A very recent example is Eric Larson's 2018 dissertation *The maximal rank conjecture* [Lar1], which proves the following old conjecture:

**Conjecture.** (Maximal rank conjecture) Let $C \subseteq \mathbb P^r$ be a general Brill-Noether¹ curve. Then the restriction map
$$H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(k)) \to H^0(C, \mathcal O_C(k))$$
has maximal rank, i.e. is injective if $h^0(\mathbb P^r, \mathcal O(k)) \leq h^0(C, \mathcal O(k))$ and surjective otherwise.

**Historical remarks.**
Although I have been unable to find a definite place where this conjecture was stated, it is attributed to M. Noether by Arbarello and Ciliberto [AC83, p. 4]. Cases of the problem have been studied by M. Noether [Noe82, §8], Castelnuovo [Cas93, §7], and Severi [Sev15, §10].

In modern days, the conjecture had regained attention by 1982 [Har82, p. 79]. Partial results from around that time are mentioned in the introduction to [Lar2].

Larson's work culminates a lot of activity, including many papers by himself with other authors. An overview of the proof and how the different papers fit together is given in [Lar3].

**References.**

[AC83] E. Arbarello and C. Ciliberto, *Adjoint hypersurfaces to curves in $\mathbb P^n$ following Petri*. In: *Commutative algebra (Trento, 1981)*. Lect. Notes Pure Appl. Math. **84** (1983), p. 1-21. ZBL0516.14024.

[Cas93] G. Castelnuovo, *Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica*. Palermo Rend. VII (1893), p. 89-110. ZBL25.1035.02.

[Har82] J. D. Harris, *Curves in projective space*. Séminaire de mathématiques supérieures, **85** (1982). Les Presses de l’Université de Montréal. ZBL0511.14014.

[Lar1] E. K. Larson, *The maximal rank conjecture*. PhD dissertation, 2018.

[Lar2] E. K. Larson, *The maximal rank conjecture*. Preprint, arXiv:1711.04906.

[Lar3] E. K. Larson, *Degenerations of Curves in Projective Space and the Maximal Rank Conjecture*. Preprint, arXiv:1809.05980.

[Noe82] M. Nöther, *Zur Grundlegung der Theorie der algebraischen Raumcurven*. Abh. d. K. Akad. d. Wissensch. Berlin (1882). ZBL15.0684.01.

[Sev15] F. Severi, *Sulla classificazione delle curve algebriche e sul teorema d’esistenza di Riemann*. Rom. Acc. L. Rend. **24**.5 (1915), p. 877-888, 1011-1020. ZBL45.1375.02.

¹ Brill-Noether curves form a suitable component of the Kontsevich moduli space $\overline M_g(\mathbb P^r, d)$ of stable maps $\phi \colon C \to \mathbb P^r$ from a genus $g$ curve whose image has degree $d$.

allPhD theses solve open problems...? $\endgroup$ – Najib Idrissi Apr 24 '18 at 7:42a precise mathematical question that at least some researcher(s) have previously articulated and tried to answer, e.g. “what is the dimension of such-and-such space?”. Most theses I know only solve open problems only in a much broader sense: open-ended questions that researchers may have wondered about, like “what can we say about the homology of such-and-such space?”, or “can we develop a useful theory of homotopy-coherent diagrams in such-and-such setting?”. $\endgroup$ – Peter LeFanu Lumsdaine Apr 24 '18 at 9:04