequences and their “tangent sequences,” expressed in terms of absolute Kähler differentials. More generally, we place Green and Griffiths’ concrete geometric approach to the infinitesimal theory of Chow groups in a natural and formally rigorous structural context, expressed in terms of nonconnective K-theory, negative cyclic homology, and the relative algebraic Chern character