In 1992, Xiao-Song Lin constructed an invariant h of knots in the 3-sphere via a signed count of the conjugacy classes of irreducible SU(2)-representations of the fundamental group of the knot exterior with trace-free meridians. Lin showed that h equals one-half times the knot signature. Using methods similar to Lin's, we construct an invariant of two-component links in the 3-sphere. Our invariant is a signed count of conjugacy classes of projective SU(2)-representations of the fundamental group of the link exterior with a fixed 2-cocycle and corresponding non-trivial second Stiefel--Whitney class. We show that our invariant is, up to a sign, the linking number. We further construct, for a two-component link in an integral homology sphere, an instanton Floer homology whose Euler characteristic is, up to sign, the linking number between the components of the link. We relate this Floer homology to the Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for two-component links in the 3-sphere, the Floer homology does not vanish unless the link is split.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

Despite having similar brains, related species can display divergent behaviors. Investigating the neural basis of such behavioral divergence can elucidate the neural mechanisms that allow behavioral change and identify neural mechanisms that influence the evolution of behavior. Fewer than three percent of Nudipleura (Mollusca, Opisthobranchia, Gastropoda) species have been documented to swim. However, Tritonia diomedea and Pleurobranchaea californica express analogous, independently evolved swim behaviors consisting of rhythmic, alternating dorsal and ventral flexions. The Tritonia and Pleurobranchaea swims are produced by central pattern generator (CPG) circuits containing homologous neurons named DSI and C2. Homologues of DSI have been identified throughout the Nudipleura, including in species that do not express a dorsal-ventral swim. It is unclear what neural mechanisms allow Tritonia and Pleurobranchaea to produce a rhythmic swim behavior using homologous neurons that are not rhythmic in the majority of Nudipleura species. Here, C2 homologues were also identified in species that do not express a dorsal-ventral swim. We found that certain electrophysiological properties of the DSI and C2 homologues were similar regardless of swim behavior. However, some synaptic connections differed in the non-dorsal-ventral swimming Hermissenda crassicornis compared to Tritonia and Pleurobranchaea. This suggests that particular CPG synaptic connections may play a role in dorsal-ventral swim expression. DSI modulates the strength of C2 synapses in Tritonia, and this serotonergic modulation appears to be necessary for Tritonia to swim. DSI modulation of C2 synapses was also found to be present in Pleurobranchaea. Moreover, serotonergic modulation was necessary for swimming in Pleurobranchaea. The extent of this neuromodulation also correlated with the swimming ability in individual Pleurobranchaea; as the modulatory effect increased, so too did the number of swim cycles produced. Conversely, DSI did not modulate the amplitude of C2 synapses in Hermissenda. This indicates that species differences in neuromodulation may account for the ability to produce a dorsal-ventral swim. The results indicate that differences in synaptic connections and neuromodulatory dynamics allow the expression of rhythmic swim behavior from homologous non-rhythmic components. Additionally, the results suggest that constraints on the nervous system may influence the neural mechanisms and behaviors that can evolve from homologous neural components.

Invariants of involutions on central simple algebras have been extensively studied. Many important results have been collected and extended by Knus, Merkurjev, Rost and Tignol in "The Book of Involutions" [BI]. Among those invariants are, for example, the (even) Clifford algebra for involutions of the first kind and the discriminant algebra for involutions of the second kind on an algebra of even degree. In his preprint "Triality, Cocycles, Crossed Products, Involutions, Clifford Algebras and Invariants" [S05], Saltman shows that the definition of the Clifford algebra can be generalized to Azumaya algebras and introduces a special cohomology, the so-called G-H cohomology, to describe its structure. In this dissertation, we prove analogous results about the discriminant algebra D(A; [tau]), which is the algebra of invariants under a special automorphism of order two of the [lambda]-power of an algebra A of even degree n = 2m with involution of the second kind, [tau]. In particular, we generalize its construction to the Azumaya case. We identify the exterior power algebra as defined in "Exterior Powers of Fields and Subfields" [S83] as a splitting subalgebra of the m-th [lambda]-power algebra and prove that a certain invariant subalgebra is a splitting subalgebra of the discriminant algebra. Assuming well-situatedness we show how this splitting subalgebra can be described as the fixed field of an S[subscript n] x C₂- Galois extension and that the corresponding subgroup is [Sigma] = S[subscript m] x S[subscript m] [mathematic symbol] C2. We give an explicit description of the corestriction map and define a lattice E that encodes the corestriction as being trivial. Lattice methods and cohomological tools are applied in order to define the group H²(G;E) which contains the cocycle that will describe the discriminant algebra as a crossed product. We compute this group to have order four and conjecture that it is the Klein 4-group and that the mixed element is the desired cocycle. / text

Not available / text

We define the symplectic rational blow-up operation, for a family of rational homology balls \(B_n\), which appeared in Fintushel and Stern's rational blow-down construction. We do this by exhibiting a symplectic structure on a rational homology ball \(B_n\) as a standard symplectic neighborhood of a certain 2-dimensional Lagrangian cell complex. We also study the obstructions to symplectically rationally blowing up a symplectic 4-manifold, i.e. the obstructions to symplectically embedding the rational homology balls \(B_n\) into a symplectic 4-manifold. First, we present a couple of results which illustrate the relative ease with which these rational homology balls can be smoothly embedded into a smooth 4-manifold. Second, we prove a theorem and give additional examples which suggest that in order to symplectically embed the rational homology balls \(B_n\), for high \(n\), a symplectic 4-manifold must at least have a high enough \(c^2_1\) as well. / Mathematics

No description available.

No description available.

No description available.

No description available.