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)

TBA

Millikan 2099, Pomona College

Briana Foster-Greenwood (Cal Poly Pomona)

TBA

Millikan 2099, Pomona College

Bianca Thompson (HMC)

TBA

Millikan 2099, Pomona College

Bryce McLaughlin (HMC)

TBA

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)

TBA

TBA

This week only the seminar meets on Thursday instead of the usual Tuesday.

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