Dilogarithm Identities in Conformal Field Theory

Dilogarithm identities for the central charges and conformal dimensions exist for at least large classes of rational conformally invariant quantum field theories in two dimensions. In many cases, proofs are not yet known but the numerical and structural evidence is convincing. In particular, close relations exist to fusion rules and partition identities. We describe some examples and ideas, and present some conjectures useful for the classification of conformal theories. The mathematical structures seem to be dual to Thurston's program for the classification of 3-manifolds.Comment: 14 pages, BONN-preprint. (a few minor changes, two major corrections in chapter 3, namely: (3.10) only holds in the case of the A series, Goncharovs conjecture is not an equivalence but rather an implication and a theorem

Josiah Parsons Cooke Jr.: Epistemology in the Service of Science, Pedagogy, and Natural Theology

Josiah Parsons Cooke established chemistry education at Harvard University, initiated an atomic weight research program, and broadly impacted American chemical education through his students, the introduction of laboratory instruction, textbooks, and influence on Harvard's admissions requirements. The devoutly Unitarian Cooke also articulated and defended a biogeochemical natural theology, which he defended by arguing for commonalities between the epistemologies of science and religion. Cooke's pre-Mendeleev classification scheme for the elements and atomic weight research were motivated by his interest in numerical order in nature, which reflected his belief in a divine lawgive

The K\"ahler-Ricci flow on surfaces of positive Kodaira dimension

The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized K\"ahler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for K\"aher surfaces with a numerical effective canonical line bundle by the K\"ahler-Ricci flow. In general, we propose a program of finding canonical metrics on canonical models of projective varieties of positive Kodaira dimension

Pseudolattices, del Pezzo surfaces, and Lefschetz fibrations

Motivated by the relationship between numerical Grothendieck groups induced by the embedding of a smooth anticanonical elliptic curve into a del Pezzo surface, we define the notion of a quasi del Pezzo homomorphism between pseudolattices and establish its basic properties. The primary aim of the paper is then to prove a classification theorem for quasi del Pezzo homomorphisms, using a pseudolattice variant of the minimal model program. Finally, this result is applied to the classification of a certain class of genus one Lefschetz fibrations over discs.Comment: v2: Minor revisions at the request of the referee. Remark 3.24 is new, as is the discussion of the noncommutative setting in the final sectio

On the classification of orbifold del Pezzo surfaces

Chapter 1 is devoted to outlining the problem. We introduce the background material, namely define orbifold del Pezzo surfaces, qG_deformations and graded ring methods. Following this, we use the properties of these classes of surfaces to compute numerical invariants and find a bound for the number of singularities. Ultimately, we obtain a list of numerical candidates for our surfaces. In Chapter 2 we recall some aspects of the Mori theory for surfaces, we define the notion of minimal surfaces and we find surfaces with singularity content (n; k1_1 5(1; 2)+k2_1 3(1; 1)) having Picard rank _ = 1. Later we establish a Directed Minimal Model Program for our class of surfaces and by analysing our numerical candidates we find the isomorphism classes of our del Pezzo surfaces. In Chapter 3 we discuss the toric case: we find all of the possible mutation classes of our orbifolds and we introduce the formalism of T_varieties. We then show how to link qG_deformations to equivariant complexity 1 deformations. We give a couple of enlightening examples to better understand the complexity 1 environment and deformations. In Chapter 4 we finally construct the cascades from the representatives of the qG_classes and we give a complete count of all the deformation classes for our type of surfaces. Chapter 5 contains tables representing a summary of the MMP outcomes and the classification of toric surfaces representing the mutation classes. Lastly, in the Appendix we report the calculations that lead us to the classification of the isomorphism classes in Chapter 2