A brief survey of symmetry in mathematics

This paper presents a brief survey of the idea of symmetry in mathematics, as exemplified by some particular developments in algebra, differential equations, topology, and number theory.

Regulation of number and size of digits by posterior Hox genes: A dose-dependent mechanism with potential evolutionary implications

The proper development of digits, in tetrapods, requires the activity of several genes of the HoxA and HoxD homeobox gene complexes. By using a variety of loss-of-function alleles involving the five Hox genes that have been described to affect digit patterning, we report here that the group 11, 12, and 13 genes control both the size and number of murine digits in a dose-dependent fashion, rather than through a Hox code involving differential qualitative functions. A similar dose–response is observed in the morphogenesis of the penian bone, the baculum, which further suggests that digits and external genitalia share this genetic control mechanism. A progressive reduction in the dose of Hox gene products led first to ectrodactyly, then to olygodactyly and adactyly. Interestingly, this transition between the pentadactyl to the adactyl formula went through a step of polydactyly. We propose that in the distal appendage of polydactylous short-digited ancestral tetrapods, such as Acanthostega, the HoxA complex was predominantly active. Subsequent recruitment of the HoxD complex contributed to both reductions in digit number and increase in digit length. Thus, transition through a polydactylous limb before reaching and stabilizing the pentadactyl pattern may have relied, at least in part, on asynchronous and independent changes in the regulation of HoxA and HoxD gene complexes.

The operator algebra approach to quantum groups

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.

Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

Randomness and degrees of irregularity.

The fundamental question "Are sequential data random?" arises in myriad contexts, often with severe data length constraints. Furthermore, there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness--e.g., to evaluate the efficacy of therapy in medicine. We address both these issues from a computable framework via a quantification of regularity. ApEn (approximate entropy), defining maximal randomness for sequences of arbitrary length, indicating the applicability to sequences as short as N = 5 points. An infinite sequence formulation of randomness is introduced that retains the operational (and computable) features of the finite case. In the infinite sequence setting, we indicate how the "foundational" definition of independence in probability theory, and the definition of normality in number theory, reduce to limit theorems without rates of convergence, from which we utilize ApEn to address rates of convergence (of a deficit from maximal randomness), refining the aforementioned concepts in a computationally essential manner. Representative applications among many are indicated to assess (i) random number generation output; (ii) well-shuffled arrangements; and (iii) (the quality of) bootstrap replicates.

Organismal duplication, inclusive fitness theory, and altruism: understanding the evolution of endosperm and the angiosperm reproductive syndrome.

For almost a century, events relating to the evolutionary origin of endosperm, a unique embryo-nourishing tissue that is essential to the reproductive process in flowering plants, have remained a mystery. Integration of recent advances in phylogenetic reconstruction, comparative reproductive biology, and genetic theory can be used to elucidate the evolutionary events and forces associated with the establishment of endosperm. Endosperm is shown to be derived from one of two embryos formed during a rudimentary process of "double fertilization" that evolved in the ancestors of angiosperms. Acquisition of embryo-nourishing behavior (with accompanying loss of individual fitness) by this supernumerary fertilization product was dependent upon compensatory gains in the inclusive fitness of related embryos. The result of the loss of individual fitness by one of the two original products of double fertilization was the establishment of endosperm, a highly modified embryo/organism that reproduces cryptically through behavior that enhances the fitness of its associated embryo within a seed. Finally, although triploid endosperm remains a synapomorphy of angiosperms, inclusive fitness analysis demonstrates that the embryo-nourishing properties of endosperm initially evolved in a diploid condition.