No abstract

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperkähler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi-Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families.

We present Copland, a language for specifying layered attestations. Layered attestations provide a remote appraiser with structured evidence of the integrity of a target system to support a trust decision. The language is designed to bridge the gap between formal analysis of attestation security guarantees and concrete implementations. We therefore provide two semantic interpretations of terms in our language. The first is a denotational semantics in terms of partially ordered sets of events. This directly connects Copland to prior work on layered attestation. The second is an operational semantics detailing how the data and control flow are executed. This gives explicit implementation guidance for attestation frameworks. We show a formal connection between the two semantics ensuring that any execution according to the operational semantics is consistent with the denotational event semantics. This ensures that formal guarantees resulting from analyzing the event semantics will hold for executions respecting the operational semantics. All results have been formally verified with the Coq proof assistant.

We describe in terms of spherical twists the Serre functors of many interesting semiorthogonal components, called residual categories, of the derived categories of projective varieties. In particular, we show the residual categories of Fano complete intersections are fractional Calabi-Yau up to a power of an explicit spherical twist. As applications, we compute the Serre dimensions of residual categories of Fano complete intersections, thereby proving a corrected version of a conjecture of Katzarkov and Kontsevich, and deduce the nonexistence of Serre invariant stability conditions when the degrees of the complete intersection do not all coincide.