In pure mathematics, usually questions can be reduced to solving systems of equations. Systems of equations of the form f1=…=fN=0 in real variables can be reduced to solving global optimization of a single function g=f12+…+fN2. The latter more generally covers also the least square fit problem in statistics theory. The relation between optimization and systems of equations go both ways. If one has a good method to solve systems of equations then one can find zeros of the gradient of the function g, which are candidates for being (local) optima of g.
There are very powerful symbolic tools to solve systems of equations in case the functions involved are polynomials, such as Groebner Basis. However they can be very slow for interesting cases one needs to solve in practice (in particularly, for real life applications such as robotics, where decisions must be made in a matter of seconds). Hence, numerical methods are usually employed, such as the Bertini software or iterative optimization methods. Also, there are cases where the concerned functions are not polynomial, such as meromorphic functions which are of interest in number theory.
When using iterative optimization methods, one needs to assure that the method behaves well. This usually boils down to two facts: i) showing that each cluster point of the sequence constructed by the method is a critical point of the function g, and ii) show that each limit point is not a saddle point. (Moreover, a very challenging question nowadays is to be able to converge to “good” local minima. This is the most one can get, given that even for very simple functions finding global minima is NP-hard.)
The theory of dynamical systems can help to resolve the question ii) above. It turns out that one has to deal with some form of randomness in the associated dynamical systems. Dynamical systems are also known to have more direct relations to solving of systems of equations. For example, the well known Weil’s Riemann hypothesis, on solutions in finite fields, can be recasted in terms of counting the number of fixed points of the Frobenius map. For the Riemann hypothesis itself, there is an attractive approach which, among other things, attempts to mimic the proof of Weil’s Riemann hypothesis using foliations and new cohomology theories.
This conference aims at being a forum where experts from the above various different fields can inform the audience about big insights and new developments, as well as have meaningful interactions with people from other fields. The conference hopes to foster cross collaboration between different fields whose ideas could be combined to solve major and useful questions.
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