## Applied Analysis

Seminar information archive ～11/01｜Next seminar｜Future seminars 11/02～

Date, time & place | Thursday 16:00 - 17:30 002Room #002 (Graduate School of Math. Sci. Bldg.) |
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### 2013/06/06

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

The Geometry of Critical Points of Green functions On Tori (ENGLISH)

**Chang-Shou Lin**(National Taiwan University)The Geometry of Critical Points of Green functions On Tori (ENGLISH)

[ Abstract ]

The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.

We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.

We can generalize our results to a sum of two Green functions.

The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.

We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.

We can generalize our results to a sum of two Green functions.