# Seminar

## Rocky Mountain Algebraic Combinatorics Seminar

### Multisets of Values of Functions over Finite Fields

Eric Moorhouse
University of Wyoming

Let F be a finite field of order q. Given a q-multiset S of values in F, in general there are many choices of function FF having S as its multiset of values. The question arises: what is the smallest degree of any polynomial in F[t] realizing S as its multiset of values? I will say what I know about this problem (which is less than I would like!).

Among the first difficult cases is the following: Fix a,b in F with b ≠ 0. Let S consist of a with multiplicity q−2, and a±b as the remaining two values in S. Then any polynomial in F[x] realizing S has degree q−2. Is the converse true?
This problem may be reformulated as follows: Let f:FF be any function satisfying ∑aF σ(a)f(a) ≠ 0 for every permutation σ of F. Show that f is constant on a set of size q−2.
If time permits, I might say something about the original geometric motivation for considering such problems.

### Representing finite lattices as congruence lattices of finite algebras

William DeMeo

We discuss various aspects of the longstanding open problem of representing a finite lattice either as the congruence lattice of a finite algebra, or as an interval in the subgroup lattice of a finite group. We explore constructive methods that yield concrete representations, as well as some nonconstructive ways to prove existence of a representation. We also give a brief demo of the computer programs-GAP and UACalc-that we use to search for representations of finite lattices. A combination of these methods has enabled us to prove that every lattice with at most seven elements, with only one possible exception, is representable as a congruence lattice of a finite algebra. This is joint work with Ralph Freese (U Hawaii) and Peter Jipsen (Chapman U).

Weber 223
4–6 pm
Friday, October 20, 2017
(Refreshments in Weber 117, 3:30–4 pm)

This is a joint Denver U / UC Boulder / UC Denver / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.

### Previous Seminars:

October 6, 2017
Spencer Gerhardt, Joshua Grochow
September 22, 2017
Amanda Schaeffer Fry, Henry Kvinge
September 8, 2017
Peter Mayr, James B. Wilson
May 5, 2017
Ferdinand Ihringer, Gavin King
April 21, 2017
April 7, 2017
Jason Williford, Anton Betten
March 24, 2017
Isabella Novik, Peter Brooksbank
March 3, 2017
Jintai Ding, Curtis Bennett
February 17, 2017
Fatma Karaoglu, Eric Moorhouse
February 3, 2017
Tim Penttila, James B. Wilson
December 2, 2016
Jim Fowler, Andrew Kelley
November 11, 2016
Joseph Gersch, Joshua Maglione
October 28, 2016
October 14, 2016
JM Landsberg, James B. Wilson
September 30, 2016
Alexander Hulpke, Oscar Levin
September 16, 2016
Delaram Kahrobaei, Amit Patel
June 23, 2016
April 29, 2016
Nick Loehr, Jason Williford
April 15, 2016
Alexander Hulpke, Klaus Lux
April 1, 2016
Eamonn O'Brien, Izabella Stuhl
February 19, 2015
James Wilson, Anton Betten
December 4, 2015
Maria Monks Gillespie, Dane Flannery
November 13, 2015
Richard Green, Tim Penttila
October 23, 2015
Christina Boucher, Sylvia Hobart
October 9, 2015
Josh Maglione, Ghodratollah Aalipour
September 25, 2015
September 11, 2015
James B. Wilson, Tim Penttila
May 8, 2015
Amanda Schaeffer Fry, Peter Brooksbank
April 24, 2015
March 6, 2015
Felice Manganiello, Eric Moorhouse
February 20, 2015
Anton Dzhamay, Anton Betten
February 6, 2015
Alexander Hulpke, Morgan Rodgers
December 5, 2014
Stefaan De Winter, Gretchen Matthews
November 14, 2014
Greg Coxson, Tom Dorsey
October 31, 2014
Octavio Paez Osuna, Sylvia Hobart
October 10, 2014
Takunari Miyazaki, Eric Moorhouse
September 26, 2014
Elissa Ross, Anton Betten
September 12, 2014
Petr Vojtěchovský, Alexander Hulpke
May 9, 2014
Philip DeOrsey, Tim Penttila
April 25, 2014
William J Martin, Jason Williford
April 11, 2014
Victor Pambuccian, George Shakan
March 7, 2014
Nathan Lindzey, Jens Harlander
February 21, 2014
Ross McConnell, Anton Betten
November 22, 2013
Justin Hughes, Josh Maglione

Department of Mathematics