# Potential Topics

Here is a list of potential topics. You may also consider browsing topics by tag. First-year students may wish to browse our list of first-year-friendly topics. We have 76 potential topics for you to choose from, or you could create your own! Once you have found a topic, whether a suggested topic or one of your own choosing, please fill out the form to submit a talk proposal.

Remember these topics are meant as suggestions to spark ideas! Feel free to take any topic mentioned here and make it your own. Click each topic for more information.

### Abstraction in Technical Computing

Scientists, mathematicians, and engineers have an increasing need to write efficient computer programs but they tend to use specific languages for technical computing like Matlab or Mathematica rather than general purpose languages. The suggested reference discusses the properties a general purpose language would need to be able to also handle technical computing – one possibility for allowing efficient technical computing involves a powerful type system and dispatch system to enable generic, staged, and higher-order programming.

Required Background: First year CS at the level of CS 135 and 136.

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### Almost Integers and Pisot Numbers

Sometimes some non-integers like $e^{π} - π = 19.9990999791...$ are curiously close to integers. While often this is just a numerical coincidence, sometimes there is a clear mathematical reason for certain numbers to be almost integers. Pisot numbers, which are special types of algebraic integers, provide a systematic means to construct almost integers that approximate integers at an exponential rate. Of course, Pisot numbers are interesting on their own right and an interested speaker can also venture into the connections to Beta expansions (see the 2nd and 3rd references below), or fractals (see the 4th reference below).

Required Background: Basic analysis at the level of 147 and 138, algebra at the level of 145. Any other background depends on the direction the speaker takes with this topic.

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### Automated Verification of Computational Systems

It was realized in the 1960s that mathematical logic provides a suitable language to formulate properties of programs. This idea was formalized by Hoare who developed Hoare logic as a means to prove program correctness. However, this technique didn't apply directly to concurrent programs. An alternative approach called Temporal Logic was proposed by Amir Pneuli in 1977 which did provide a means to reason about concurrent programs. This topic studies this and then delves into the fundamental ideas behind model checking.

Required Background: Basics of computer science at the level of CS 146, Logic at the level of CS 245.

Possible reference materials for this topic include

• Grumberg, O., & Veith, H. (Eds.). (2008). 25 years of model checking: history, achievements, perspectives (Vol. 5000). Springer.

### Completeness, Incompleteness, and Turing Machines

This topic studies the connection between Gödel’s incompleteness theorems and turing machines and how each implies the other

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### Complex Event Processing Systems

The ever-increasing amount of information that needs to be processed has led to the development of Complex Event Processing systems such as Apache Storm or Twitter Heron. These systems distribute a workload over many machines in a cluster, and offer both efficiency and fault-tolerance.

Possible reference materials for this topic include

• Kulkarni, S., Bhagat, N., Fu, M., Kedigehalli, V., Kellogg, C., Mittal, S., ... & Taneja, S. (2015, May). Twitter heron: Stream processing at scale. In Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data (pp. 239-250). ACM. doi:10.1145/2723372.2742788

### Complexity of Matrix Multiplication

Volker Strassen showed that $n^3$ matrix multiplication was not optimal in 1969. Since then, new algorithms such as Coppersmith-Winograd have further improved the time complexity of matrix multiplication. It is conjectured that matrix multiplication is possible in $O(n^{2+ɛ})$ for any $ɛ>0$, however small. This is one of the few remaining open problems in finite-dimensional linear algebra.

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### Compressed Sensing

Compressed sensing is about minimizing the information gathered and stored by sensors, reducing the need for file compression later on for transmission. This can reduce costs for certain applications, such as non-visible wavelength cameras.

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### Constructive Mathematics

Constructive mathematics, or mathematics without the law of the excluded middle, is becoming more popular thanks to connections with computer science, category theory, and topology. Its logical foundations may be initially difficult to grasp for those used to a classical system.

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### Continued Fractions and Hyperbolic Geometry

This topic studies the beautiful connection between continued fractions and the famous Farey tessekation of the hyperbolic plane.

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### Conway’s Game of Life

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### Costas Arrays

Permutation matrices for which all $\frac{n(n-1)}{2}$ displacements between all pairs of 1s are distinct are known as Costas arrays, and these have important applications in sonar. The Welch construction of Costas arrays uses concepts from number theory, whereas the Lempel-Golomb construction uses concepts from field theory.

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### Curry–Howard–Lambek Correspondence

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### Dealing with Missing Data

Data are rarely perfect. Robust data science tools must have ways to deal with missing data. However, this is not always easy. A balance must be struck between performance and convenience.

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### Differential Equations on Fractals

Possible reference materials for this topic include

• Robert Strichartz : Differential Equations on Fractals - A tutorial

### Dirichlet's Theorem on Primes in Arithmetic progression

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### Distribution of Primes Modulo p

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### Down with Determinants

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### Entropy in Mathematics and Information Theory

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### Expander Graphs and Applications

Expander graphs are sparse graphs with strong connectivity properties and finds applications in complexity theory, designing of computer networks and the theory of error-correcting codes.

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### Fermi Estimation

Physicist Enrico Fermi was known for his impressive order-of-magnitude estimation. He was famously able to estimate the yield of the Trinity test atomic bomb to about a factor of two. Why is Fermi estimation so useful and accurate? What are the techniques to make better approximations? How can and do scientists, mathematicians, engineers, and others use Fermi estimation in their work?

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### Galois Field Arithmetic

A Galois field is a finite field and are used in a variety of applications, including in classical coding theory and cryptography algorithms. This topic studies how to efficiently optimize arithmetic in such fields.

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### Geographic Authentication Schemes

Users often choose passwords that are easy to remember but also easy to guess. Both textual and graphical passwords have failed to provide a viable solution to this usability-security tension. It thus remains a critical challenge in password research to design an authentication scheme that is resilient to guessing attacks while offering good memorability.

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### Geometry and Billiards

The second topic studies which billiard table shapes are optimal.

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### Gillespie Algorithm

The Gillespie Algorithm for stochastic equations is used heavily in a number of fields of applied mathematics, in particular computational systems biology.

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### Hindley–Milner Type System

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### Hyperbolic Geometry

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### IEEE 754-2008: Floating Point Arithmetic

Most programming languages provide a floating-point type. What is floating point, and how does it work? What caveats should programmers be aware of?

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### Integrals and Graph Theory

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### Iterated function systems

Aesthetically appealing fractal art can be created through interesting mathematical processes, including contraction mappings on metric spaces. Several well known fractals can be formalized mathematically and also appreciated visually. A talk on this subject would explore both mathematical and artistic aspects.

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### Jupyter Notebooks

Jupyter Notebooks are a must-have for any data scientist or engineer. They are available for a wide variety of programming languages, particularly Python.

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### Lattices, Linear Codes, and Invariants

This topic studies sphere packing and the connection to lattices and linear codes.

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### Lindenmeyer systems

Originally developed to model plant growth, L-systems are a logical approach to various scientific questions, as well as one of many ways to generate artistically pleasing fractals.

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### Lindstrom-Gessel-Viennot Lemma

This topic studeis an interesting connection between combinatorics and determinants.

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### Low Dimensional Geometry

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### Markov Equation and Irrational Numbers

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### Molecular Set Theory

In 1960, A.F. Bartholomay wrote about a mathematical model of chemical reaction mechanisms using set theory.

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### Monsky's Theorem

This topic studies the proof of Monsky's theorem that it is not possible to dissect a square into an odd number of triangles of equal area.

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### Optimal Stopping Theory and the Secretary Problem

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### P vs. NP

Possible reference materials for this topic include

• Arora, Bazak: Computational Complexity

### Patterns in Primes

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### Penrose Tilings

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### Plausible Conjectures

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### Predicate Dispatch

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### Presburger Arithmetic

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### Principle of Least Action

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### Protein Structure Prediction

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### Purely Functional Data Structures

Persistent (or purely function) data structures are those whose operations do not mutate the existing data structure, but rather create a new data structure that might share some state. These data structures are useful because there is no need to copy memory; rather the data structure can be shared among multiple objects. Techniques for making these data structures fast, memory-efficient, or real-time are an active area of research in functional programming.

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### Queueing Theory

Queueing theory is the mathematical study of waiting lines, or queues. In queueing theory, a model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

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### Quivers and Platonic Solids

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### Random Walks and Self-Avoiding Walks

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### Rational Tangles and Conway's Theorem

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### Rental Harmony Theorem

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### Representation Theory and Voting Theory

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### Separating Words Problem

Given two strings, we can write a computer program to tell them apart. A special kind of very simple computer program that accepts or rejects strings, using only a finite number of states while reading the string, is called a deterministic finite automaton. (These deterministic finite automata can also be expressed as a regular expression.) For any two strings of length $n$, there is some deterministic finite automaton (or regular expression) that recognizes one and not the other, and there is some smallest one. In the worst case, it is known (Robson 1989) that any two strings of length $n$ can be separated by an automaton of size $O(n^{2/5} (\log n)^{3/5})$, but this is not known to be optimal. The optimal upper bound remains an open problem.

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### Sparse Matrices

Sparse matrices are frequently used in scientific and numerical computation. How do they work? What new findings have there been? Speakers are encouraged to specialize this broad topic to a particular subfield of interest.

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### Sphere Packing, Lattices, and Groups

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### Symmetric Groups, Young Tableaux and Representation Theory

Possible reference materials for this topic include

• Bruce Sagan : The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions

### Symmetries and their Role in Physics

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### The Angel Problem

A game is played by two players called the angel and the devil. It is played on an infinite chessboard (or equivalently the points of a 2D lattice). The angel has a power k (a natural number 1 or higher), specified before the game starts. The board starts empty with the angel at the origin. On each turn, the angel jumps to a different empty square which could be reached by at most k moves of a chess king, i.e. the distance from the starting square is at most k in the infinity norm. The devil, on its turn, may add a block on any single square not containing the angel. The angel may leap over blocked squares, but cannot land on them. The devil wins if the angel is unable to move. The angel wins by surviving indefinitely.

The angel problem is: can an angel with high enough power win?

Required Background: Basic analysis at the level of 147 and algebra at the level of 145.

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### The Hyperbolic Crochet Coral Reef

Naturally growing coral and crochet share an interesting trait in common: they help visualize hyperbolic planes. This talk might explore aspects of geometry, art and craft, or biology.

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### The Joy of Factoring

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### The Mathematics of Gerrymandering

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### The Riemann Hypothesis

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### The Stochastic Lotka-Volterra Equation

The famous Lotka-Volterra equations model the population of a predator and a prey, which are of course closely related. Stochastic variations on the Lotka-Volterra equations can be an illuminating introduction to the methods used to solve stochastic differential equations.

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### The Strong Law of Small Numbers

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### Tropical geometry and Computational Biology

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of twonumbers is their minimum and the product is their sum. This turnspolynomials into piecewise-linear functions, and their zero sets into polyhedralcomplexes.

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### Understanding LLVM

The LLVM Compiler Infrastructure Project now powers a wide variety of software, including major programming languages including Rust and Julia, as well as compilers for the popular C and C++ programming languages. Understanding how LLVM works, and being able to read some LLVM bytecode, is extremely useful for optimizing code.

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### Unique Games Conjecture

The Unique Games Conjecture, if true, implies that many problems that lack exact polynomial time solutions also lack approximate polynomial time solutions. It was formulated by Subhash Khot in 2002 and is still unsolved.

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### Waring's Problem and writing numbers as sums of powers

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### What is Area?

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### Why are some functions not integrable in terms of elementary functions?

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### Wilf zeilberg method to computer verify combinatorial identities

One of the most exciting mathematical discoveries in the early 1990s was the Wilf-Zeilberger (WZ) algorithm that can be used for proving, evaluating, and discovering identities involving hypergeometric terms automatically by computer. And unlike computerized proof techniques in other areas, the computerized proofs generated by this method provides a certificate for the validity of a combinatorial identity, called a WZ-pair.

Required Background: Combinatorics at the level of Math 249, Calculus at the level of Math 147.

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