Barry Balof (Whitman College)

The Golden Ratio, $\phi$, arises naturally from the Fibonacci Numbers, and is a well-studied constant with many diverse applications. Perhaps less well-known is the 'Plastic Constant', $\psi$, which arises from the Padovan Sequence. Both constants share some interesting algebraic and geometric properties, which we will explore. We will also look at some diverse applications of $\psi$ to measurement and architecture, and some combinatorial extensions of classic tiling problems.

Millikan 2099, Pomona College

Sam Dittmer (UCLA)

The study of linear extensions of posets has applications ranging from combinatorics to computer science to voting theory. In 1991, Brightwell and Winkler showed that the number of linear extensions of a poset is #P-complete. We extend this result to posets with certain restrictions. First, we prove that the number of linear extensions of a poset of height two is #P-complete. This resolves the Brightwell-Winkler conjecture. Next, we present an overview of a proof that the number of linear extensions of a poset of dimension two is #P-complete. This problem is equivalent to computing the size of a principal ideal in the weak Bruhat order, a fundamental object in algebraic combinatorics. The proof includes a brute force computer search used to construct certain families of logic gates. It is, to the best of our knowledge, the first computer assisted proof of #P-completeness, and the first proof to use algebraic systems to encode logic gates. Joint work with Igor Pak.

Millikan 2099, Pomona College

Briana Foster-Greenwood (Cal Poly Pomona)

Many noncommutative algebras arise as deformations of commutative algebras. For instance, the Weyl algebra is a deformation of the polynomial ring R[x,y] obtained by replacing the commutative relation yx-xy=0 with the noncommutative relation yx-xy=1. For a twist, we consider skew group algebras formed as the semi-direct product of a polynomial algebra and a group algebra. Gradings, filtrations, and degrees play an important role in the deformation theory of skew group algebras. Graded Hecke algebras (including symplectic reflection algebras and rational Cherednik algebras), for instance, are deformations in which commutators of vectors of degree 1 are set to elements of the group algebra, which have degree 0. Work of Shepler and Witherspoon extends the theory to deformations where commutators of vectors of degree 1 are set to combinations of elements of degrees 1 and 0. In this talk we use invariant theory and cohomology to explore such deformations for skew group algebras arising from single and doubled permutation representations of the symmetric group. Joint work with Cathy Kriloff (Idaho State University).

Millikan 2099, Pomona College

Bianca Thompson (HMC)

In arithmetic dynamics we are interested in classifying and counting points that live in cycles or eventually live in cycles for a family of maps over a given field. We refer to each of these types of points as periodic or preperiodic, respectively. Over a finite field, everything is preperiodic, so the questions becomes what can we say about the number points that are periodic. We'll talk about a family of rational maps associated to supersingular elliptic curves and give preliminary results on what we can say about the number of periodic points over \FF_{p^n}, for some prime p.

Millikan 2099, Pomona College

Bryce McLaughlin (HMC)

In 1946, Erdős posed the distinct distances problem, which asks for the minimum number of distinct distances that any set of n points in the real plane must realize. Erdős showed that any point set must realize at least Ω(n^{1/2}) distances, but could only provide a construction which offered Ω(n/√log(n)) distances. He conjectured that the actual minimum number of distances was Ω(n^{1-ε}) fo any ε>0, but that sublinear constructions were possible. This lower bound has been improved over the years, but Erdős' conjecture seemed to hold until in 2010 Larry Guth and Nets Hawk Katz used an incidence theory approach to show any point set must realize at least Ω(n/log(n)) distances. In this talk we will explore how incidence theory played a role in this process and expand upon recent work by Adam Sheffer and Cosmin Pohoata who use geometric incidences to get bounds on the bipartite variant of this problem. A consequence of our extensions on their work is that the theoretical upper bound on the original distinct distances problem of Ω(n/√log(n)) holds for any point set which is structured such that half of the n points lies on an algebraic curve of arbitrary degree.

Millikan 2099, Pomona College

Mark Huber (CMC)

Given a partially ordered set (poset), a linear extension is a total ordering of the elements that respects the partial ordering. For example, if $1\preceq 2$ and $3 \preceq 4$, then $1 \sqsubseteq 3 \sqsubseteq 2 \sqsubseteq 4$ is a valid linear extension (write $1324$ for simplicity) while $3 \sqsubseteq 2 \sqsubseteq 4 \sqsubseteq 1$ (3241 for short) is not because it does not respect $1 \preceq 2$. All partial orders have at least one linear extension, but in general counting the number of linear extensions of a poset is a #P complete problem. Dittmer & Pak recently showed that this holds even for posets where no three different elements form a chain $i \prec j \prec k$. This is called a height 2 poset. So counting exactly is very difficult, even in this restrictive case. However, if one can sample uniformly from the set of linear extensions, then they can approximately count the number of linear extensions. In this talk I will present a method for sampling uniformly from the linear extensions of a height-2 poset that runs in time $O(n \ln(n) \Delta^2)$, where $\Delta$ is an upper bound on the number of elements comparable to an arbitrary element $i$. This improves upon the previously best known running time of $O(n^3 \ln(n))$.

Millikan 2099, Pomona College

Samuel Yih (Pomona College)

A numerical semigroup is a cofinite subset of the nonnegative integers closed under addition. Extreme factorization invariants such as the maximum and minimum factorization lengths are well understood, but intermediate invariants such as the mean, median, and mode factorization lengths are more subtle. In this talk we will first study these invariants for semigroups of embedding dimension 3, in which a number of interesting phenomena already occur. We will then broaden our perspective to semigroups of any dimension, for which we have a general result for the mean factorization length and open conjectures for the remaining invariants. Joint work with Stephan Garcia (Pomona College) and Christopher O'Neill (UC Davis).

Millikan 2099, Pomona College

Sam Nelson (CMC)

Twisted virtual handlebody-links are regular neighborhoods of trivalent graphs embedded in manifolds of the form $\Sigma times [0,1]$ where $\Sigma$ is a compact surface which may or may not be orientable. In this talk we will introduce combinatorial moves for representing these knotted objects and describe an algebraic structure called twisted virtual bikeigebras for distinguishing them by counting colorings. This is joint work with former CMC student Yuqi Zhao.

Millikan 2099, Pomona College

Matt Papanikolas (Texas A&M University)

In 2012 Pellarin defined a new class of L-series over the polynomial ring over a finite field that take values in the Tate algebra of the closed unit disk for the place at infinity. These L-series interpolate both special values of Dirichlet L-functions as well as other special values of Goss L-series related to the Carlitz module. Subsequent multivariable versions were studied by Anglès, Pellarin, and Tavares Ribeiro, with applications to analogues of Stark units. In this talk we will discuss approaches to Pellarin's theory over more general rings, in particular over coordinate rings of elliptic curves over finite fields. Using the theory of shtuka functions, we prove special value formulas for Pellarin L-series that take values in the affinoid algebra of the elliptic curve. Joint with N. Green.

Millikan 2099, Pomona College

Taoufiq Mohamed (Aalto University, Finland)

The action of the diagonal group A (i.e. the group of real diagonal matrices with positive entries and determinant one) on a lattice L was extensively studied in the context of the Minkowski conjecture. Proving non-emptiness of the intersection of it's orbit AL with the set W_n of well-rounded lattices in dimension n is considered to be a big step toward Minkowski conjecture in that dimension. The aim of this talk is to highlight some links between this intersection AL \cap W_n and applications in wireless communication. We will then AL \cap W_n when L is arising from a real quadratic number field in more detail.

Millikan 2099, Pomona College

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