▁be ▁seen ▁as ▁ 2 3 , ▁each ▁tang ency ▁condition ▁impos ing ▁a ▁quadratic ▁condition ▁on ▁the ▁space ▁of ▁circles . ▁ ▁However , ▁for ▁special ▁arrang ements ▁of ▁the ▁given ▁circles , ▁the ▁number ▁of ▁solutions ▁may ▁also ▁be ▁any ▁integer ▁from ▁ 0 ▁( no ▁solutions ) ▁to ▁six ; ▁there ▁is ▁no ▁arrangement ▁for ▁which ▁there ▁are ▁seven ▁solutions ▁to ▁Ap oll on ius ' ▁problem . ▁ ▁Key ▁tools ▁A ▁number ▁of ▁tools , ▁r anging ▁from ▁the ▁elementary ▁to ▁the ▁more ▁advanced , ▁include : ▁ ▁Dim ension ▁counting ▁ ▁Bé z out ' s ▁theorem ▁ ▁Sch ub ert ▁calculus , ▁and ▁more ▁generally ▁characteristic ▁classes ▁in ▁coh om ology ▁ ▁The ▁connection ▁of ▁counting ▁inter sections ▁with ▁coh om ology ▁is ▁Po inc ar é ▁du ality ▁ ▁The ▁study ▁of ▁mod uli ▁spaces ▁of ▁curves , ▁maps ▁and ▁other ▁geometric ▁objects , ▁sometimes ▁via ▁the ▁theory ▁of ▁quantum ▁coh om ology . ▁The ▁study ▁of ▁quantum ▁coh om ology , ▁G rom ov – W itten ▁invari ants ▁and ▁mirror ▁symmetry ▁gave ▁a ▁significant ▁progress ▁in ▁C lem ens ▁conject ure . ▁ ▁En umer ative ▁geometry ▁is ▁very ▁closely ▁tied ▁to ▁intersection ▁theory . ▁ ▁Sch ub ert ▁calculus ▁En umer ative ▁geometry ▁saw ▁spect ac ular ▁development ▁towards ▁the ▁end ▁of ▁the ▁nin ete enth ▁century , ▁at ▁the ▁hands ▁of ▁Hermann ▁Sch ub ert . ▁He ▁introduced ▁for ▁the ▁purpose ▁the ▁Sch ub ert ▁calculus , ▁which ▁has ▁proved ▁of ▁fundamental