I will discuss my ongoing project towards a version of the Modular Andre-Oort Conjecture incorporating the derivatives of the j function.  The work originates with Jonathan Pila, who formulated the first "Modular Andre-Oort with Derivatives" conjecture.  The problem can be approached via o-minimality; I will discuss two categories of result.  The first is a weakened version of Jonathan's conjecture.  Under an algebraic independence conjecture (of my own, though it follows from standard conjectures), the result is equivalent to the statement that Jonathan's conjecture holds.  
The second result is conditional on the same algebraic independence conjecture - it specifies more precisely how the special points in varieties can occur in this context.  
If time permits, I will discuss my most recent work towards making the two results uniform in algebraic families.

Several optimization problems combine nonlinear constraints with the integrality of a subset of variables. For an important class of problems  called Mixed Integer Second-Order Cone Optimization (MISOCO), with applications in facility location, robust optimization, and finance, among others, these nonlinear constraints are second-order (or Lorentz) cones.

For such problems, as for many discrete optimization problems, it is crucial to understand the properties of the union of two disjoint sets of feasible solutions. To this end, we apply the disjunctive programming paradigm to MISOCO and present conditions under which the convex hull of two disjoint sets can be obtained by intersecting the feasible set with a specially constructed second-order cone. Computational results show that such cone has a positive impact on the solution of MISOCO problems.

  Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with even first Betti number. However, not all solutions of the Einstein-Maxwell equations on such manifolds arise in this way. In this lecture, I will describe a construction of new compact examples that are Hermitian, but not Kaehler.