Finite FF-representation type for homogeneous coordinate rings of non-Fano varieties

Finite $F$-representation type is an important notion in characteristic-$p$ commutative algebra, but explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings in positive characteristic will fail to have finite $F$-representation type. To do so, we prove a connection between differential operators on the homogeneous coordinate ring of $X$ and the existence of global sections of a twist of $(\mathrm{Sym}^m \Omega_X)^\vee$. By results of Takagi and Takahashi, this allows us to rule out FFRT for coordinate rings of varieties with $(\mathrm{Sym}^m \Omega_X)^\vee$ not ``positive''. By using results positivity and semistability conditions for the (co)tangent sheaves, we show that several classes of varieties fail to have finite $F$-representation type, including abelian varieties, most Calabi--Yau varieties, and complete intersections of general type. Our work also provides examples of the structure of the ring of differential operators for non-$F$-pure varieties, which to this point have largely been unexplored.Comment: 15 pages; comments welcome

Singularities of Birational Geometry via Arcs and Differential Operators

We study singularities of algebraic varieties, in particular those arising in birational geometry, from several points of view. The first is that of arc schemes: arc schemes parametrize “infinitesimal curves” on a variety, and their geometry reflects properties of singularities. We show that morphisms of arc schemes (more precisely, of “local” arc schemes) can detect local isomorphisms of varieties. More precisely, we use the triviality of a certain ideal-closure operation to show that if a morphism induces an isomorphism of local arc schemes then it must be an isomorphism on local rings. We then use arc schemes, in conjunction with the theory of determinantal rings, to verify the semicontinuity conjecture for the behavior of the minimal log discrepancy (a subtle invariant of singularities) in the case of determinantal varieties. In particular, we calculate the Nash ideal of a generic square determinantal variety, which then allows us to give an explicit formula for the minimal log discrepancies of pairs of determinantal varieties and determinantal subvarieties. This allows us to verify the semicontinuity conjecture for such pairs. We then take another point of view, via the study of differential operators on singular rings. At least since [Levasseur and Stafford 1989], the question had been asked of whether one can characterize singularities of rings via certain properties of their rings of differential operators. In particular, one question is whether a ring with mild singularities is a simple module under the action of its ring of differential operators. While an answer in characteristic p had been provided by [Smith 1995], no answer had been forthcoming in characteristic 0. We provide a counterexample showing that the expected connection does not exist, through the study of the global geometry of Fano varieties. More specifically, we show that certain del Pezzo surfaces do not have big tangent bundles, and thus their homogeneous coordinate rings are not simple under the action of their rings of differential operators, despite having “mild” singularities.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169625/1/malloryd_1.pd

Which Exterior Powers are Balanced?

A signed graph is a graph whose edges are given ±1 weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal ±1 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1,..., n − 1, is a graph ∧k Σ whose adjacency matrix is given by A ( ∧k † Σ) = P ∧A(Σ □k)P∧, where P ∧ is the projector onto the anti-symmetric subspace of the k-fold tensor product space (C n) ⊗k and Σ □k is the k-fold Cartesian product of Σ with itself. The exterior power creates a signed graph from any graph, even unsigned. We prove sufficient and necessary conditions so that ∧ k Σ is balanced. For k = 1,..., n − 2, the condition is that either Σ is a signed path or Σ is a signed cycle that is balanced for odd k or is unbalanced for even k; for k = n − 1, the condition is that each even cycle in Σ is positive and each odd cycle in Σ is negative

PERFECT STATE TRANSFER ON SIGNED GRAPHS

We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative 2-clique with any positive (n,3)-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as n increases. Next, we prove that a signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for creating perfect state transfer since no complete graph (except for the 2-clique) has perfect state transfer. Also, we show that the double-cover of a signed graph has perfect state transfer if the positive subgraph has perfect state transfer and the negative subgraph is periodic. Here, signing is useful for constructing unsigned graphs with perfect state transfer. Finally, we study perfect state transfer on a family of signed graphs called the exterior powers which is derived from a many-fermion quantum walk on graphs

Bannayan-Riley-Ruvalcaba syndrome: report of a family

Bannayan–Riley–Ruvalcaba (BRR) syndrome is a rare inherited condition. We describe the protean orofacial manifestations of this syndrome in one family and consider their management. The dental surgeon should be aware of this entity, its orofacial connotations and the possible association with Cowden’s syndrome

Context-aware experience sampling reveals the scale of variation in affective experience

Emotion research typically searches for consistency and specificity in physiological activity across instances of an emotion category, such as anger or fear, yet studies to date have observed more variation than expected. In the present study, we adopt an alternative approach, searching inductively for structure within variation, both within and across participants. Following a novel, physiologically-triggered experience sampling procedure, participants’ self-reports and peripheral physiological activity were recorded when substantial changes in cardiac activity occurred in the absence of movement. Unsupervised clustering analyses revealed variability in the number and nature of patterns of physiological activity that recurred within individuals, as well as in the affect ratings and emotion labels associated with each pattern. There were also broad patterns that recurred across individuals. These findings support a constructionist account of emotion which, drawing on Darwin, proposes that emotion categories are populations of variable instances tied to situation-specific needs