An Experimental Road to Mathematical Truth/TITLE> </HEAD> <BODY BGCOLOR=#FFFFFF>  <A HREF="7.html"><IMG SRC="images/back.gif"></A> <A HREF="0.html"><IMG SRC="images/up.gif"></A> <font size=+3><B>The Shapiro Conjecture and Beyond</B></FONT>  <HR> <TABLE> <TR valign=top> <TD> <font size=+2> The Shapiro conjecuture generalizes this example, and it concerns all questions of enumerating linear subspaces that are incident on other, fixed subspaces in any vector space; If the fixed subspaces are chosen tangent to a normal curve, then all solutions will be real.<br> When it was first made, this Shapiro conjecture seemed to good to be true (it is in fact false in its original generality). Nevertheless, proofs of special cases and a few <font color=red>million</font> computatons together yielded overwhelming evidence for it.<br> Work of Eremenko and Gabrielov, as well as Mukhin, Tarasov, and Varchenko (three papers in the Annals of Mathematics and JAMS) eventually proved the conjecture in the generality it could be true.<br> Nevertheless, there are several appealing generalizations and corrections which are open. These have been formulated and studied by my research team over the past decade; These projects have used over <font color=red>1.2 teraHertz-years</font> of computation, looking at over <font color=red>2.4 billion cases</font>. This large-scale experimentation has involved 12 different people, including undergraduate students. </TD> </TR> </TABLE>  <HR> </BODY> </HTML>