Hi. I'm Arkady.
Who are you? Fundamentally, I mean - what part of you defines you as "you"?
This is a question that for many years has been answered simply - the immortal soul. This neatly gets around the Theseus Paradox by creating a part of you that isn't constantly replaced (like our cells) and which doesn't decay or change, like the arrangement of our cells.
However, for an atheist or a skeptic who doesn't otherwise believe in the immortal soul this isn't a satisfying answer.
Part of the the problem with answering this is that we are ever-changing beings. You could try defining yourself by saying "I am defined as a person who does ..." This is impossible to do in practice - the list would essentially be infinite - but such problems never stopped mathematicians working with "the set of all real numbers", so there's no real conceptual problem.
However, even this isn't satisfactory - fifteen years ago I believed the greatest thing to ever happen to music was the Spice Girls. This is something that has fundamentally changed about me, yet I'm clearly the same person, if a little embarrassed about it.
A stand-up comic I've seen* jokes about this, saying he went to a motivational speaker, who told the group "every moment of every second you can reinvent yourself - become a totally new person, who you were not before." At the end of the session, as he was leaving he was asked to pay for the session, and replies "I think you've mistaken me for someone else."
Try using the excuse for not paying a speeding fine and you'll land yourself in more trouble and laughed out. Clearly we do have some innate sense of who a person is, and it is strong enough to cope with the fact that we change both materially and behaviourally. Otherwise the joke wouldn't work.
For anyone who is now thoroughly confused about who they are (you know who you are) I'd like to propose an answer: you are the sum of the actions you take.
When taken literally, and openly, there are several things this implies about how we must approach the world ethically. It means that we must take full responsibility for all outcomes of what we do - intended and unintended. It means that you have to live with your mistakes, but it means taking pride in your accomplishments. It means never saying "I'm a great writer, even though I don't write anything" - you aren't a writer unless you actually write. It means not worrying about thoughts we never act on (some people beat themselves up a lot over these) because no-one will ever know, and it means you change yourself by doing the things that the sort of person you want to be will do. It means reinventing yourself doesn't in any way "betray" your old self, or kill them.
So this week, do something you always wanted to do, but never quite got around to.
* I'm pretty sure he was called Christopher White, but there appears to be another stand-up comic by that name who is more famous, and dominates the Google results.