## Morphic numbers, measurement, sequences, and tilings

### Speaker

Barry Balof (Whitman College)

### Abstract

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.

### Where

Millikan 2099, Pomona College

## TBA

### Speaker

Sam Dittmer (UCLA)

TBA

### Where

Millikan 2099, Pomona College

## TBA

### Speaker

Briana Foster-Greenwood (Cal Poly Pomona)

TBA

### Where

Millikan 2099, Pomona College

## TBA

### Speaker

Bianca Thompson (HMC)

TBA

### Where

Millikan 2099, Pomona College

## TBA

### Speaker

Bryce McLaughlin (HMC)

TBA

### Where

Millikan 2099, Pomona College

## Near-linear time uniform sampling from some height-2 linear extensions

Mark Huber (CMC)

### Abstract

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 $O(n^3 \ln(n))$.

### Where

Millikan 2099, Pomona College

## Factorization length distribution in numerical semigroups

### Speaker

Samuel Yih (Pomona College)

### Abstract

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).

### Where

Millikan 2099, Pomona College

## Twisted virtual bikeigebras

Sam Nelson (CMC)

### Abstract

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.

### Where

Millikan 2099, Pomona College

## TBA

### Speaker

Matt Papanikolas (Texas A&M University)

TBA

TBA

### Misc. Information

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

## TBA

### Speaker

Taoufiq Mohamed (Aalto University, Finland)

TBA

### Where

Millikan 2099, Pomona College

### Misc. Information

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