cSplash 2015

Zero Knowledge Proofs
Here you will learn about a different type of proof - zero knowledge proofs. Such a proof leaves you certain of the fact being proved but nothing more. For example, I might convince you I know a password, but
without revealing it. How can this be possible?
I will start by convincing you that two digits written on two cards
are equal without revealing the digits. To do this I will use some
additional cards as helpers.

PolyPasswordHasher: Using Math to Gain the Higher Ground Against Password Crackers
Large companies are repeatedly attacked and their password databases are leaked to hackers or the public. The stored passwords are typically protected with hashing and key stretching. However, password hashing and key stretching offer very limited protections against cracking and have turned password storage into an arms race between supposedly secure servers and the specialized password cracking hardware used by attackers. An attacker, using the same hardware as the defender, can try N password guesses in N times the effort it takes the defender to verify a password.

How to Discover Viral Content for Fun and Profit!
Ever wondered what kind of content goes viral on the Internet? In this talk, we will use computers and math to answer exactly this question. The work is derived from years of tinkering with content and is composed in an easy to understand format to share the findings.

Programming in an Eggshell
In 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 translated into this very simple game. To understand how this works, we will study Church encodings (no praying involved - I promise) and fixpoint 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.

 Fundamental and Elegant Theorems in Economics
This class will introduce a series of theorems from economics which are both fundamental in terms of their economic content and elegant in terms of their mathematical formulation and proof. In particular, we shall discuss the First welfare theorem, a preference representation theorem, Arrow's impossibility theorem, and Gibbard-Satterthwaite theorem.
 

Game Theory
In this session, I will give an introduction to game theory. We will start with two-player zero-sum games (such as Chess, Checkers, or Tic-Tac-Toe) and the Minimax Theorem for such games, proved by the famous mathematician John von Neumann in 1928. We will then look at games which are general-sum, i.e., which can involve win-win and lose-lose outcomes, and which may involve more than two players. These games can be studied in various ways --- one way is where players can form groups and act collectively, while another way is where players act individually.

Cosmology
The talk will begin with an introduction to the ideas of kinetic theory and suggest how the stress energy tensor is derived therefrom. I'll give a notion of what a tensor is and say why the version of the stress-energy tensor I favor is an improvement on the standard one. Whether I do it in terms of Newtonian physics or not will depend on the audience. I may give a brief (<10 minutes) explanation of general relativity.

Economics and Finance in the Vacuum
This class will introduce a series of results from economics and finance which consist in elegant mathematical formulations not unlike physics in the vacuum. We shall discuss a proposition called Ricardian Equivalence (regarding the effect of taxes and government expenditures on economic activity when economic agents are not borrowing constrained) and the Modigliani-Miller theorem (regarding the effect of corporate financing on stock prices).

Mathematics of a Rubik's Cube
Everyone at some point has tried to solve a Rubik's cube. In this class, we will look at some elementary group theory behind a 3 by 3 Rubik's cube. As a consequence, we will see that there are only very few elementary set of operations one needs to perform to solve the Rubik's cube. At the end of the class, everyone will be able to solve a 3 by 3 Rubik's cube starting from any initial configuration.

Love on Top(ology)
Topology is considered one of the "big three" when it comes to mathematics. In general, topology is the study of structures and the properties that they share. For this class, we will get an introductory feel for pointset topology, the defining axioms, and some basic results redefined in topological terms.

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