Research and Mathematical Interests
George T. Gilbert
Jeremy Smith's TCU dissertation, completed in 2018 under my supervision, was
titled Indices of Algebraic Integers in Cubic Fields.
My own research centered on how much one has to know about an L-function to
determine a modular or automorphic form.
- Cuspidal coverings for pairs of congruence subgroups, jointly with
M. P. Richey, Journal of Mathematical Physics, 36 no. 1 (1995), pp.
426-434 (MR 1308655).
- Strong multiplicity theorems for GL(n), Transactions of the
American Mathematical Society, 302 no. 2 (1987) pp. 561-576 (MR
0891635).
- Multiplicity theorems for GL(2) and GL(3), in The Selberg Trace
Formula and Related Topics, Contemporary Mathematics 53, 1986, pp. 201-206
(MR 0853559).
- An evolving, selectively rigorous, set of notes on
the
derivation and properties of the standard probability
distributions. I use them for optional, supplemental readings for my Math
30803, Math 30853, and Math 40603 courses. (Last revised on
April 11, 2019.)
With the Riemann Hypothesis and orthogonal polynomials as motivation, we
searched for families of functions whose zeros lie, for instance, on a
line.
- Lecture
notes on orthogonal polynomials from TCU's internal seminar. An
introduction to orthogonal polynomials. Monotone interpolation,
the Boutroux-Cartan Lemma, Mellin transforms of orthogonal polynomials. (February 2011)
- Vector spaces of functions with mostly real zeros, jointly with M.
K. Oberle, S. L. Scott, R. L. Hatcher, and D. F. Addis, Journal of
Mathematical Analysis and Its Applications, 188 no. 1 (1994), pp. 203-208
(MR 1301726).
- Mellin transforms of a generalization of Legendre polynomials,
jointly with M. K. Oberle, S. L. Scott, R. L. Hatcher, and D. F. Addis,
Journal of Computational and Applied Mathematics, 45 no. 3 (1993), pp.
367-369 (MR 1216080).
- Zeros of symmetric, quasi-definite, orthogonal polynomials, Journal
of Mathematical Analysis and Its Applications, 157 no. 2 (1991), pp.
346-350 (MR 1112321).
- Sliding piece puzzles with oriented tiles, jointly with J.
Berenbom, J. Fendel, and R. L. Hatcher, Discrete Mathematics, 175 no. 1-3
(October 1997), 23-33 (MR 1475835).
- A sliding block problem, jointly with L. C. Larson, The College
Mathematics Journal, 23 no. 4 (1992), pp. 315-319.
I am interested in the aggregate movement of chips in poker tournaments
and in developing models for expected values in tournaments that reflect
differing skill levels.
- The Independent Chip Model and risk aversion. I consider
the Independent Chip Model (ICM) for expected value in poker
tournaments. The first result is that participating in a fair bet with one
other player
will always lower one's expected value under this model.
The second result is that the expected value for players not participating
in
a fair bet between two players always increases. I
show that neither result necessarily holds for a fair bet among three
or more players. (arXiv 0911.3100)
- Slides from my
Knoxville Talk on Modeling Expected Value in Poker
Tournaments.
-
Racing early in tournaments, Two Plus Two Internet Magazine, 2:3 (March
2006) (draft version). I make a preliminary examination of several models for
expected value in poker tournaments and strategic decisions based on these
models.
- Proximal locking screws for nailing femoral shaft fractures: Should
cephalomedullary screws be the standard?, jointly with Cory Collinge, Alexander Guyott,
Frank Liporace, Journal of Orthopaedic Trauma, 24 no. 12 (December 2010),
pp. 717-722. I estimated some life expectancies
for the analysis.
- Wagering in Final Jeopardy!, jointly with R. L. Hatcher,
Mathematics Magazine, 67 no. 4 (1994), pp. 268-277 (MR
1300566).
- Positive definite matrices and Sylvester's criterion, American
Mathematical Monthly, 98 no. 1 (1991), pp. 44-46 (MR 1083614).
George Gilbert's Home Page