# Difference between revisions of "AMS Student Chapter Seminar"

(→Spring 2015: Added title and abstract for Eric's talk next week.) |
(→April 8, James Waddington: title and abstract) |
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===April 8, James Waddington=== | ===April 8, James Waddington=== | ||

− | Title: | + | Title: Goodstein's Theorem |

− | Abstract: | + | Abstract: One of the most important results in the development of mathematics are |

+ | Gödel's Incompleteness theorems. The first incompleteness theorem shows that no | ||

+ | list of axioms one could provide could extend number theory to a complete and | ||

+ | consistent theory. The second showed that one such statement was no | ||

+ | axiomatization of number theory could be used to prove its own consistency. | ||

+ | Needless to say this was not viewed as a very natural independent statement | ||

+ | from arithmetic. | ||

+ | |||

+ | Examples of non-metamathematical results that were independent of PA, but true | ||

+ | of second order number theory, were not discovered until much later. Within a | ||

+ | short time of each three such statements that were more "natural" were | ||

+ | discovered. The Paris–Harrington Theorem, which was about a statement in Ramsey | ||

+ | theory, the Kirby–Paris theorem, which showed the independence of Goodstein's | ||

+ | theorem from Peano Arithmetic and the Kruskal's tree theorem, a statement about | ||

+ | finite trees. | ||

+ | |||

+ | In this talk I shall discuss Goodstein's theorem which discusses the end | ||

+ | behavior of a certain "Zero player" game about k-nary expansions of numbers. | ||

+ | I will also give some elements of the proof of the Kirby–Paris theorem. | ||

==Fall 2014== | ==Fall 2014== |

## Revision as of 23:43, 2 April 2015

The AMS Student Chapter Seminar is an informal, graduate student-run seminar on a wide range of mathematical topics. Pastries (usually donuts) will be provided.

**When:**Wednesdays, 3:00 PM – 3:30 PM**Where:**Van Vleck, 9th floor lounge**Organizers:**Laura Cladek, Ryan Julian, Xianghong Chen, Daniel Hast

Everyone is welcome to give a talk. To sign up, please contact the organizers with a title and abstract. Talks are 30 minutes long and should avoid assuming significant mathematical background beyond first-year graduate courses.

## Spring 2015

### January 28, Moisés Herradón

Title: Winning games and taking names

Abstract: So let’s say we’re already amazing at playing one game (any game!) at a time and we now we need to play several games at once, to keep it challenging. We will see that doing this results in us being able to define an addition on the collection of all games, and that it actually turns this collection into a Group. I will talk about some of the wonders that lie within the group. Maybe lions? Maybe a field containing both the real numbers and the ordinals? For sure it has to be one of these two!

### February 11, Becky Eastham

Title: A generalization of van der Waerden numbers: (a, b) triples and (a_1, a_2, ..., a_n) (n + 1)-tuples

Abstract: Van der Waerden defined w(k; r) to be the least positive integer such that for every r-coloring of the integers from 1 to w(k; r), there is a monochromatic arithmetic progression of length k. He proved that w(k; r) exists for all positive k, r. I will discuss the case where r = 2. These numbers are notoriously hard to calculate: the first 6 of these are 1, 3, 9, 35, 178, and 1132, but no others are known. I will discuss properties of a generalization of these numbers, (a_1, a_2, ..., a_n) (n + 1)-tuples, which are sets of the form {d, a_1x + d, a_2x + 2d, ..., a_nx + nd}, for d, x positive natural numbers.

### February 18, Solly Parenti

Title: Chebyshev's Bias

Abstract: Euclid told us that there are infinitely many primes. Dirichlet answered the question of how primes are distributed among residue classes. This talk addresses the question of "Ya, but really, how are the primes distributed among residue classes?" Chebyshev noted in 1853 that there seems to be more primes congruent to 3 mod 4 than their are primes congruent to 1 mod 4. It turns out, he was right, wrong, and everything in between. No analytic number theory is presumed for this talk, as none is known by the speaker.

### February 25, David Bruce

Title: Mean, Median, and Mode - Well Actually Just Median

Abstract: Given a finite set of numbers there are many different ways to measure the center of the set. Three of the more common measures, familiar to any middle school students, are: mean, median, mode. This talk will focus on the concept of the median, and why in many ways it's sweet. In particular, we will explore how we can extend the notion of a median to higher dimensions, and apply it to create more robust statistics. It will be awesome, and there will be donuts.

### March 4, Jing Hao

Title: Error Correction Codes

Abstract: In the modern world, many communication channels are subject to noise, and thus errors happen. To help the codes auto-correct themselves, more bits are added to the codes to make them more different from each other and therefore easier to tell apart. The major object we study is linear codes. They have nice algebraic structure embedded, and we can apply well-known algebraic results to construct 'nice' codes. This talk will touch on the basics of coding theory, and introduce some famous codes in the coding world, including several prize problems yet to be solved!

### March 10 (Tuesday), Nathan Clement

*Note: This week's seminar will be on Tuesday at 3:30 instead of the usual time.*

Title: Two Solutions, not too Technical, to a Problem to which the Answer is Two

Abstract: A classical problem in Algebraic Geometry is this: Given four pairwise skew lines, how many other lines intersect all of them. I will present some (two) solutions to this problem. One is more classical and ad hoc and the other introduces the Grassmannian variety/manifold and a little intersection theory.

### March 25, Eric Ramos

Title: Braids, Knots and Representations

Abstract: In the 1920's Artin defined the braid group, B_n, in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. In fact, Jones was able to show that knot invariants can often be realized as characters of special representations of the braid group.

The purpose of this talk is to give a very light introduction to braid and knot theory. The majority of the talk will be comprised of drawing pictures, and nothing will be treated rigorously.

### April 8, James Waddington

Title: Goodstein's Theorem

Abstract: One of the most important results in the development of mathematics are Gödel's Incompleteness theorems. The first incompleteness theorem shows that no list of axioms one could provide could extend number theory to a complete and consistent theory. The second showed that one such statement was no axiomatization of number theory could be used to prove its own consistency. Needless to say this was not viewed as a very natural independent statement from arithmetic.

Examples of non-metamathematical results that were independent of PA, but true of second order number theory, were not discovered until much later. Within a short time of each three such statements that were more "natural" were discovered. The Paris–Harrington Theorem, which was about a statement in Ramsey theory, the Kirby–Paris theorem, which showed the independence of Goodstein's theorem from Peano Arithmetic and the Kruskal's tree theorem, a statement about finite trees.

In this talk I shall discuss Goodstein's theorem which discusses the end behavior of a certain "Zero player" game about k-nary expansions of numbers. I will also give some elements of the proof of the Kirby–Paris theorem.

## Fall 2014

### September 25, Vladimir Sotirov

Title: The compact open topology: what is it really?

Abstract: The compact-open topology on the space C(X,Y) of continuous functions from X to Y is mysteriously generated by declaring that for each compact subset K of X and each open subset V of Y, the continous functions f: X->Y conducting K inside V constitute an open set. In this talk, I will explain the universal property that uniquely determines the compact-open topology, and sketch a pretty constellation of little-known but elementary facts from domain theory that dispell the mystery of the compact-open topology's definition.

### October 8, David Bruce

Title: Hex on the Beach

Abstract: The game of Hex is a two player game played on a hexagonal grid attributed in part to John Nash. (This is the game he is playing in /A Beautiful Mind./) Despite being relatively easy to pick up, and pretty hard to master, this game has surprising connections to some interesting mathematics. This talk will introduce the game of Hex, and then explore some of these connections. *As it is a lot more fun once you've actually played Hex feel free to join me at 3:00pm on the 9th floor to actually play a few games of Hex!*

### October 22, Eva Elduque

Title: The fold and one cut problem

Abstract: What shapes can we get by folding flat a piece of paper and making (only) one complete straight cut? The answer is surprising: We can cut out any shape drawn with straight line segments. In the talk, we will discuss the two methods of approaching this problem, focusing on the straight skeleton method, the most intuitive of the two.

### November 5, Megan Maguire

Title: Train tracks on surfaces

Abstract: What is a train track, mathematically speaking? Are they interesting? Why are they interesting? Come find out!

### November 19, Adrian Tovar-Lopez

Title: Hodgkin and Huxley equations of a single neuron

### December 3, Zachary Charles

Abstract: An addition chain is a sequence of numbers starting at one, such that every number is the sum of two previous numbers. What is the shortest chain ending at a number n? While this is already difficult, we will talk about how addition chains answer life's difficult questions, including: How do we compute 2^4? What can the Ancient Egyptians teach us about elliptic curve cryptography? What about subtraction?