IU Math Club
IU Math Club
In-person at Rawles Hall 105, Lounge
In-person at Rawles Hall 105, Lounge
Some related problems from complex analysis and the calculus of variations
Terence Harris (Mathematics, UW Madison)
I will introduce some (still open) problems from complex analysis and the calculus of variations, and explain why they are related. Then I will discuss how ideas from this area are related to the problem of constructing a function with a prescribed finite set of derivatives.
The Dragon and The Paperfolding Sequence: Math Behind the Math Department T-Shirt
Larry Moss (Mathematics)
I will be discussing the 'dragon curve' (also called the 'Heighway dragon') which you may find at https://math.indiana.edu/research/gallery/heighway.html This curve is also printed on a t-shirt that our department once printed.
The talk is also a participatory study of the 'paperfolding sequence', a sequence of 1's and -1's that is related to the dragon curve. What I mean by 'participatory' here is that to attend the talk, you need to bring a pencil or pen along. I'll provide several strips of paper to fold repeatedly, and then you'll write on the strips and ponder the pattern that emerges.
The talk will be accessible to all math-curious students. At some points, I will connect matters to automata theory and to induction, but you don't really need to have seen those topics. If you like the picture, I promise that the talk will be fun.
Special Halloween event
Game night
An introduction to projection theory
Shengwen Gan (Mathematics, UW Madison)
Given a set of points A in R^2, we want to project A to lines in R^2 that pass through the origin. For a generic line, the projection of A onto that line has the same cardinality as A. A natural question is to ask about the property of those lines so that the projection of A onto them has a cardinality much smaller than A. We will talk about this problem and its relationship with other problems, like the Szemeredi-Trotter theorem, and the Furstenberg set problem.
First meeting, game night
Tour to the McCala house
Seeking for sequences
Dylan Thurston (Mathematics)
Given a sequence of integers that you suspect follows a pattern, perhaps a polynomial one, how can you find the pattern? The technique of repeated differences elegantly gives a solution to this, and admits several generalizations.
Embedding complex manifolds in C^k
Vitor Braga (Mathematics)
When considering embeddings of smooth manifolds, $f : M → N$, one can always find such an embedding of any smooth manifold to a Euclidean space
$\mathbb{R}^k$, for $k$ big enough. This does not follow nicely to the complex case, where we need very specific constraints in a complex manifold $M$ for it to admit an embedding to a complex space $ \mathbb{C}^k$, for $k$ big enough. In this talk, I will present some of the foundations of complex manifold theory and show the necessary conditions for a complex manifold M to admit such an embedding, while also comparing it and motivating it from the real case.
Math behind the theory of black holes
Shouhong Wang (Mathematics)
We shall give a brief introduction to Einstein’s theory of general relativity. Then we derive two special solutions of the Einstein equations: the Schwarzschild solution, and the Tolman-Oppenheimer-Volkoff solution. We will then discuss the notion and theories of black holes.
NO EVENT
Game night: math contest
N-tuplewise independence and the Central Limit Theorem
Richard Bradley (Mathematics)
In basic probability theory and statistical inference, arguably the most important theorem is the classic Central Limit Theorem, which states in essence that under reasonable conditions, the probability distribution of the sum or average (suitably normalized) of a ``large sample'' of ``independent'' observations from a given ``population'' will approximately follow a standard bell curve. It turns out that for any given fixed positive integer N chosen beforehand (no matter how large), that theorem would fail to hold if its assumption of ``independence'' were replaced by the (weaker) assumption of ``N-tuplewise independence'' (that is, that every N of the observations are independent). This talk will give a gentle exposition of the Central Limit Theorem as well as a few key ideas in the construction (for N chosen beforehand, no matter how large), in a joint paper with Alexander Pruss in 2009, of a particular counterexample that satisfies N-tuplewise independence.
Game night: math jeopardy
Callout meeting
Fractals and the Kakeya needle problem
Shukun Wu (Mathematics)
Fractals are geometric objects that look self-similar at arbitrarily small scales. I will first discuss some examples of fractals including the Cantor sets and the Sierpiński triangle.
Then I will introduce the 100-year-old Kakeya needle problem, a fundamental real-geometry problem. The target object of the Kakeya needle problem is the Kakeya set, which is a set that contains a unit line segment for every direction. There are only a few constructions for "interesting" Kakeya sets, and they all possess certain fractal structures. For the Kakeya needle problem, it basically asks: Must a Kakeya set be "large"? I will formulate the question in a natural way that I like most, and discuss how it is connected to other branches of mathematics.
Coverings and complexity: two interpretations of Hausdorff dimension
Jacob Fiedler (Mathematics, UW Madison)
How can you tell apart the size of two objects, both of which have zero volume? Fractal dimension is one of the most useful tools for this purpose, and Hausdorff dimension is one of the most robust notions of fractal dimension. Hausdorff dimension is classically defined in terms of efficient coverings of a set. However, an alternative, equivalent definition using tools from algorithmic information theory has recently been established. This new definition has already been used to make progress on a number of problems related to the size of interesting sets. We will discuss both definitions, some of the major problems in the study of dimension, and some recent applications of this new work.
Study session (Putnam exercise)
Game night
Amenable groups and the failure of the Banach-Tarski paradox in two dimensions
Matt Bainbridge (Mathematics)
In this talk, we'll find out what is an amenable group and how they can be used to show that the Banach-Tarski paradox fails in two dimensions. This talk is a sequel to last week's talk, but you will be able to follow even if you didn't attend last week.
Amenable groups and the Banach-Tarski paradox
Matt Bainbridge (Mathematics)
In this talk, we'll find out what is an amenable group and how they are related to the famous Banach-Tarski paradox: a three-dimensional ball can be cut into five pieces and then reassembled to form two copies of the same ball.
All (categorical) concepts are Kan extensions
Yun Liu (Mathematics)
Category theory is a language that mathematicians use to build abstract framework to understand and compare different mathematical structures. Categories consists of information of a class of objects that shares similar properties and how they interact with each other (morphisms between objects), functors between categories provides us a "machine" to transform from one category to another, and natural transformations connects two functors in a compatible way.
In this talk we are going to review basic notions of category, functors and natural transformations, and discuss how to view Kan extension as a way of best approximation to an extension of a functor F:C->E along another functor K:C->D.
Saunders Mac Lane states in his book Categories for Working Mathematician that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory". To justify the slogan, we will discuss several examples, including group representations, adjunctions, and derived functors.
What is a Circle
Michael Larsen (Mathematics)
I'll quickly recall the standard definitions; the equivalences between them involve some of the most famous theorems in mathematics, like the Pythagorean Theorem and Euler's formula for e^{ix}. The trig functions pop out, and I'll describe an alternative version of history in which inverse trig functions were discovered first, then ordinary trig functions, and finally the circle. This is what actually happened in the analogous situation of elliptic curves.
Like elliptic curves, circles are groups as well as curves. I'll recall what that means and examine the group law from the various points of view and look at some consequences for trig, geometry, and number theory.