As per Karma theory, certain events of my current life are the fruits of my actions in one or more earlier births.
Whose action's fruits am I facing in this birth?
Given that it cannot be my current body or mind, it follows that it has to be my Atman.
By this reasoning you are saying you are not responsible for anything you did yesterday or anything before current moment.
In particular, the seemingly different people we see are all Brahman and seem different only because of Maya. Then how is it possible to determine the me from a previous birth?
If you can identify yourself as doer of a task among many people, you performed yesterday, then you may very well also identity yourself among many people of your previous birth tasks.
If one can indeed determine the me from the previous birth then does that not directly imply duality contradicting Advaita?
This contradiction seems to even arises if you think you can identify yourself as doer of a task you performed yesterday among many. Because according to Advaita, only one exists. Even in this every moment this contradiction arises. You may simple say, if Advaita is true, why are there so many people.
In fact I do not see how rebirth is even possible in the context of Advaita. Who is getting reborn?
Again, I really do not see your argument here. What's the problem with rebirth? It's just like going from yesterday to today or childhood to youth to old age. If you take yourself as the same person yesterday and today, what is wrong with rebirth? Like Krishna says:
देहिनोऽस्मिन्यथा देहे कौमारं यौवनं जरा।
तथा देहान्तरप्राप्तिर्धीरस्तत्र न मुह्यति।।2.13।।
Are Advaita and Karma theory completely contradictory?
There's nothing like 'complete' contradiction or 'slight' contradiction. It's either contradiction or no contradiction. Anyhow, answering the question. No there is no contradiction in karma theory and Advaita.
What is Karma really?
Karma essentially are the vasanas because one only performs karma out of vasanas and not without it. Vasistha also says:
जन्तुर्_यद्_वासनो राम तत् कर्ता भवति क्षणात् ।
(14, 9, 2 YV)
Objection: This is not always the case. Some actions are performed based on circumstances (or bhagya).
Answer: This is not true. The circumstances (or bhagya) also are a result of past karmas.
यद्_एव तीव्र~संम्वेगाद्_दृढम् कर्म कृतम् पुरा ।
तद्_एव दैव-शब्देन पर्यायेण_इह कथ्यते ॥१६॥
(16, 9, 2 YV)
What are vasanas really? Vasanas are nothing other than Manas.
यद्_दैवम् यानि कर्माणि कर्म साधो मनो हि तत् ।
मनो हि पुरुषस्_तस्माद्_दैवम् न_अस्ति_इति निश्चयः ॥१८॥
(18, 9, 2 YV)
The above mentioned theory is karma theory. This is as true as saying area of square is side^2.
Now, Advaita only says that this Manas is unreal. And by neglecting this Manas everything gets neglected from Manas down to Karma and area of square. There is simply no contradiction. It seems there's a contradiction because we try to question the unreality of karmas while presuming the existance of Manas. When you say karmic theory is false, its as good as saying area of square is not side^2. Karma theory is said to be false according to Advaita only because of lack of better words. And when it is said false, it is assumed you also neglect the Manas along with it.
So, when you say "Why am I getting the fruits of karma? The one that performed karma in the past life was not me because everything is one", you are using Advaita to claim "Only Atman exist" and the same time saying "I enjoy". You cannot say "I enjoy" in light of Advaita because Atman never enjoys nor does action. Remember all the Manas idea gets neglected in light of Advaita. On the other hand if you take the responsibility of fruits you are enjoying, you also need to acknowledge your actions of the past.
Objection of OP: I can determine that the me of yesterday and the me of today are the same person by purely empirical methods (DNA test, and if you want to take care of twins, radioactive tracers, etc.)
Answer: You are associating yourself with the body. Body can not be the doer of actions as it is just a piece of biological mass. When you come out of a dream and say I did such and such thing, you feel you did that even though your body didn't went along with you. You feel the exact some touch, sound and smell that you had felt with your body.
Shankracharya also says, body can not be the knower
न देहादिसंघातस्य विद्वत्ता
Are you suggesting that just like there is continuity and memory in this birth, there is some sort of continuity in the 'mind' across rebirths?
Yes during rebirth the mind doesn't die, it continues like our mind takes a different body during dream. Whatever be the desires at the time of death, according the adequate body and world is manifested at the time of death just like a dream is manifested depending upon the thoughts one is having before sleeping. The mind continues in the form of vasanas. Once the vasanas of mind is completely annihilated there's no world to be seen. This is called liberation.
विकरोत्यपरान्भावानन्तश्चित्ते व्यवस्थितान् ।
नियतांश्च बहिश्चित्त एवं कल्पयते प्रभुः ॥ १३ ॥
- The Lord (Ātman), with his mind turned outward, variously imagines the diverse objects (such as sound, etc.), which are already in his mind (in the form of Vāsanas or Saṅkalpas or desires). The Ātman again (with his mind turned within), imagines in his mind various (objects of) ideas.
(Mandukya Karika, verse 2.13)
आत्मसत्यानुबोधेन न संकल्पयते यदा ।
अमनस्तां तदा याति ग्राह्याभावे तदग्रहम् ॥ ३२ ॥
- When the mind does not imagine on account of the knowledge of the Truth which is Ātman, then it ceases to be mind and becomes free from all idea of cognition, for want of objects to be cognised.
(Mandukya Karika, verse 3.32)