The shortlist

1
Matrix positivity preservers in fixed dimension. I (pdf) Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai Adv. Math. 298 (2016), 325-368 MR3505743   MSC:47B65 (15B48,26C05,62H99,65F15)
4
Knots, BPS states, and algebraic curves (pdf) Garoufalidis, Stavros; Kucharski, Piotr; Sulkowski, Piotr Comm. Math. Phys. 346 (2016), no. 1, 75-113 MR3528417   MSC:57M27 (11B65,14H81,33C90,57R56)
5
Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry (pdf) Müller, Stefan; Feliu, Elisenda; Regensburger, Georg; Conradi, Carsten; Shiu, Anne; Dickenstein, Alicia Found. Comput. Math. 16 (2016), no. 1, 69-97 MR3451424   MSC:92E20 (12D10,34A34,37C10,52C40,80A30)
7
On rational connectedness of globally F-regular threefolds (pdf) Gongyo, Yoshinori; Li, Zhiyuan; Patakfalvi, Zsolt; Schwede, Karl; Tanaka, Hiromu; Zong, Runhong Adv. Math. 280 (2015), 47-78 MR3350212   MSC:14E30 (13A35,14D06,14M22)
11
The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients (pdf) Hofmann, Steve; Kenig, Carlos; Mayboroda, Svitlana; Pipher, Jill Math. Ann. 361 (2015), no. 3-4, 863-907 MR3319551   MSC:35J25 (35B30,35B65)
17
Free energy fluctuations for directed polymers in random media in $1+1$ dimension (pdf) Borodin, Alexei; Corwin, Ivan; Ferrari, Patrik Comm. Pure Appl. Math. 67 (2014), no. 7, 1129-1214 MR3207195   MSC:82D10 (60B20,60H15,60K35,60K37)
18
A case of the dynamical Andre-Oort conjecture (pdf) Ghioca, Dragos; Krieger, Holly; Nguyen, Khoa Int. Math. Res. Not. IMRN 2016, no. 3, 738-758 MR3493432   MSC:37P55 (14H45)
19
Rank-finiteness for modular categories (pdf) Bruillard, Paul; Ng, Siu-Hung; Rowell, Eric C.; Wang, Zhenghan J. Amer. Math. Soc. 29 (2016), no. 3, 857-881 MR3486174   MSC:18D10 (57R56,81R50)
23
Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves (pdf) Bhargava, Manjul; Kane, Daniel M.; Lenstra, Hendrik W., Jr.; Poonen, Bjorn; Rains, Eric Camb. J. Math. 3 (2015), no. 3, 275-321 MR3393023   MSC:11G05 (14G25)