Undergraduate Colloquium

Fall 2016
Wednesdays 12:55 - 1:45
LCB 225

Pizza and discussion after each talk
Receive credit for attending
Past Colloquia

August 24    No Talk

August 31     Andrejs Treibergs
Helly’s Theorem with Applications in Combinatorial Geometry
Abstract: Helly’s Theorem is a surprising simple statement about convex sets that deserves to be better known. Given a collection of compact convex sets in the plane, if any three of them have a point in common then all of them have a point in common. We shall indicate a proof and discuss generalizations and applications. The diagram illustrates one application, a theorem of Rey, Pastor and Santalo: Consider a set of parallel line segments in the plane. If any three of them have a common transversal then all of them have a common transversal.

Lecture Slides

September 7     Evelyn Lamb
Visualizing Hyperbolic Geometry
Abstract: For two thousand years, mathematicians tried to prove that Euclidean geometry, the geometry you probably learned in high school, was all there was. But it's not! In the early nineteenth century, János Bolyai and Nikolai Lobachevsky independently discovered that by tweaking one of Euclid's postulates, geometry can look totally different. We will explore the rich world of hyperbolic geometry, one of the new and beautiful systems of geometry that results from this tweak. Our guides on the adventure will be mathematically inspired artists and artistically inspired mathematicians, including M.C. Escher, Daina Taimina, and Henry Segerman.

September 14     Adam Boocher
Are Fair Elections Possible?
Abstract: When a third party candidate enters a political race in the US, they are often blamed for changing the outcome. In response, many propose alternative systems, e.g. instant runoff voting. Astonishingly, in the 1950s, Kenneth Arrow proved that in an election with three or more candidates there is no ``fair'' way to count the votes! In this talk we'll say what we mean by fairness and what sort of vote-counting methods we allow. We'll discuss this theorem - give a proof, and talk about whether or not this theorem matters. We'll even have a mini-election and you can witness some election-fixing in action!

September 21     Adam Kodeda, Goldman Sachs
Exploring Quantitative Roles at Goldman Sachs
Abstract: Goldman Sachs’ Strategies business unit is a world leader in developing quantitative and technological techniques to solve complex business problems. Working with the firm’s trading, sales, banking, and investment management divisions, strategists (“strats”) use their mathematical and scientific training to create financial products, advise clients on transactions, measure risk, and identify market opportunities. We will provide an overview of systematic market-making and its open-ended challenges, discuss our summer internship program, and present previous intern projects.

Bio: Adam Kodeda is a Vice President in Goldman Sachs' Fixed Income, Currencies, and Commodities Systematic Market-Making group. Adam joined the firm in 2008, and has developed various facets of the firm's electronic and algorithmic currency trading platform. Prior to Goldman Sachs, Adam helped firms optimize equity derivative pricing computations at a boutique consultancy, and developed embedded operating systems and formally verified type-1 virtual machines at Microsoft. Adam earned a BMath from University of Waterloo.

September 28     Karl Schwede - Director of Graduate Studies
Applying for and attending graduate school
Abstract: This weeks Undergraduate Colloquium will be aimed at helping undergraduates answer the following questions:
  • Should I apply to graduate school?
  • How do I apply to graduate school?
  • What will it be like when I'm in graduate school?
There will be a short presentation followed by a panel discussion. Faculty, postdocs and current graduate students from all areas of the department will be there to give their points of view and to answer your questions. This discussion should be useful both for students who will be applying this fall and students who are just starting to think about going to graduate school and may be applying in future years.

October 5     Thomas Goller
The Gory Harp
Abstract: What do knight's tours, map coloring, and the internet have in common? Their internal music is graphic.

October 12     No Talk - Fall Break

October 19     Sean McAfee
Basics of Game Theory
Abstract: Developed by John von Neumann in the 1940's, game theory is the study of competitive decision making. It is an incredibly useful tool that has found applications in many diverse subjects such as psychology, economics, and computer science. In this talk, we will introduce some of the main ideas behind game theory, and give some interesting (and fun!) applications. Absolutely no math background besides basic sarithmetic is required!

October 26     Kelly MacArthur
Equivalent Inequalities
Abstract: Start with a < b . Does this necessarily mean f(a) < f(b)? What about g(a) < g(b)? Or is h(a) < h(b) true? It's likely that you learned to “flip” the sign in an inequality when you multiply or divide both sides of an inequality by a negative number. Perhaps you were even left believing that's the only operation applied to both sides of an inequality that requires switching the sign. But, is that really the only time we need to switch the sign? What is the underlying mathematical reason that we switch the sign anyway? And, is there a mathematical operator that we could apply to both sides of the inequality where we can't determine whether to switch the sign or not? Are there operators that we are allowed to “do to both sides” in an equation that are off limits in inequalities? We'll explore these questions and uncover the answers together. Be prepared to think and work!

November 2     Christel Hohenegger
Calculus of bubbles and drops
Abstract: Why do small bubbles in a soda pop louder than larger bubbles? Why does water pinching off a thin jet of fluid take a circular shape? How do falling raindrops maintain their spherical shapes? Why does mercury rise in the narrow tube of a thermometer? How do small insects walk on water? Surprisingly, we can use tools of multivariable calculus to take a first step towards answering these questions once we learned some of the underlying physics principles.

November 9     Ethan Levien
Adding needles to the haystack: an introduction to Monte Carlo simulation of rare events.
Abstract: The Monte Carlo method is the simplest method for estimating statistics of random processes, yet it remains one of the most widely used tools in scientific computing. This talk will introduce the Monte Carlo method in the context of a simple random walk. We will see that in its simplest form the method is ill-suited for obtaining information about rare events and explore some avenues for improvement. No specific background knowledge is necessary.

November 16     Anna Romanova
A Glimpse of the Fourth Dimension: Exploring Higher Dimensional Polytopes
Abstract: Platonic solids are the most symmetric objects that we can construct in three dimensions. Built from regular polyhedra, they look the same at each vertex. One of the major revelations of classical Greek mathematics was that the fact that there are only five solids possessing this maximal symmetry - the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. But what happens if we expand our view to four dimensions? In this talk, I will introduce the four dimensional analogues to the platonic solids and discuss how we can understand objects that we cannot see. Along the way, we'll encounter a woman with a special ability to "see" four dimensions, a hypercube of monkeys, and the view from inside a hyperdodecahedron. Bring your tablets and smart phones for full immersion.

November 23     Donald Robertson
Szemerédi's theorem and the Green-Tao theorem
Abstract: The Green-Tao theorem states that there are arbitrarily long arithmetic progressions whose terms are prime numbers. A key ingredient in the proof is Szemerédi's theorem on arithmetic progressions in large sets of integers. In this talk I will discuss Szemerédi's theorem and its role in the proof on the Green-Tao theorem.

November 30     Peter Alfeld
Hotel Infinity
Abstract: You are the owner of Hotel Infinity. It has infinitely many rooms, and it's full. A new guest arrives and insists you give her a room. How do you accommodate her? The next day, a family with infinitely many members arrives, each of whom wants a private room. The next day infinitely many families, each with infinitely many members, arrive. Each family member insists on a private room. You can do it! Infinity is different.

December 7     James Gossell
Avoiding Right Triangles in Square Lattices
Abstract: Imagine a game in which your goal is to select as many points as you can from an n by n square lattice. There is just one rule: No three points in your selected set may form a right triangle. For n > 1, you will find that you can pick 2n - 2 points from the lattice without forming any right triangles. But try as you may, it is impossible to avoid forming a right triangle if you pick at least 2n - 1 points.

In this talk, we will prove this result and examine several variations to this game. This talk will be highly interactive with no special mathematical background required.