My Research
My research is focused on the interplay between geometry, topology, and fixed point theory. What is fixed point theory? Let's try to answer that question by way of a few examples:
1. If you have a cup of coffee or water on a table, pick it up and gently swirl it around (don't do anything too crazy to it...), then return it to the table where it was originally sitting and let it come to rest. There will be at least one drop of coffee which has returned precisely to its original location in the cup.
2. If you take two identical sheets of paper, place one flat on a table, crumple the other sheet and place it on top of the first sheet. Then there will be at least one point on the crumpled sheet which is directly above its corresponding point on the flat sheet. That is, there was at least one point that wasn't "moved" by the crumpling.
3. If you are standing on campus and holding a campus map parallel to the ground, then there will be precisely one point on the map which is exactly above the corresponding point on campus (think of it like a ``you are here'' pin).
4. We can prove the existence of fractals (and use computers to generate them!) using fixed point theory. That is, we can make pictures like these:
Because of the work of Stefan Banach and Luitzen Brouwer (to whom all of the above facts are due) in the early 20th century, essentially everything is known about fixed point theory in finite dimensions (like the three-dimensional world we live in). Because of this, recent research in fixed point theory has moved to the infinite dimensional setting. If you have questions or want to chat about math, feel free to stop by my office!
Publication list
``New fixed point free nonexpansive mappings in K(l^2) and related spaces'' (with Maria Japon Pineda, Chris Lennard, and Adam Stawski). In Preparation.
``Selection type results and fixed point property for affine bi-Lipschitz maps'' (with Cleon Barroso). In preparation.
The nonexpansive and mean nonexpansive fixed point properties are equivalent for affine mappings (with Maria Japon Pineda and Chris Lennard). Journal of Fixed Point Theory and Applications, 22(4) (2020).
Rocky Mountain Journal of Mathematics, 47 (2017), pp. 2167-2178.
Mean nonexpansive mappings are mean isometries if and only if they are isometries (with Chris Lennard).
Journal Nonlinear Convex Analysis, 18 (2017), pp. 73-94.
Advances in Operator Theory, 2 (2017), pp. 1-16.
Averaging and fixed points in Banach spaces (Ph.D. thesis).
The demiclosedness principle for mean nonexpansive mappings.
Journal of Math Analysis and Applications, 439 (2016), pp. 832-842.
Weak compactness is not equivalent to the fixed point property in c (with Chris Lennard and Roxana Popescu).
Journal of Math Analysis and Applications, 431 (2015), pp. 471-481.