## cSplash 2018

Historical perspectives on Deep Learning for Object RecognitionWhen viewed from outside, the progress of science seems to consist of a few leaps, each of which is separated from others by many years if not decades. Similarly, looking at the exploding media coverage of so-called successful progress in AI these days, it is easy to think that there was a recent leap in our understanding of AI that led to this explosion. When we look more carefully, it becomes painfully obvious that each of such leaps is not a leap out of nowhere but an accumulation of hundreds, thousands or even more scientists working for decades tirelessly. This talk aims to be a brief and incomplete but still useful tour going through these small, incremental advances the whole field of computer science and AI has made in order to arrive at the modern AI technology. The Math of a Zombie Outbreak!What would happen if a zombie epidemic spread across the United States? We've seen it on TV shows like “The Walking Dead”, but how can we use mathematical modeling to understand (and fight!) the outbreak? We'll model the outbreak together, i.e. decide on the rules of our zombie model. I'll demonstrate how to use agent-based models and differential equations to describe zombie infections and zombie movement and show computer simulations of more complicated zombie models. Effectively, this will be a cross of disease and population dynamics modeling. This course can be viewed as an introduction to Mathematical Modeling. Zombies, beware! Projective Geometry and Pappus' TheoremThe projective plane is an amazing place. Every pair of lines meets in a point. Circles, ellipses, parabolas, and hyperbolas are all the same shape; and all quadrilaterals are the same shape. You can turn all the points into lines and all the lines into points, and nobody will notice. The price you pay is that you can't compare two distances or two angles, and, if you have three points on a line, you can't say which one is in the middle. This lecture will give an introduction to projective geometry, one of the most important examples of a non-Euclidean geometry. We will explain how projective geometry relates to perspective in drawing and images and how you can use projective geometry to prove the astonishing and important Pappus' theorem. Pappus' theorem: Draw two lines L and L'. Choose three points A, B, C on L and D, E, F on L'. Let X be the intersection of AE with BD; let Y be the intersection of AF with CD; and let Z be the intersection of BF with CE. Then X, Y, and Z are collinear. Erdos MagicPaul Erdos was a giant of twentieth century mathematics. We explore Erdos Magic, aka The Probabilistic Method. Here to show the existence of combinatorial objects one examines the (appropriately defined) random object. Example: There exists a FINITE tournament (every pair plays, no ties) where for every set of 10 players there is a player that beats all of them. Erdos had a unique personality and Prof. Spencer will discuss working with "Uncle Paul." Money MattersMost of the fiscal decisions we make every day are determined by mathematics, as are fiscal policies made by governments and firms. How do we choose what to invest in when there are multiple variables and risks? This talk will explore the basics of optimization in the context of economics and finance to provide an insight into how economies and firms try to achieve the optimum in terms of profit, portfolio returns or budgets. Concepts covered include Lagrangian and constrained optimization, discounting, projecting future income streams using calculus and statistics, and how we can account for risks in our projections. Bundling: Empirical Analysis and Mathematical ModelingBundling refers to the sale of several goods in a package. The talk focuses on the empirical analysis of a particular market, retail DVD sales; and then proposes a dynamic mathematical model to explain demand for, and sales of, DVDs. The model is calibrated to replicate several features of the actual data; and is then used to answer several questions, including the determination of the optimal bundling discount (the difference between the price of A plus the price of B minus the price of the AB bundle). Programming in an EggshellIn this lecture, you will learn a simple puzzle game called Alligators and Eggs. What does this have to do with programming, you might ask? Quite a lot! Surprisingly, every computer program can be encoded into this very simple game. To understand how this works, we will study Church encodings (no praying involved - I promise) and fix-point combinators. Along the way, we will touch on fundamental questions in computer science such as "What is computable?". I will end by giving you a glimpse of a real programming language. You will see that programming is actually like playing alligators and eggs in disguise. The Maths Behind BitcoinWe will show some of the mathematical ideas behind cryptocurrencies such as bitcoin, namely the mining of new bitcoin and the role of the blockchain. Mining is how bitcoin is introduced in the market and the blockchain is the unalterable public record of all bitcoin transactions. We will explain the mathematical concepts that ensure the security of bitcoin, and why it is hard to counterfeit. We will also play a simple mining game to illustrate the underlying algorithms. Your Crazy Uncle, That Weird Neighbor and Some Other 433 Complete Strangers: Why the House of Representatives Should be Replaced by 435 Randomly Chosen CitizensThis short talk will explore the history of random selection (sortition) in politics and put forward the case that it presents several advantages over elections. Mathematics of Cell DivisionBiology and medicine starting to be mathematical is a great scientific revolution of our days, similar to mathematics revolutionizing physics in the last few centuries. One of the most fundamental biological processes is cell division. Understanding molecular machines of the dividing cell requires very simple but profound mathematical estimates employing concepts from calculus and probability theory. I will show how simple calculations help understanding remarkable speed and accuracy of the cell division. |
Nuclear Fusion: The Energy of the Future?Nuclear fusion has long been hailed as the ultimate solution to our current energy crisis. However, to understand how nuclear fusion works, we must first understand how plasmas behave. This remains a very difficult problem, for several reasons. The first, is that our understanding of the motion of fluids in general is still developing, with a million dollar prize available to anyone who can prove that the equations for motion of ordinary fluids are well-formulated. Furthermore, our understanding of plasma physics is still being rapidly developed, both by physicists and mathematicians. In this presentation, we will discuss the basic scientific and mathematical ideas behind fusion power and plasma physics, as well as its prospects for use as a future energy source. :Predicting the outcome of an electionThe outcome of an election can be predicted very precisely by polling a small number of voters. The goal of this talk is to explain why this is the case using basic tools from probability theory. In addition, we will discuss why in practice some elections are difficult to predict. The only prerequisites are basic calculus and some curiosity. How Strangers Cooperate - An Introduction To BitcoinIn this talk, we will explore the mathematics behind cryptocurrencies, by looking at the Bitcoin protocol. Following a brief introduction to money, I will discuss the fundamental problem that Bitcoin solved, and how it solved it. I will focus on the inherently mathematical nature of this system - so you know what the hype and speculation is all about! Math and OrigamiOrigami is the art of paper folding from a single sheet of square paper without gluing or cutting, and has many applications to mathematical modeling in fields like architecture, robotics, and molecular biology. It has surprising ease in providing solutions to difficult math problems like doubling a cube or trisecting an angle. Students will have the opportunity to fold various origami constructions, play around with the models, and learn about its connections to graph theory, notably how crease patterns are related to the Four Color Theorem. No prior experience in graph theory is required, just knowledge of vertices, angles, and some geometry. The New Approach to the Doppler TheoryThe Doppler Theory is part of most High School Physics courses. Many students and professors remember the Doppler formula, however almost nobody understands it properly. Whether you are a student or a professor, you will learn something new at this presentation. Logical Discourse in Mathematics and ArtMathematics exists in a liminal space between science, philosophy, and art: what exactly is it, where does it come from, and what is its mode of expression? Several interesting parallels obtain in comparing mathematical proofs and theories to art (from painting to sculpture to music), most notably a privileging of logic. Some questions that will be explored include - in Western music, why have major and minor scales proven to be more enduring than others?
- Why has symmetry been privileged by many branches of mathematics?
- What opportunities are afforded by using particular ratios in the visual arts?
- spend some time outlining the provenance of mathematics and art respectively
- examine various theoretical similarities between the methods of mathematics and those of painting/music/etc.
- delve into a few specific examples of shared traits amongst the two domains with special regard for logical discourse
Problem Solving & Paradoxes within Probability TheoryProbability Theory is an amazing branch of mathematics that applies to everything from card games to quantum physics. We will learn how to calculate likelihood, and deconstruct 4 classical paradoxes in probability theory. The Central Limit Theorem and Law of Large Numbers will be examined from a high level and can proved if there is interest & time. Ramsey Theory: Order in DisorderRamsey theory answers the question, "under what circumstances can we find order in disorder?" For example, suppose six people are at a party and each pair of guests is either friends or strangers. Can we always find three people at the party who are mutual friends (or strangers)? How many people must we invite to the party so that four people are mutual friends? Five? The Ramsey number is the smallest number of people we must invite to the party to ensure a certain number of guests are mutual friends. We will generalize The Party Problem in the context of Ramsey's Theorem for graphs and discuss bounds for Ramsey numbers. Join us for one. BIG. party! Networks in the BrainWhile neural networks have found a number of uses in a wide variety of fields (indeed, it's become quite a buzzword), in this talk we will focus on neurally plausible neural networks. In particular, we will try and answer the following questions: - What is a neural network?
- What makes a neural network neurally plausible?
- What are some examples of neurally plausible neural networks?
- What questions about the brain have neural networks helped answer?
- What are some of the open questions in biologically plausible neural networks?
How to Create Jammed Packings of M&Ms on a Computer How many M&M candies are in a jar that has been shaken for a long time and filled until no more candies could fit? We can answer this question by simulating the process of packing ellipsoids in a container using a computer. Specifically, I will describe how we generated packing of ellipsoids in spherical containers using hard-particle molecular dynamics, and tell you what shape of ellipsoid packs the best. I will also discuss what makes a packing random or not, and what makes it jammed, and whether mathematics can answer these questions precisely or not. |