In this paper, we investigate the exactness of the Grassmannian Bernstein-Gel’fand-Gel’fand complexes introduced in a previous work of the author, and obtain some inequalities between some Hodge numbers of some irregular varieties. In particular, we obtain sharp lower bounds for the Hodge numbers of smooth subvarieties of Abelian varieties, as well as some improvements of results of Lazarsf…

Let A be an m × m complex matrix with zero trace. Then there are m × m matrices B and C such that A = [B,C] and |B||C|2 <= (logm + O(1))1/2|A|2 where |D| is the norm of D as an operator on m2 and |D|2 is the Hilbert–Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely, there is a zero trace m × m matrix A such that whenever A = [B,C], |B||C|2 >= | logm − …

Genomic approaches are revolutionizing the biomedical sciences, providing new ways in which to approach the development of therapeutic practices and the understanding of the causal bases of disease. So, too, do genomics approaches have the potential to transform the way in which ecological studies can be conducted by providing powerful tools in which genotype and phenotype data of diverse “wi…

The ethical position underpinning decision making is an important concern for conservation biologists when setting priorities for interventions. The recent debate on how best to protect nature has centered on contrasting intrinsic and aesthetic values against utilitarian and economic values, driven by an inevitable global rise in conservation conflicts. These discussions have primarily been tar…

Society’s growing demand for clean and abundant energy has repercussions for biodiversity and human well-being. Directives for renewable energy, energy security, and technological advancements such as horizontal drilling in conjunction with hydraulic fracturing have spurred a rapid increase in alternative and unconventional energy production over the last decade. Given the projected increases…

We discuss definability of henselian valuation rings in the Macintyre language LMac, the language of rings expanded by nth power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly ∃-∅-definable in LMac, and henselian valuation rings with value group Z are uniformly ∃∀-∅-definable in the ring langu…

We correct an error in the proof of Lemma 2.4.

It is known that in the complex case the Fatou set of a polynomial has no wandering components. This is not true in the non-Archimedean case; there exist examples of polynomials having a wandering Fatou component. These examples are related to a phenomenon called wild ramification, a property that does not occur in the complex numbers. In this work, we will use combinatorics related to the Fib…

We construct two infinite sequences of immersions of the 3-sphere into 4-space, parameterized by the Dynkin diagrams of types A and D. The construction is based on immersions of 4-manifolds obtained as the plumbed immersions along the weighted Dynkin diagrams. We compute their Smale invariants and bordism classes of immersions by using Ekholm–Takase’s formula in terms of singular Seifert su…

Colleges and universities need to provide students with training and experience in (a) interdisciplinary research, (b) the fuzzy areas of responsible conduct of research, and (c) the mentor–mentee relationship. We developed workshops that combine the three content objectives— interdisciplinary research, responsible conduct of research, and mentor–mentee relationship, therefore promoting e…