[CvGmt News] avviso Seminario di Matematica prof. Fabrizio Catanese (31.03.2015)

[Valeria Giuliani]

SEMINARIO DI MATEMATICA



Martedì 31 marzo 2015

ore 16:00



*Scuola Normale Superiore*

Pisa

Aula Mancini



*Fabrizio Catanese*

*University of Bayreuth*



Terrà un seminario dal titolo:



*“Higher dimensional lemniscates: m particles in n-space with logarithmic
potentials via complex analysis”*



*Abstract:*

*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|.** 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) is a
global Morse function. 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 the set of connected components (`chambers')
of  the open set of the  configuration space which parametrizes the  Morse
functions. 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
Morse functions in one real variable. With Ingrid Bauer we extended the
results to Riemann surfaces of higher genus, using the tight connection
with Riemann's existence theorem, and braid group actions. Recently,
together with Bauer and Di Scala, we made a breakthrough in higher
dimensions, showing via elementary complex analysis that the critical
points of F have Hessian of positivity at least (n-1). This implies that
if  F is a Morse function, then it has only local minima and saddle points
with negativity 1. 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 other local minima (we show this by using symmetry and symmetry
breaking). The chambers in configuration space are divided by two type of
walls, those of quantitative type (where the number of local minima
changes), and those of qualitative type (where the configuration may
change). Major open questions are: bounding effectively the number of
minima in terms of the dimension n and the number of points m, finding
generating functions for the number of configurations, and the number of
chambers in configuration space. The study of analogous questions with
different potentials is also an interesting  question.*

*Tutti gli interessati sono invitati a partecipare.*


Valeria Giuliani
Scuola Normale Superiore
Servizio alla Didattica e Allievi
tel. 050 509260
Piazza dei Cavalieri, 7
56126 Pisa
E-mail: valeria.giuliani at sns.it
E-mail: classi at sns.it
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