From May 22^{nd} till May 26^{th} the 5^{th} International Conference on Elliptic and Parabolic Problems took place in Gaeta, Italy. With 17 invited talks and 184 short talks in 19 minisymposia distributed on four and a half days the amount of talks at the conference was quite impressive. Besides the more than 200 participants from all over the world there were six researchers from the CRC: Wolfgang Reichel who had the honour to give the opening talk; Janina Gärtner, Andreas Hirsch, Piotr Idzik, Rainer Mandel and Martin Spitz, all of them contributing by a short talk to the minisymposium “PDEs arising in nonlinear optics”. In this minisymposium consisting of 14 talks, which was organised by Jarosław Mederski and Wolfgang Reichel, we were kept up to date with the ongoing research of colleagues who are well-known to our working groups in Karlsruhe.

Apart from the “PDEs arising in nonlinear optics” the topics of the talks were widely spread (as the title of the conference suggests) and therefore we got in touch with subjects which are not closely related to ones own field of research. Parallel sessions and the conference programme reaching from 8:30 a.m. to 7:30 p.m. raised the problem of making an initial selection of talks. This was done with mixed success. For instance we learned that even though the title of a talk might look very interesting it can have nothing to do with its content, or, as the speaker admitted on his first slide: the title was only there for attracting undecided audience. As the conference proceeded we were able to figure out the talks which were worthwhile. Thus, finally we enjoyed some inspiring talks on Morse indices, uniqueness of the Lane-Emden problem, non-autonomous evolution equations and free boundary problems, to highlight only four of them. In addition to all the theoretical investigations we could also explore wave phenomena in practice: at the beach nearby the hotel.

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Our plenary speakers are

- Martina Chirilus-Bruckner (Univ. Leiden, Netherlands)
- Maria Lopez-Fernandez (La Sapienza Univ. Rome, Italy)
- Marie Rognes (Simula Research Laboratory, Norway)
- Claire Scheid (Côte d’Azur Univ., France)
- Birgit Schörkhuber (Univ. Vienna, Austria)
- Susanna Terracini (Univ. Turin, Italy)
- Chrysoula Tsogka (Univ. Crete, Greece)

Female participants in all stages of their career are invited to present posters on their research or current projects. Registration is free, but will be required for attendance.

More information can be found on the workshop’s webpage http://womeninpdes.waves.kit.edu Details on the program and news will be posted there.

]]>Consider the equation $$ \lambda x – x^3 = 0$$ where \(x,\lambda\) are real numbers. This equation has the feature that the trivial solution is always a solution, but there are nontrivial solutions as well! Factoring \(x\) out we find that all nontrivial solutions satisfy \(\lambda=x^2\). Hence we observe that nontrivial solutions bifurcate from \(\lambda^*=0\) which may be illustrated as follows:

Here it is comparatively easy to illustrate all solutions and the above picture, a so-called *bifurcation diagram*, shows why we speak of *bifurcations*. Notice that the latin word *furca* means *fork* (*Gabel* in German) so that a literal translation of *bifurcation* to German language would be *Zweigabelung*. However, mathematicians say *Verzweigung*. The reader may amuse him-/herself by replacing the exponent 3 by 4: How does the picture change?

The next equation represents one of the fundamental problems of linear algebra: Which are the eigenvalues of a given quadratic matrix \(A\)? In other words: Find the nontrivial solutions \((\lambda,x)\) of the equation $$ Ax – \lambda x=0.$$ After two semesters of linear algebra the solution is pretty clear: If \(\lambda\) is a real eigenvalue of the matrix then it is a bifurcation point. More precisely, if the geometric multiplicity is bigger than 2 then there is even a surface of nontrivial solutions (spanned by the eigenvectors) that bifurcates from the trivial solution. The bifurcation diagram below illustrates the case of 3 simple eigenvalues.

Example 1 and 2 show that the phenomenon occurs, but we have to clarify why it is worth a theory if we can calculate all nontrivial solutions by hand? The simple answer to that question is that the above examples are artificial and made for expository purposes. In most applications from analysis we have to deal with **nonlinear problems** in **infinite-dimensional (Banach) spaces**. Here, it is in general impossible to calculate explicit solutions that bifurcate from the trivial solution so that one has to use some nontrivial tools from nonlinear analysis and functional analysis that allow to treat such a problem. Such tools can be used, e.g., for the study of boundary value problems for partial and ordinary differential equations.

Finally, I would like to mention the pendulum equation $$\theta”(t) + \lambda \sin(\theta(t)) = 0,\qquad \theta(0)=\theta(T/2)=0.$$ Here \(\theta\) denotes the angle with respect to the vertical (rest) position and \(T\) is the oscillation period while the parameter lambda is proportional to the inverse of the length of the pendulum. In the first lectures of my course I will provide a method that allows to show that the bifurcation diagram is, qualitatively, the following:

Every point on some curve corresponds to a solution of the above boundary value problem and the vertical component measures its maximum. In particular we see that for any preassigned period we get more and more oscillating solutions the larger \(\lambda\) is, i.e. the smaller the pendulum is. This matches our real-life experience, doesn’t it? Of course, a more realistic model should include friction effects and also such problems can be treated.

Next semester (summer term 2017) I will give a 2+2-course on *Bifurcation theory* where we will develop a functional analytical framework to study problems like the ones above and much more complicated ones. The topics will cover the Implicit Function Theorem in Banach spaces, bifurcation from a simple eigenvalue (Crandall–Rabinowitz Theorem), bifurcation from infinity, Hopf bifurcation for periodic solutions of ordinary differential equations. The prerequisites are a thorough knowledge of analysis, ordinary differential equations and the basics of functional analysis or boundary and eigenvalue problems.

In the winter term 2017/2018 I am going to offer two options to broaden your expertise in bifurcation theory. There will be a seminar based on the lecture and another lecture where topological methods and/or variational will be applied to bifurcation problems. These methods allow to detect so-called *global bifurcations*, which is much better but also much more advanced than the local bifurcations we find in the up-coming lecture. Either course can be a starting point for a master thesis.

Contact information: Rainer.Mandel@kit.edu

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