<div dir="ltr"><p class="" style="text-align:center"><span style="font-size:18pt;font-family:'Times New Roman',serif">SEMINARIO
DI MATEMATICA</span></p>
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<p class="MsoNormal" align="center" style="margin-left:9pt;text-align:center;line-height:18pt"><span style="font-size:16pt;font-family:'Times New Roman',serif">Martedì 31 marzo 2015</span><span style="font-size:16pt;font-family:'Times New Roman',serif"></span></p>
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<p class="MsoNormal" align="center" style="margin-left:9pt;text-align:center;line-height:18pt"><u><span style="font-size:16pt;font-family:'Times New Roman',serif">Scuola Normale Superiore</span></u></p>
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<pre style="text-align:center"><b><span style="font-size:22pt;font-family:'Times New Roman',serif;background-image:initial;background-repeat:initial">Fabrizio Catanese</span></b></pre><pre style="text-align:center"><i><span style="font-size:12pt;font-family:'Times New Roman',serif">University of Bayreuth</span></i></pre><pre style="text-align:center"><i><span style="font-size:12pt;font-family:'Times New Roman',serif;background-image:initial;background-repeat:initial"> </span></i></pre>
<p class="MsoNormal" align="center" style="text-align:center"><span style="font-size:16pt;font-family:'Times New Roman',serif">Terrà un
seminario dal titolo:</span></p>
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<p class="MsoNormal" align="center" style="text-align:center"><b><span lang="EN-US" style="font-size:20pt;font-family:'Times New Roman',serif">“<span style="background-image:initial;background-repeat:initial">Higher
dimensional lemniscates: m particles in n-space with logarithmic potentials via
complex analysis</span>”</span></b></p>
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<p class="MsoNormal" style="text-align:justify"><b><span lang="EN-US" style="font-family:'Times New Roman',serif">Abstract:</span></b></p>
<p class=""><i><span lang="EN-GB" style="font-size:11pt;font-family:'Times New Roman',serif;background-image:initial;background-repeat:initial">What is a lemniscate? Given m points w_j
in n-dimensional space, consider the function F(x) which is the product of the
distances |x-w_j|.</span></i><i><span lang="EN-GB" style="font-size:11pt;font-family:'Times New Roman',serif"> <span style="background-image:initial;background-repeat:initial">A lemniscate is defined to
be a singular level set of the function F, and the configuration of
lemniscates is said to be generic if f= log(F)</span></span> <span style="background-image:initial;background-repeat:initial">is a global Morse function.</span> <span style="background-image:initial;background-repeat:initial">In my work with Paluszny 25 years ago we
completely classified the possible lemniscate configurations in the plane,
showing that there is a bijection with</span> <span style="background-image:initial;background-repeat:initial">the set of connected components (`chambers') of the open set of
the configuration space which parametrizes the Morse functions.</span>
<span style="background-image:initial;background-repeat:initial">And we showed that the number of configurations
leads to the generating function (1-sin t)^{-1}, the same which was found
by Arnold for</span> <span style="background-image:initial;background-repeat:initial">Morse functions in
one real variable.</span> <span style="background-image:initial;background-repeat:initial">With Ingrid
Bauer we extended the results to Riemann surfaces of higher genus, using the
tight connection with Riemann's existence theorem,</span> <span style="background-image:initial;background-repeat:initial">and braid group actions.</span> <span style="background-image:initial;background-repeat:initial">Recently, together with Bauer and Di Scala, we made a
breakthrough in higher dimensions, showing via elementary complex analysis</span>
<span style="background-image:initial;background-repeat:initial">that the critical points of F have Hessian of
positivity at least</span> <span style="background-image:initial;background-repeat:initial">(n-1). This
implies that if F is a Morse function, then it has only local minima and
saddle points with negativity 1.</span> <span style="background-image:initial;background-repeat:initial">The
critical points lie in the convex span of the points w_j (these are absolute
minima): but we made the discover that F can also have</span> <span style="background-image:initial;background-repeat:initial">other local minima (we show this by using symmetry and
symmetry breaking).</span> <span style="background-image:initial;background-repeat:initial">The chambers
in configuration space are divided by two type of walls, those of quantitative
type (where the number of local minima changes),</span> <span style="background-image:initial;background-repeat:initial">and those of qualitative type (where the configuration
may change).</span> <span style="background-image:initial;background-repeat:initial">Major open questions
are: bounding effectively the number of minima in terms of the dimension n and
the number of points m, </span><span style="background-image:initial;background-repeat:initial">finding
generating functions for the number of configurations, and the number of
chambers in configuration space. </span><span style="background-image:initial;background-repeat:initial">The study of analogous questions with different
potentials is also an interesting question.</span></i><i><span lang="EN-US" style="font-size:11pt;font-family:'Times New Roman',serif"></span></i></p>
<p class=""><font face="Times New Roman, serif"><span style="font-size:14.6666669845581px"><b>Tutti gli interessati sono invitati a partecipare.</b></span></font></p><p class=""><font face="Times New Roman, serif"><span style="font-size:14.6666669845581px"><b><br></b></span></font></p><div><div class="gmail_signature"><div dir="ltr"><div>Valeria Giuliani</div><div>Scuola Normale Superiore</div><div>Servizio alla Didattica e Allievi</div><div>tel. 050 509260</div><div>Piazza dei Cavalieri, 7</div><div>56126 Pisa</div><div>E-mail: <a href="mailto:valeria.giuliani@sns.it" target="_blank">valeria.giuliani@sns.it</a></div><div>E-mail: <a href="mailto:classi@sns.it" target="_blank">classi@sns.it</a></div></div></div></div>
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