The graduate student seminar meets from 2:00 to 3:20 pm Thursdays in Goldsmith 226. Want to give a talk? Email Steve Hermes at srhermes@etc... for details.
The question of how to hold a fair election has plagued thinkers since the Enlightenment. In the last century, this question has been subjected to mathematical rigor; in his 1950 Ph.D. thesis, however, Kenneth Arrow gave a startlingly straightforward proof that, in some sense, no fair election procedure can exist. In this talk, I'll work through some historical proposals for voting methods, motivate and prove Arrow's theorem, and, if time permits, explore some ideas for how to pick up the pieces.
Mirror symmetry first made the leap from the world of theoretical physics into mathematics in 1991 when Candellas, de la Ossa, Green, and Parkes used the phenomenon to calculate the number of rational curves of a given degree on a generic quintic. To this day the mirror symmetry is mysterious from the eyes of Mathematics. In his 1994 ICM address, Maxim Kontsevich proposed a category theoretic interpretation of mirror symmetry called the Homologcial Mirror Symmetry Conjecture. In this talk I will attempt to explain the conjecture and develop the ingredients necessary for its formulation. The material should be accessible to everyone willing to take some of the rudiments of manifolds and differential forms on faith.
Can knotted surfaces be studied using knots? Yes. Can knot Floer homology, a very powerful method of studying knots, be used to study surfaces? That is the subject of my dissertation and the subject of the talk. I will explain the movie move theorem and a generalization to grid diagrams. I will also outline my dissertation project and then present my partial results.
Recently, Sarkar showed that a smooth marked cobordism between two knots induces a map between the knot Floer homology groups of the two knots. It has been conjectured that this map is well defined (with respect to smooth marked cobordisms). After outlining what needs to be shown to prove this conjecture, I will present my current progress towards showing this result for the combinatorial version of knot Floer homology. Specifically, I will present a generalization of Carter and Saito's movie move theorem to grid diagrams, give a very brief introduction to combinatorial knot Floer homology and then present a couple of the required chain homotopies needed for the proof of the conjecture.
I hope to paint a picture for the audience of the lands in which Fermat's Last Theorem, and the activity of the past twenty years, lives. The story begins in two places: modular forms and elliptic curves. After discussing the broad (arithmetic) theory of these two objects our goal will be to connect them, first via L-functions and second via Galois representations. Pizza will be provided, allegedly.
Modern number theory revolves a lot around so called Langlands Program. It is a huge set of conjectures, some well formulated and some yet to be precisely formulated . The main idea behind these sets of conjectures is to classify finite extensions of Q in a systematic manner. The main idea of Langlands was these algebraic objects can be classified in terms of a class certain analytic object, called automorphic representation. Automorphic representations are generalization of modular forms in a nontrivial way. Modular forms are functions on upper half plane of complex numbers, which satisfies certain properties. Another part of this conjectures is about relating automorphic representation of two groups, known as Langlands functoriality.
In this talk we will treat automorphic representation as a black box, but to give some idea what kind of objects are these, we will see how we can modular forms as automorphic representation. My goal is to give an introduction to this huge field of study. Mostly we will talk about Local Langlands Correspondence, if time permits we will talk about Global Langlands Conjectures and Langlands Functoriality.
For this talk I will assume that you are familiar with basic representation theory.
Why has modern theory become a gorgeous disaster? In this talk I explain why the current situation is not completely number theorists' fault. We will prove that all recursively enumerable sets are Diophantine and exhibit some interesting related results, including the existence of a universal Diophantine equation and an inequality in 26 variables whose positive values are precisely the prime numbers.
No previous knowledge of computability theory will be assumed.
The goal of this talk is to give an outline of Einstein’s General Relativity. Basically this theory states that mass curves the spacetime (the 4-dimensional manifold we live in) and therefore gravitation occurs. First of all we will talk about special relativity and flat Minkowski spacetime to get familiar with the basic concepts of relativity. Then we will spend a good amount of time for the definition of curvature. For this, we will introduce several differential geometrical objects such as metrics and connections on manifolds. From this point on, we can write down the Einstein equation, which involves the curvature, metric and mass distribution of spacetime and completely determines its geometry. In the end, we should be able to write down solutions for this equation and understand why the universe is expanding!
Although the talk will mainly be about differential geometry, I will try to stay as less technical as possible and rather explain the intuitive meanings behind the formulas. It should be accessible for anyone who likes pictures and knows multivariable calculus.
In the fulfillment of my minor requirement, I will give a talk on the theory of random walks on hyperbolic groups, and then move to the "more general" context of random walks on groups acting on CAT 0 spaces. In particular, I will discuss a new boundary recently defined by Ruth Charney which seeks to capture hyperbolic properties, and recent work on random walks whose endpoints lie in it.
Cluster algebras are commutative rings with certain canonical sets of generators called clusters. The clusters of a cluster algebra are related to one another by a combinatorial procedure known as mutation. Surprisingly, this mutation phenomenon shows up in a variety of seemingly unrelated contexts, such as the geometry of the moduli space of surfaces and the representation theory of associative algebras. In this talk I will describe attempts under way to use this induced relationship between representation theory and geometry to construct characteristic classes of surface bundles.
Let C be a non-singular curve of genus g defined over a number field k. In 1922, Mordell conjectured that if g > 1 then C(k) has finitely many points. This famous conjecture had been open for more than 60 years before finally taken down by Gerd Faltings in 1983, using highly sophisticated tools from algebraic geometry. Few years later, more elementary proofs of the theorem were given by Paul Vojta, Bombieri, and Faltings himself. In this talk, I would like to discuss how the theory of height functions and techniques in Diophantine approximation were involved in the second line of proofs of Faltings' theorem.
The Seiberg-Witten Invariants are a tool to study smooth four dimensional smooth manifolds. They were introduced in 1994 by Nathan Seiberg and Edward Witten, and were followed by numerous results after their introduction. The invariants are setup in a differential geometric framework, and make use of gauge-transformations among other geometric tools.
These invariants arise from a set of differential equations, in a context similar to one developed by Donaldson years before. However they turn out to be more powerful and easier to compute.
In the talk I will describe its construction, and I will conclude by showing an application.