## Mathematics## 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 *F*→ *F* 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!).

*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*:

*F*→

*F*be any function satisfying ∑

_{a ∈ F}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

University of Colorado, Boulder

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)

Colorado State University

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.