Schur Q-functions and degeneracy locus formulas for morphisms with symmetries

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle, and is based on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear in the Proceedings of Intersection Theory Conference in Bologna, "Progress in Mathematics", Birkhause

Local unitary invariants for multipartite quantum systems

A method is presented to obtain local unitary invariants for multipartite quantum systems consisting of fermions or distinguishable particles. The invariants are organized into infinite families, in particular, the generalization to higher dimensional single particle Hilbert spaces is straightforward. Many well-known invariants and their generalizations are also included.Comment: 13 page

On pointed Hopf algebras associated to unmixed conjugacy classes in S_n

Let s in S_n be a product of disjoint cycles of the same length, C the conjugacy class of s and rho an irreducible representation of the isotropy group of s. We prove that either the Nichols algebra B(C, rho) is infinite-dimensional, or the braiding of the Yetter-Drinfeld module is negative

Tertullian’s Adversus Judaeos: a Tale of Two Treatises

Tertullian’s Adversus Judaeos is a controversial text of disputed origins. Until recently, it was not given much scholarly attention, because it was unclear whether or not Tertullian wrote it as an integral, finished work, intended for publication. Two aspects of the text are especially problematic. Sections of chapters 9-14 appear to be taken whole cloth from Tertullian’s Adversus Mariconem, suggesting that Adversus Judaeos, as preserved, may be a composite of two works. Also, the work is disjointed, digressive, and repetitious, unlike Tertullian’s usual standards of authorship. Nonetheless, the most recent scholarly assessment of Adversus Judaeos, based on a comprehensive rhetorical analysis, argues strongly for the work’s authenticity and integrality. My thesis, a rebuttal of this most recent position, is that Adversus Judaeos is indeed a poorly collated composite of two of Tertullian’s works: 1/ an original, rhetorically-complete, two-book Christian apology, and 2/ passages ripped (later) from Book III of Adversus Marcionem. I argue further that the original apology is grounded in issues which arose in Carthage when Septimius Severus assumed power as undisputed Emperor of Rome in 197 c.e. A comprehensive analysis of Adversus Judaeos is presented to demonstrate: 1/ that Parts I and II were written for different (although related) purposes; 2/ that the argument of Part I is not dependent upon the argument of Part II and vice versa; 3/ that a recent proposal for the rhetorical structure of Adversus Judaeos – advanced in defense of the work’s unity – omits many observable rhetorical elements; 4/ that Parts I and II have independent rhetorical structures; and 5/ that Parts of Adversus Marcionem, Book III were redacted to form a significant part of Adversus Judaeos, Part II, and not vice versa. As a whole, the results of analysis make a strong case for the composite nature of the treatise as preserved, and facilitate a proposed reconstruction of the work as originally written, most likely as part of Tertullian’s apologetic program. The original text addresses the “charge” of Christian novelty by grounding the Church securely in ancient Jewish tradition. The unfortunate redaction came later, when someone – not Tertullian – collated the original treatise with sections of Adversus Marcionem, Book III. The result adds little in the way of argument to the original treatise, and therefore the purpose of the composite, as preserved, remains a mystery

The regulation of capillary blood flow .

Thesis (Ph.D.)--Boston University

On the 2D zero modes' algebra of the SU(n) WZNW model

A quantum group covariant extension of the chiral parts of the Wess-Zumino-Novikov-Witten model on a compact Lie group G gives rise to two matrix algebras with non-commutative entries. These are generated by "chiral zero modes" which combine in the 2D model into "Q-operators" which encode information about the internal symmetry and the fusion ring. We review earlier results about the SU(n) WZNW Q-algebra and its Fock representation for n=2 and display the first steps towards their generalization to higher n.Comment: 10 pages, Talk presented by L.H. at the International Workshop LT10 (17-23 June 2013, Varna, Bulgaria

Two generalizations of the PRV conjecture

Let G be a complex connected reductive group. The PRV conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for "the same reason" as the PRV ones

Inclusion-exclusion and Segre classes

We propose a variation of the notion of Segre class, by forcing a naive `inclusion-exclusion' principle to hold. The resulting class is computationally tractable, and is closely related to Chern-Schwartz-MacPherson classes. We deduce several general properties of the new class from this relation, and obtain an expression for the Milnor class of a scheme in terms of this class.Comment: 8 page

Stringy K-theory and the Chern character

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natural group grading. We prove there is a ring isomorphism, the stringy Chern character, from stringy K-theory to stringy cohomology, and a ring homomorphism from full orbifold K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy Grothendieck-Riemann-Roch for etale maps. We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's construction. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results simplify the definitions of Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of Abramovich-Graber-Vistoli's orbifold Chow. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler Resolution Conjecture holds for symmetric products. Our results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.Comment: Exposition improved and additional details provided. To appear in Inventiones Mathematica