Barriers to show $P=NP$
Mihalis Yannakakis has shown that the traveling salesman problem cannot be solved in polynomial time by using a symmetric linear program.

See the paper Expressing combinatorial optimization problems by Linear Programs, by Yannakakis.

This result was improved recently by  Fiorini,  Massar, Pokutta,  Tiwary, and De Wolf to drop the "symmetric" requirement in Yannakakis' result.