First, a little history - not long after Einstein published his theory of special relativity in 1905, scientists Henri Poincare and Hermann Minkowski pointed out that the equations in the theory could be rewritten with a geometric interpretation, where the weird effects of relativity (time slowdown, length contraction, etc.) were explained as a rotation in 4-dimensional spacetime, with the time coordinate normal (perpendicular) to all 3 space coordinates. Einstein assumed that this was simply a mathematical formalism; in fact he once described Minkowski's work as "superfluous erudition." Mayer takes the spacetime theory one step further, by saying we've failed to recognize that Minkowski spacetime reveals something about the true nature of the universe - that time is a vector normal to space. In other words, time isn't flowing in the same "direction" everywhere in the universe. I'll first explain with an easier to understand analogy.
If you are standing on the surface of the Earth and don't already know it's a sphere, then you can make the naive assumption that it's a flat plane and that gravity points in the same direction for everyone on Earth. Of course we know this isn't true - the gravity vector on the other side of the Earth is pointing in the opposite direction. If you are in the US, people in Europe are approximately "sideways" relative to you. But nobody notices anything strange because everyone has their own local idea of "up" and "down".
Mayer argues that time is similar - that the universe is a gigantic hypersphere, and the space we live is the "surface volume" (4 dimensions are weird aren't they?!?) of the sphere. Time is always normal to space, so the time vector in any part of the universe effectively points along the radius of the hypersphere. In practice, this doesn't mean much for a small localized part of the universe (e.g. a galaxy). Just like gravity in a single city on Earth, everyone's time vector in one galaxy is pretty much pointing in the same direction.
The kicker comes when you start to get into deep-space astronomy. When you observe a distant galaxy, you are seeing photons from a region where the time vector points in a significantly different direction. In Mayer's theory, this causes a time dilation effect exactly like what happens with special relativity. So, guess what, all those photons from the distant galaxy are redshifted because of the difference in the local and remote time vectors. People who understand Big Bang theory should be getting wide-eyed right about now because the primary reason we think the Big Bang happened is that all distant galaxies are redshifted. The standard way to interpret that redshift is to assume it must be present because the galaxy is moving away from us. If all galaxies are moving away from us, then at some time in the past everything must have "exploded" from a central point, hence the Big Bang. But Mayer's theory throws all of that out the window, because those redshifts aren't caused by motion, they are caused by geometric time dilation.
That's all well and good, but where's the evidence? Mayer spends a whole lot of time talking about the SDSS (Sloan Digital Sky Survey) in his slides. This is a very cool astronomy project that has produced a huge catalog of galaxies, including their luminosity and redshift. Mayer applies statistics to this collection to support his theory. The Big Bang theory uses the Hubble law, where galaxy redshift scales linearly with distance to the galaxy. But with geometric time, that relationship is not linear; it's based on a trigonometric function. Mayer argues that his geometric time equations produce a far better fit to the statistical distribution of galaxy redshifts than the linear Hubble law does. There are lots of slides showing this. Also important, those equations were not made to fit the data in any way - they follow directly from the geometric interpretation of time.
Whew, that's long and I barely scratched the surface, but I hope it's at least a little helpful.