Quantitative results on quadratic semi-algebraic sets

Quantitative semi-algebraic geometry studies accurate bounds on topological invariants (such as the Betti numbers) of semi algebraic sets in terms of the number of equations, their degree and their number of variables. For general semialgebric sets, these bounds have an exponential dependance in the number of variables. In contrast, for semi-algebraic sets defined by quadratic equation, the dependance is polynomial in the number of variables. The talk will include a survey of the main results known for general semi-algebraic sets before concentrating on the quadratic case. The lecture will use material from joint work with Saugata Basu and Dimitri Pasechnik.