We show that smooth curves in the same biliaison class on a hypersurface in ${\mathbf{P}}^3$ with ordinary singularities are linearly equivalent. We compute the invariants $h^0({\mathcal{I}}_C(d))$, $h^1({\mathcal{I}}_C(d))$, and $h^1({\mathcal{O}}_C(d))$ of a curve C on such a surface X in terms of the cohomologies of divisors on the normalization of X. We then study general projections in ${\mathbf{P}}^3$ of curves lying on the rational normal scroll $S(a,b)\subset {\mathbf{P}}^{a+b+1}$. If we vary the curves in a linear system on $S(a,b)$ as well as the projections, we obtain a family of curves in ${\mathbf{P}}^3$. We compute the dimension of the space of deformations of these curves in ${\mathbf{P}}^3$ as well as the dimension of the family. We show that the difference is a linear function in a and b which does not depend on the linear system. Finally, we classify maximal rank curves on ruled cubic surfaces in ${\mathbf{P}}^3$. We prove that the general projections of all but finitely many classes of projectively normal curves on $S(1,2)\subset {\mathbf{P}}^4$ fail to have maximal rank in ${\mathbf{P}}^3$. These give infinitely many classes of counter-examples to a question of Hartshorne.

The Fourier–Mukai transform from algebraic geometry may be formulated in KK-theory as the map of composition with a certain topological correspondence in the sense of Connes and Skandalis. The goal of this note is to analyze this correspondence and to describe the induced map in terms of certain natural Baum–Douglas cycles and cocycles for tori. This leads to a purely geometric description of the Baum–Connes assembly map for free Abelian groups.

For a finite graph G, we study the maximum 2-edge colorable subgraph problem and a related ratio $\frac{\mathrm{\mu }(G)}{\mathrm{\nu }(G)}$, where $\mathrm{\nu }(G)$ is the matching number of G, and $\mathrm{\mu }(G)$ is the size of the largest matching in any pair $(H,H^{\prime })$ of disjoint matchings maximizing $|H|+|H^{\prime }|$ (equivalently, forming a maximum 2-edge colorable subgraph). Previously, it was shown that $\frac{4}{5}\le \frac{\mathrm{\mu }(G)}{\mathrm{\nu }(G)}\le 1$, and the class of graphs achieving $\frac{4}{5}$ was completely characterized. In this paper, we first show that graph decompositions into paths and even cycles provide a new way to study these parameters. We then use this technique to characterize the graphs achieving $\frac{\mathrm{\mu }(G)}{\mathrm{\nu }(G)}=1$ among all graphs that can be covered by a certain choice of a maximum matching and H, $H^{\prime }$ as above.

In this paper, I compute the regularity of the Rees algebra of binomial edge ideals of closed graphs. I obtain a lower bound for the regularity of the Rees algebra of binomial edge ideals. I also present some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. I obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.

We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups. Then we will see how these results together with the existing tools related to isometries can be applied to groups of dimension 4 and 5 in particular. Thus, we take steps toward determining all the equivalence classes of groups up to isometry and quasi-isometry. We completely solve the classification up to isometry for simply connected solvable groups in dimension 4 and for the subclass of groups of polynomial growth in dimension 5.