When thinking about the gender and pay gaps in the corporate world, what problems do women face today when trying to advance in their careers?  For allies, what do you think women could do better to advocate for themselves when trying to advance their careers?

The book “Athena Rising” is helpful at developing the ROI in cross gender mentoring. Agree with Lean In and also suggest Radical Candor.

Qualitative from a bystander there are about four examples; woman who are helping risers, woman who are paragons of example far beyond a gender, woman leadership that appear more critical of the next generation (as they don’t have to walk up the same snowy hill both ways they did), and a types of cronyism where a sub group build us/them community with wide generalizations similar to the ineffective culture they are attempt to displace.

So trying to highlight more of the positive examples while limiting or tackles more the more destructive models.

Cultural systems are the biggest challenges to advancing a career regardless of gender. Some organization's cultures favor men and some favor women. It depends on the environment and culture.

As long as organizations continue using favor or often an unintentional bias evaluation toward any person instead of setting a transparent and fair exam/test both for technical, characteristics, ethics, and morals then this issue will continue to be debated.

The right-mind women and men already know what they need to do or trade off what they bargained for.

Besides, what's the point of trying to do better or differently for advancing career when being themselves and true is the way we all know the most comfortable way to live our lives in general?

From my experience and observations:
One challenge is not knowing what the pay gaps actually are, come companies try to keep employees from talking about salary and that makes it hard for women to act on facts. 

For the women who do ask or try to negotiate pay, they stumble not because of a lack of confidence, but from how to handle and answer the pushback they get and present their accomplishments in a way that men don't perceive as pushy or aggressive. I have found having an ally who will sing your praises helps to counter that.

First, women need a strong foundation to build from. Women can establish themselves by focusing on their personal brand and secure a sponsor to advocate for them when they're not in the room. Consider launching a "Women's Leadership Initiative" (if you don't have one already) that provides educational opportunities, allyship and mentors with the ultimate goal of helping women pursue leadership roles. One tactic to address pay gaps is to educate male leaders. Bring them into the discussion and lean on them as allies (upstanders, sponsors, etc.). Explain the important role they play in advancing the career goals of women. Hope this helps.

Unfortunately, an aspect not in the control of women is men. Too many men still discriminate (yes the glass ceiling still exists), and while an aggressive / confident male is admired, an aggressive / confident women is too often referred to by an unkind word. This is something that we as men must not tolerate!

But there is an item in the control of women - marketing! When a man does something good we pat ourselves on the back and tell everyone how great we are. But one of the great aspects of women is their ability to work in a team and in sharing success with others. This an admirable, but does not focus their success in the eyes of senior management. 

My advice is not to stop being who you are, but to find a way to ensure management appreciates your confidence, and sees and appreciates your successes! Makes sure you are always viewed as a leader and not as just a manager.