It is simply a matter of getting the date objects in a sensible order, getting the labels on the right date objects, and removing the option AxesOrigin -> Center.
TimelinePlot[{
Interval[{DateObject[{2015, 1, 15}], DateObject[{2015, 9, 9}]}],
DateObject[{2015, 6, 1}] ->
Row[{"Start ", displayLaTeX["\\sum_{i=0}^{10} f(x_i)"]}],
Interval[{DateObject[{2015, 6, 1}], DateObject[{2016, 2, 29}]}],
DateObject[{2016, 3, 6}] -> "End ",
Interval[{DateObject[{2016, 7, 27}], DateObject[{2016, 8, 6}]}]},
PlotTheme -> "Classic"]
