Interleaving distances are used widely in Topological Data Analysis (TDA) as a tool for comparing topological signatures of datasets. The theory of interleaving distances has been extended through various category-theoretic constructions, enabling its usage beyond standard constructions of TDA, while clarifying certain observed stability phenomena by unifying them under a common framework. Inspired by metrics used in the field of statistical shape analysis, which are based on minimizing energy functions over group actions, we define three new types of increasingly general interleaving distances. Our constructions use ideas from the theories of monoidal actions and 2-categories. We show that these distances naturally extend the category with a flow framework of de Silva, Munch and Stefanou and the locally persistent category framework of Scoccola, and we provide a general stability result. Along the way, we give examples of distances that fit into our framework which connect to ideas from differential geometry, geometric shape analysis, statistical TDA and multiparameter persistent homology.

Trisections of closed 4-manifolds, first defined and studied by Gay and Kirby, have proved to be a useful tool in the systematic analysis of 4-manifolds via handlebodies. Subsequent work of Abrams, Gay, and Kirby established a connection with the algebraic notion of a group trisection, which strikingly defines a one-to-one correspondence. We generalize the notion of a group trisection to the non-closed case by defining and studying relative group trisections. We establish an analogous one-to-one correspondence between relative trisections and relative group trisections up to equivalence. The key lemma in the construction may be of independent interest, as it generalizes the classical fact that there is a unique handlebody extension of a surface realizing a given surjection. Moreover, we establish a functorial relationship between relative trisections of manifolds and groups, extending work of Klug in the closed case.

Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories. Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Thélène. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places.

We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric

The purpose of this article is to give a small sampling of the theory of arithmetic toric varieties, focusing on simple examples in low dimension using explicit polynomial equations. A second goal is to highlight the class of arithmetic toric varieties as the most natural testing ground to study the limitations of algebro-geometric invariants in the arithmetic setting.

We develop a generalization of the Q-construction of the first author, Diemer, and the third author for Grassmann flips. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. The idempotent kernel, after restriction, induces a semi-orthogonal decomposition which compares the flipped varieties. Furthermore its image, after restriction to the geometric invariant theory semistable locus, “opens” a canonical “window” in the derived category of the quotient stack. We check this window coincides with the set of representations used by Kapranov to form a full exceptional collection on Grassmannians.

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a slight generalization of this conjecture, where the endomorphism algebras of the exceptional objects are allowed to be separable field extensions of the base field. We show this generalization is false by exhibiting a geometrically rational, smooth, projective threefold over the the field of rational numbers that possesses a full étale-exceptional collection but not a rational point. The counterexample comes from twisting a non-retract rational variety with a rational point and full étale-exceptional collection by a torsor that is invisible to Brauer invariants. Along the way, we develop some tools for linearizing objects, including a group that controls linearizations.

We compute the group of K1-zero-cycles on the second generalized involution variety for an algebra of degree 4 with symplectic involution. This description is given in terms of the group of multipliers of similitudes associated to the algebra with involution. Our method utilizes the framework of Chernousov and Merkurjev for computing K1-zero-cycles in terms of R-equivalence classes of prescribed algebraic groups. This gives a computation of K1-zero-cycles for some homogeneous varieties of type C2.

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections, making it possible to give concrete descriptions of their derived categories. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type A. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly nontoric) varieties over nonclosed fields.

We combine the Bondal-Uehara method for producing exceptional collections on toric varieties with a result of the first author and Favero to expand the set of varieties satisfying Orlov's Conjecture on derived dimension.

In this manuscript, it is shown that the group of K1-zero-cycles on the second generalized Severi–Brauer variety of an algebra A of index 4 is given by elements of the group K1(A) together with a square-root of their reduced norm. Utilizing results of Krashen concerning exceptional isomorphisms, we translate our problem to the computation of cycles on involution varieties. Work of Chernousov and Merkurjev then gives a means of describing such cycles in terms of Clifford and spin groups and corresponding R-equivalence classes. We complete our computation by giving an explicit description of these algebraic groups.

Given a scheme X, there are two important techniques one may use to study X. On one hand, there is the theory which utilizes subschemes or divisors and algebraic cycles and its corresponding cohomology theory is represented by the Chow ring A(X). On the other hand, there is the method of studying bundles over X, utilizing vector bundles and coherent and quasicoherent sheaves, and its corresponding cohomology theory is K-theory. This is analogous to the methods used in differential geometry relating submanifolds and vector bundles. Our main goal will be to completely characterize the Grothendieck group of a nonsingular algebraic curve in terms of its Picard group. We begin with a few definitions.

In the algebraic setting, one main method used in analyzing a given object is to investigate maps into and out of it. For example, the study of abelian categories consists of considering functors from a given abelian category into a well-behaved category such as Ab, the category of abelian groups and group homomorphisms. In this context, one property we can investigate is the exactness of such a functor. One example of an abelian category is the category of modules over a ring Λ, denoted Mod(Λ). In fact, this is really the only example (see the Freyd-Mitchell Theorem or Mitchell’s Embedding Theorem). In studying these categories, homological methods can be used to determine how far certain functors are from being exact. The objects which measure this non-exactness are called derived functors. Unfortunately, these categories can be much too large to be the subject of a fruitful study (and may cause set-theoretic difficulties), so often our analysis must be reduced to the study of a more easily understood subcategory which still captures enough information to be useful. For example, in the algebraic K-theory of rings, to analyze a ring Λ, one restricts themselves to the study of finitely generated projective modules over Λ. This category is a much more tractable substitute for the full module category but still possesses a great deal of information about Λ. Category O is a similar substitution for the module category Mod(U(g)). We restrict our attention to those modules which have a certain finiteness condition so that many computations may be done explicitly. However, in restricting our attention to this much smaller category there is the possibility that we have lost many of the properties that allow

It was conjectured in \cite{Namikawa_ExtendedTorelli} that the Torelli map $M_g\to A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}_g^{\rm cent}$. A counterexample in genus $g=9$ was found in \cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $g\le8$, thus completely solving the problem in every genus.

AbstractWe present a study of the temporal changes in the sensitivities of the frequencies of the solar p-mode oscillations to corresponding changes in the levels of solar activity during Solar Cycle 23. From MDI and GONG++ full-disk Dopplergram three-day time series obtained between 1996 and 2008 we have computed a total of 221 sets of m-averaged power spectra for spherical harmonic degrees ranging up to 1000. We have then fit these 284 sets of m-averaged power spectra using our WMLTP fitting code and both symmetric Lorentzian profiles for the peaks as well as the asymmetric profile of Nigam and Kosovichev to obtain 568 tables of p-mode parameters. We then inter-compared these 568 tables, and we performed linear regression analyses of the differences in p-mode frequencies, widths, amplitudes, and asymmetries as functions of the differences in as many as ten different solar activity indices. From the linear regression analyses that we performed on the frequency difference data sets, we have discovered a new signature of the frequency shifts of the p-modes. Specifically, we have discovered that the temporal shifts of the solar oscillation frequencies are positively correlated with the changes in solar activity below a limiting frequency. They then become anti-correlated with the changes in activity for a range of frequencies before once again becoming positively-correlated with the activity changes at very high frequencies. We have also discovered that the two frequencies where the sensitivities of the temporal frequency shifts change sign also change in phase with the average level of solar activity.

We present a study of the temporal changes in the sensitivities of the frequencies and widths of the solar p-mode oscillations to corresponding changes in the levels of solar activity during Solar Cycle 23. From MDI and GONG++ full-disk Dopplergram three-day time series obtained between 1996 and 2008 we have computed a total of 221 sets of m-averaged power spectra for spherical harmonic degrees ranging up to 1000. We have then fit these 221 sets of m-averaged power spectra using our WMLTP fitting code and both symmetric Lorentzian profiles for the peaks as well as the asymmetric profile of Nigam and Kosovichev to obtain 442 tables of p-mode parameters. We then inter-compared these 442 tables which comprise in excess of 5.3 million p-mode parameters, and we performed linear regression analyses of the differences in p-mode frequencies and widths as functions of the differences in as many as ten different solar activity indices. From these linear regression analyses we have discovered new signatures of the frequency shifts of the p-modes and a similar, but slightly different, signature of the temporal shifts in the widths of the oscillations.