• Forming spaces by dragging a sphere cover by its branch points.
• A complete description of Schur (or exceptional covers) covers given by rational functions; a special case of a general problem.
• Schur cover descriptions as equivalent to Serre's Open Image Theorem.

2. AltGroups-LiftInvariants04-21-15: Geometry Talk 1: Alternating groups and Lift Invariants: The talk starts by finishiing computing the Geometric monodromy group of the moduli space covers of the j-line arising from prime (and prime-squared) degree exceptional covers.

• Inner versus absolute Hurwitz spaces, and their modular curve analog.
• Fried-Serre Lift Invariant applied to 3-cycle alternating group covers. Main Theorem: These spaces have one (resp. two) component(s) if r = n-1 (resp. r ≥ n). By improving a Fried-Serre result, we can explicitly recognize in which component a 3-cycle cover belongs.
• Harbater-Mumford components of Hurwitz spaces, and appearance of even and odd theta functions.

3. GoalsForAnOIT04-21-15: Geometry Talk 2: Goals for an OIT: Based on a a prime p, a p-perfect group G and conjugacy classes C for which C is p', we form a sequence of spaces. We conjecture for them properties like those of modular curves.

• Main Conjecture: High tower levels should have general type and have no points over a given number field K.
• We explain the towers for the Alternating group of Talk 1. Then, we show how the conjecture – known for r = 4 – reveals the difficulty in the Inverse Galois problem.
• The towers should have geometric monodromy over the j-line that is eventually p-Frattini. This translates to a weak version of Serre's OIT.
• We develop a case based on three groups, listing the connected components in this case. As expected from interpreting a statement of Grothendieck from the '60s, this requires new ideas to organize how an OIT theorem might work.
• We outline how the collection group idea works in this case to decide if the p-Frattini property holds.