Alan Roche Department of Mathematics University of Oklahoma Norman, OK 73019-3103 Office: PHSC 1112 Email: aroche (at) ou (dot) edu

Welcome! The picture is not me: it's Aoibhe, pronounced A V as in the letters. Mine was merely the mortal hand and eye that dared to frame her fearful symmetry.

Aoibhe studies CATegory theory and purr‒obability. My research area is algebra, mainly representation theory. And in representation theory, I've mainly studied representations of p‒adic groups, a subject with close ties to number theory. Recently, however, I've found myself writing semi-expository pieces on more elementary topics. Some are listed below.

Some Recent Notes and Papers

• A Note on Dual Modules and the Transpose (with T. Madsen and C. R. Vinroot)
Arch. Math. (Basel) 114 (2020), no. 3, 247-257.
• Four Group-theoretic Proofs of Wedderburn's Little Theorem
Irish Math. Soc. Bulletin 88 (2021), 57-68.
• A Tale of Two Cyclicities: Counting in Finite Cyclic Groups and Finite Fields (with J. Cohen)
Amer. Math. Monthly 129 (2022), no. 7, 660-676.
• A Characterization of Cyclic Groups via Indices of Maximal Subgroups
Irish Math. Soc. Bulletin 89 (2022), 65-72.
• Which Finite Rings Have the Most or Fewest Units? (with J. Cohen)
Amer. Math. Monthly 129 (2022), no. 8, 717-733.
• Another Short Proof that Rational Algebraic Integers are Integers
To appear in Math. Mag.
• There are Infinitely Many Primes: Two Ring-theoretic Variations on Euclid
To appear in Irish Math. Soc. Bulletin.

In Preparation

• A Short Proof of the Riemann Hypothesis

The paper needs some work.

Teaching

This semester I'm teaching Intro. to Abstract Algebra II (MATH 4333).

Grab Bag

Sam—A Triptych

Sam is Aoibhe's pal. Here are three sides of this sometimes risible, always delightful creature—studious Sam, singing Sam, sitting Sam. My son took the first picture. My daughter took the second. I'm just out of the frame about to hand Sam his dinner: the little guy routinely bursts into song at dinnertime. I took the picture of sitting Sam.

Can you tell we are fond of cats in our family? Too much so? Yes, some tell us. My wife and I once invited a good friend over for dinner. Her response: she would be delighted to come on one condition—no talk about cats!

A favorite poem

The Sunlight on the Garden

The sunlight on the garden
Hardens and grows cold,
We cannot cage the minute
Within its nets of gold,
When all is told
We cannot beg for pardon.

Our freedom as free lances
Advances towards its end;
The earth compels, upon it
Sonnets and birds descend;
And soon, my friend,
We shall have no time for dances.

The sky was good for flying
Defying the church bells
And every evil iron
Siren and what it tells:
The earth compels,
We are dying, Egypt, dying

And not expecting pardon,
Hardened in heart anew,
But glad to have sat under
Thunder and rain with you,
And grateful too
For sunlight on the garden.

 Louis MacNiece  (1936)