## Arithmetic of quaternion algebras and Shimura curves

### Speaker

Laia Amoros (Aalto University, Finland)

### Abstract

I will introduce quaternion algebras and show how one can construct Shimura curves with them. Quaternion algebras arise as a natural generalisation of matrix algebras. In the same way that the action of SL(2,Q) (and of all its congruence subgroups) on the complex upper half-plane give us modular curves, the action of certain subgroups of quaternion algebras will give us some algebraic structure (the so called Shimura curves). After this introduction I will explain some applications of Shimura curves and sketch how one can compute the bad reduction of certain families of Shimura curves, based on a joint work with P. Milione.

### Where

Millikan 2099, Pomona College

## Distribution of descents in matchings

Gene Kim (USC)

### Abstract

The distribution of descents in certain conjugacy classes of $S_n$ have been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.

### Where

Millikan 2099, Pomona College

## Rudin-Shapiro-like polynomials with low correlation

### Speaker

Daniel Katz (Cal State Northridge)

### Abstract

The Rudin-Shapiro polynomials are a sequence of polynomials with coefficients in {+1,-1} arising from an elegant recursive construction that gives them interesting analytic and combinatorial properties. They are known to be very flat on the complex unit circle: if f is a Rudin-Shapiro polynomial and we let z range over the unit circle, then the ratio of the maximum value of |f(z)| to the root mean square value is never larger than the square root of 2. When we identify a polynomial with the sequence of its coefficients, we find that a typical Rudin-Shapiro polynomial has very low autocorrelation: the inner products of the sequence of coefficients with nontrivial translates of itself have considerably smaller mean square magnitude than what one would obtain with a typical random sequence with entries from {+1,-1}. The fact that the inner product is only large when the sequences are perfectly aligned makes such sequences very useful for synchronization in communications networks. For multi-user networks, we want to avoid confusing one user's sequence with another's, so we demand pairs of sequences that also have low cross-correlation (inner product with each other) at all shifts. We investigate pairs of Rudin-Shapiro-Like polynomials, a generalization of Shapiro's original recursive construction, and derive a general formula for their mean square cross-correlation. Borwein and Mossinghoff had shown that there are Rudin-Shapiro-like polynomials with low autocorrelation, and now we have found pairs of polynomials such that both have low autocorrelation and the pair has low mutual cross-correlation. There are bounds on how small these quantities can be, and we have found infinite families that achieve the bounds. This is joint work with Sangman Lee, Eli Moore, and Stanislav A. Trunov.

### Where

Millikan 2099, Pomona College

### Misc. Information

There will be a short organizational meeting in the same room right before the talk at 12:00 noon.

## Reversed Dickson permutation polynomials

### Speaker

Neranga Fernando (Northeastern University)

### Abstract

Let $p$ be a prime and $q$ a power of $p$. Let $F_q$ be the finite field with $q$ elements. A polynomial $f$ in $F_q[x]$ is called a permutation polynomial of $F_q$ if the associated mapping $x --> f(x)$ from $F_q$ to $F_q$ is a permutation of $F_q$. Permutation polynomials over finite fields have important applications in coding theory, cryptography, finite geometry, combinatorics and computer science, among other fields. Recently, reversed Dickson polynomials over finite fields have been studied extensively by many for their general properties and permutation behaviour. For $a$ in $F_q$, the $n$-th reversed Dickson polynomial of the $(k+1)$-th kind $D_{n,k}(a,x)$ is defined by

$$D_{n,k}(a,x) = \sum_{i=0}^{\lfloor\frac n2\rfloor}\frac{n-ki}{n-i}\dbinom{n-i}{i}(-x)^{i}a^{n-2i},$$

and $D_{0,k}(a,x)=2-k$. In this talk, I will completely explain the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(a,x)$ when $a=0$, $n=p^l$, $n=p^l+1$, and $n=p^l+2$, where $l \geq 0$ is an integer. I will also explain the permutation behaviour of $D_{n,k}(1,x)$ when $n$ is a sum of odd prime powers. Moreover, I will present some algebraic and arithmetic properties of the reversed Dickson polynomials of the $(k+1)$-th kind. In particular, I will explain the explicit evaluation of the sum $\sum_{a \in F_q}D_{n,k}(1,a)$ which provides a necessary condition for $D_{n,k}(1,x)$ to be a permutation polynomial of $F_q$. These results unify and generalize numerous recently discovered results on reversed Dickson polynomials over finite fields.

### Where

Millikan 2099, Pomona College

## Voting methods, Fourier analysis, and Gaussian geometry

### Speaker

Steven Heilman (UCLA and University of Notre Dame)

### Abstract

Given votes for candidates, what is the best way to determine the winner of the election, if some of the votes have been corrupted or miscounted? As we saw in Florida in 2000, where a difference of 537 votes determined the president of the United States, the electoral college system does not seem to be the best voting method. We will survey some recent answers to the above question along with some open problems. These results use tools from probability, from discrete Fourier analysis and from the geometry of the Gaussian measure on Euclidean space. Answering the above voting question reveals unexpected connections to Khot's Unique Games Conjecture in theoretical computer science and to the MAX-CUT problem and its variants. We will discuss these connections and present recent results and open problems.

### Where

Millikan 2099, Pomona College

## Formal groups -- a finite history

### Speaker

Ghassan Sarkis (Pomona College)

### Abstract

A formal group is a power series that behaves like a group law and exhibits deep connections to number theoretic results from reciprocity laws to elliptic curves, and from local class field theory to dynamical systems. In this talk, I will introduce formal groups, describe their more salient features, and trace some of their unexpected cameos since (and sometimes before) their invention/discovery more than 70 years ago, ending with an answer to a recent question from the internet (and Jonathan Lubin).

### Where

Millikan 2099, Pomona College

## Special chain partitions of the normalized matching posets

### Speaker

Alex Hof (Pomona College)

### Abstract

It is well known that any partially ordered set can be partitioned into totally ordered subsets, or chains, and in particular that it is possible to obtain such a partition where the number of chains is equal to the poset's width. However, the constraints on the sizes of these chains have yet to be fully understood. In this talk, we examine the conjectures of Füredi, Griggs, and Sands, which concern this question and apply variously to the Boolean lattice and to the larger class of posets which possess the so-called normalized matching property.

### Where

Millikan 2099, Pomona College

## Tests of uniformity for discrete circular data

### Speaker

Mike Orrison (HMC)

### Abstract

Suppose you are given a data set that can be viewed as a non-negative integer-valued function defined on a finite set. A natural question to ask is whether the data can be viewed as a sample from a uniform distribution on the set, in which case you might want to apply some sort of test of uniformity to the data. In this talk, I will describe some of the recent work Anna Bargagliotti (Loyola Marymount University) and I have been doing to better understand tests of uniformity when the underlying set is a discretized circle. In particular, I will explain how the discrete Fourier transform and some simple ideas from digital signal processing can be used to create a wide variety of user-friendly tests of uniformity for discrete circular data.

### Where

Millikan 2099, Pomona College

## Prime numbers and their biases

### Speaker

Stephan Garcia (Pomona College)

### Abstract

We survey some classical and modern results about prime numbers. In particular, we highlight remarkable biases displayed by prime pairs that were discovered by Pomona undergraduates in 2017.

### Where

Millikan 2099, Pomona College

## An algebraic perspective on integer sparse recovery

### Speaker

Lenny Fukshansky (CMC)

### Abstract

A basic problem of compressed sensing is that of sparse recovery: accurately reconstruct a sparse signal x in R^d from its noisy measurement Ax+e in R^m, where m < d, A is an m x d compression matrix, and e is an error vector. The signal x is called s-sparse if it has only s nonzero coordinates, where s is usually no bigger than m. There are known recovery algorithms for the situations when m = O(s lod (d)), which usually rely on incoherence or restricted isometry properties of the matrix A. In this talk, we will discuss an approach to this problem when the sparse signal vector x is assumed to have integer coordinates, a direction with many potential applications in digital communications and related fields. We will outline a number-theoretic construction of matrices A that allow for efficient recovery over noisy channels. We will also talk about some limitations of this approach. This is joint work with Deanna Needell and Benny Sudakov.

### Where

Millikan 2099, Pomona College

### Misc. Information

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