### Lecture notes

Lecture 1 Catalog of Problems

Lecture 2 Mathematical Formulation

Lecture 3 Existence

Lecture 4 Optimal Regularity

Lecture 5 ACF-type Monotonicity Formulas

Lecture 6 Optimal Regularity (Continued)

Lecture 7 Optimal Regularity (Continued)

Lecture 8 Nondegeneracy and (n-1)-Hausdorff measure

Lecture 9 Normalized solutions, Rescalings and Blowups

Lecture 10 Weiss's Monotonicity Formulas

Lecture 11 Classification of Free Boundary Points

Lecture 12 First Results on the Regularity of the Free Boundary: A

Lecture 13 First Results on the Regularity of the Free Boundary: B

Lecture 14 First Results on the Regularity of the Free Boundary: C

Lecture 15 Global Solutions

Lecture 16 Approximation by Global Solutions

Lecture 17 Lipschitz Regularity of the Free Boundary

Lecture 18

Lecture 19

Lecture 20 Higher Regularity: Problems A and B draft

Lecture 21 The Singular Set draft

Lecture 2 Mathematical Formulation

Lecture 3 Existence

Lecture 4 Optimal Regularity

Lecture 5 ACF-type Monotonicity Formulas

Lecture 6 Optimal Regularity (Continued)

Lecture 7 Optimal Regularity (Continued)

Lecture 8 Nondegeneracy and (n-1)-Hausdorff measure

Lecture 9 Normalized solutions, Rescalings and Blowups

Lecture 10 Weiss's Monotonicity Formulas

Lecture 11 Classification of Free Boundary Points

Lecture 12 First Results on the Regularity of the Free Boundary: A

Lecture 13 First Results on the Regularity of the Free Boundary: B

Lecture 14 First Results on the Regularity of the Free Boundary: C

Lecture 15 Global Solutions

Lecture 16 Approximation by Global Solutions

Lecture 17 Lipschitz Regularity of the Free Boundary

Lecture 18

*C*^{1,\alpha}Regularity of the Free Boundary: Problems A, B updatedLecture 19

*C*^{1}Regularity of the Free Boundary: Problems CLecture 20 Higher Regularity: Problems A and B draft

Lecture 21 The Singular Set draft