Moduli spaces of stable maps to projective space are not equidimensional in higher genus. Work of J. Li, R. Vakil and A. Zinger led to a desingularization of the main component (the generic element of which represents a map from a smooth elliptic curve) and the introduction of reduced invariants for complete intersections. For threefolds these are related to ordinary Gromov-Witten invariants by the Li-Zinger's formula; they have a better enumerative meaning in the sense that they discard a degenerate contribution from rational curves. More recently M. Viscardi has introduced smaller compactifications by allowing maps from Smyth's singularities, e.g. cusps. In a joint work with F. Carocci and C. Manolache we show that reduced and cuspidal invariants coincide for the quintic threefold. We use p-fields and local equations in order to split the intrinsic cone, adapting techniques of H-L. Chang, Y. Hu and J. Li.
g=1 reduced invariants of the quintic threefold from maps with cusps Sponsored by the Meyer Fund
We define a general class of sphere packings, and study the subclass of "superintegral" such. By connecting these to the theory of hyperbolic arithmetic reflection lattices, we prove that they only exist in finitely many dimensions, and in fact in finitely many commensurability classes, in principle allowing for a complete classification.
Sphere Packings and Arithmetic - note unusual time and place!
Mar. 06, 2018 2pm (MATH 350)
Lie Theory
Spencer Gerhardt (CSU)
X
Let be conjugacy classes of the algebraic group defined over a sufficiently large field . We consider the problem of determining necessary and sufficient conditions for the existence of such that is Zariski dense in . Related questions are considered for other simple algebraic groups, and applications to generic stabilizers in linear representations of algebraic groups and random generation of finite groups of Lie type are discussed.
Topological generation of algebraic groups and applications Sponsored by the Meyer Fund
g=1 reduced invariants of the quintic threefold from maps with cusps