We present the proof theory and the model theory of a monotonic framework for default reasoning, and we extend logic programming techniques to this framework. Standard for-malizations of default reasoning do not syntactically separate between hard knowledge and conjectural knowledge. Such a separation is fundamental in our framework. To illustrate our approach, we show how it solves the Yale Sh...

We give a recursive algorithm for computing the Orlik-Terao algebra of the Coxeter arrangement of type An−1 as a graded representation of Sn, and we give a conjectural description of this representation in terms of the cohomology of the configuration space of n points in SU(2) modulo translation. We also give a version of this conjecture for more general graphical arrangements.

We prove the conjectural relation between the Stokes matrix for the quantum cohomology of X and an exceptional collection generating Dbcoh(X) when X is the Grassmannian Gr(r, n). The proof is based on the relation between the quantum cohomology of the Grassmannian and that of the projective space.

We determine the cohomology of Deligne-Lusztig varieties associated to some short-length regular elements for split groups of type F4 and En. As a byproduct, we obtain conjectural Brauer trees for the principal Φ14-block of E7 and the principal Φ24-block of E8.

We describe an isomorphism of categories conjectured by Kontsevich. If M and M̃ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on M̃. We prove this equivalence when M is an elliptic curve and M̃ is its dual curve, exhibiting the dictionary in detail.

We give a complete description of the fourth tautological group of the moduli space of pointed stable curves, Mg,n, and prove that for g ≥ 8 it coincides with the cohomology group with rational coefficients. We further give a conjectural upper bound depending on the genus for the degree of new tautological relations.

Using techniques due to Coster, we prove a supercongruence for a generalization of the Domb numbers. This extends a recent result of Chan, Cooper and Sica and confirms a conjectural supercongruence for numbers which are coefficients in one of Zagier’s seven “sporadic” solutions to second order Apéry-like differential equations.

In this paper we prove some new series for 1/π as well as related congruences. We also raise several new kinds of series for 1/π and present some related conjectural congruences involving representations of primes by binary quadratic forms.

The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3 surfaces. New results and conjectures (with D. Maulik) about descendent integration on K3 surfaces are announced. The recent proof of the Yau-Zaslow conjecture...