Its not very object oriented: One reason statics might be considered "evil" by some people is they are contrary the object-oriented paradigm. In particular, it violates the principle that data is encapsulated in objects (that can be extended, information hiding, etc). Statics, in the way you are describing using them, are essentially to use them as a global variable to avoid dealing with issues like scope. However, global variables is one of the defining characteristics of procedural or imperative programming paradigm, not a characteristic of "good" object oriented code. This is not to say the procedural paradigm is bad, but I get the impression your supervisor expects you to be writing "good object oriented code" and you're really wanting to write "good procedural code".
There are many gotchyas in Java when you start using statics that are not always immediately obvious. For example, if you have two copies of your program running in the same VM, will they shre the static variable's value and mess with the state of each other? Or what happens when you extend the class, can you override the static member? Is your VM running out of memory because you have insane numbers of statics and that memory cannot be reclaimed for other needed instance objects?
Object Lifetime: Additionally, statics have a lifetime that matches the entire runtime of the program. This means, even once you're done using your class, the memory from all those static variables cannot be garbage collected. If, for example, instead, you made your variables non-static, and in your main() function you made a single instance of your class, and then asked your class to execute a particular function 10,000 times, once those 10,000 calls were done, and you delete your references to the single instance, all your static variables could be garbage collected and reused.
Prevents certain re-use: Also, static methods cannot be used to implement an interface, so static methods can prevent certain object oriented features from being usable.
Other Options: If efficiency is your primary concern, there might be other better ways to solve the speed problem than considering only the advantage of invocation being usually faster than creation. Consider whether the transient or volatile modifiers are needed anywhere. To preserve the ability to be inlined, a method could be marked as final instead of static. Method parameters and other variables can be marked final to permit certain compiler optimiazations based on assumptions about what can change those variables. An instance object could be reused multiple times rather than creating a new instance each time. There may be compliler optimization switches that should be turned on for the app in general. Perhaps, the design should be set up so that the 10,000 runs can be multi-threaded and take advantage of multi-processor cores. If portablity isn't a concern, maybe a native method would get you better speed than your statics do.
If for some reason you do not want multiple copies of an object, the singleton design pattern, has advantages over static objects, such as thread-safety (presuming your singleton is coded well), permitting lazy-initialization, guaranteeing the object has been properly initialized when it is used, sub-classing, advantages in testing and refactoring your code, not to mention, if at some point you change your mind about only wanting one instance of an object it is MUCH easier to remove the code to prevent duplicate instances than it is to refactor all your static variable code to use instance variables. I've had to do that before, its not fun, and you end up having to edit a lot more classes, which increases your risk of introducing new bugs...so much better to set things up "right" the first time, even if it seems like it has its disadvantages. For me, the re-work required should you decide down the road you need multiple copies of something is probably one of most compelling reasons to use statics as infrequently as possible. And thus I would also disagree with your statement that statics reduce inter-dependencies, I think you will end up with code that is more coupled if you have lots of statics that can be directly accessed, rather than an object that "knows how to do something" on itself.