Inaugural Issue: Foundations & Frontiers

We give explicit counterexamples to two questions. One is asked by Pogorelov and the other is by Toponogov, at least both are stated in. These questions concern the existence of closed asymptotic curves in a saddle surface, namely a complete immersed regular surface in R^3 with nonpositive Gaussian/sectional curvature, and its geometric consequences under some topological conditions. We also modify the statements and prove a corrected version. In the appendix we include an example clarifying a conjecture of Milnor.

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A hole is an induced cycle of length at least four, and a cap is a hole together with a vertex adjacent to exactly two consecutive vertices of the hole. We prove that if q2 and G is a (cap,even hole)-free graph with no odd hole of length at most 2q-1, then _f(G)2q+12q(G). The bound is best possible, as shown by clique blowups of C_2q+1.

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Burde proved that the Lie algebra sl_2(K), over a field K of characteristic p > 2, admits compatible left-symmetric algebra structures if and only if p = 3. Since the even part of the Lie superalgebra osp(1|2) is isomorphic to sl_2(K), we ask whether Burde's structures can be extended to compatible left-symmetric superalgebra structures on osp(1|2). We verify Burde's left-symmetric structures on sl_2(K), then test their possible extensions to osp(1|2). We show that no such extension exists in characteristic 3. Together with the obstruction coming from the even part for characteristic 0 and p > 3, this gives non-existence for a field K of characteristic different from 2.

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Cox and Zhao showed that two classical families of orthogonal polynomials are naturally encoded in the coordinate ring A=C[x^1,uu^2=p(x)] of a hyperelliptic curve. The Legendre polynomials arise from the quotient A/A, where =u,ddx, while the Chebyshev polynomials describe the group of units of A. Their results are obtained in the broader context of universal central extensions for arbitrary branch data. We provide a self-contained treatment of these two phenomena in the three smallest cases, using only elementary linear algebra. We also compare the resulting dimensions with the topological quantity 2g+r-1 via the map dx/u. We record two observations that, to our knowledge, have not appeared previously: the dimension of A/A remains unchanged at parameter values where the curve develops a node, and the classical quadratic relation between the generating functions of the two polynomial families arises from a single polynomial relation---the defining equation of the curve in one realization and a Pell-type norm form in the other.

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