Graduate Student Seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars

Wednesday
January 24, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Xujia Chen, Stony Brook University
Belyi's theorem and dessin d'enfants

Every compact Riemann surface can be realized as the normalization of an algebraic curve in P^2. Belyi's theorem states that a compact Riemann surface S can be written as the normalization of an algebraic curve defined by a polynomial F, all of whose coefficients are algebraic numbers, if and only if there exists a branched covering from S to P^1 with at most three branch values. Such a Riemann surface with such a branched covering is called a Belyi pair. Belyi pairs are in one-to-one correspondence to a certain kind of graphs, dessin d'enfants (``children's drawing''), which are defined purely combinatorially. I will begin from the definition and basic properties of Riemann surfaces. Belyi's theorem will not be proved, but I will explain the general idea and give part of the proof if time permits.


Wednesday
January 31, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Saman Habibi Esfahani, Stony Brook University
Combinatorial Knot Floer Homology

Knot Floer homology is an invariant for knots and links in 3-manifolds. We will see a combinatorial description of this invariant and some of its applications in low dimensional topology, including Milnor's conjecture about Torus knots.


Wednesday
February 07, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Aleksandar Milivojevic, Stony Brook University
Computations in Cartan-de Rham homotopy theory

The sizable differential graded algebra of forms on a smooth manifold admits a tractable model which contains more homotopy information than the real cohomology algebra. We will determine this model for several manifolds, compute some higher homotopy groups modulo torsion, and discuss how to model fiber bundles.


Wednesday
February 14, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Nathan Chen, Stony Brook University
Elliptic Curves and the Monster group

There are several perspectives that one can take with regards to elliptic curves. We will first classify them up to biholomorphism using the $j$-invariant, and then explore the relationship between the $j$-invariant and the Monster group. You donut want to miss this talk!


Wednesday
February 21, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Prithviraj Chowdhury, Stony Brook University
Use of localization in commutative algebra.

One of the most frequently used tools in commutative algebra and algebraic geometry is localization. We will start from the very basics, by reviewing the construction of a field of fractions from an integral domain to motivate the idea behind localising. After this we will discuss some simple applications of
localization such as local global properties, the going up and going down theorem, and if time permits, the factorization of ideals in Dedekind domains.


Wednesday
February 28, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Jean-Francois Arbour, Stony Brook University
The fractal nature of isometry classes of Riemannian metrics

Although the space of Riemannian metrics on a manifold is merely a convex cone in a vector space, it turns out that the space of isometry classes of metrics is awesomely complicated in dimension at least 5. I will present ideas of A. Nabutovsky to that effect. Very informally, one consequence of his work is that in dimension at least 5, the graph of the diameter functional on the space of metrics with bounded curvature on any manifold has infinitely many arbitrarily deep basins at every scale. A beautiful feature of his work is that it blends together ideas from the theory of algorithms, algebraic topology and Riemannian geometry.


Wednesday
March 07, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Hang Yuan, Stony Brook University
Morse theory and A infinity categories

We will first review the classical Morse complex associated to a Morse-Smale function on a compact oriented Riemannian manifold; and then we will propose a natural way to "categorify" it. Unfortunately this way fails to cook up an ordinary category, as the composition is not associative in general. However, it is "associative up to homotopy", and this leads us to a discussion of A infinity algebras and categories. After that, we will use "gradient trees", rather than usual gradient lines, to construct the so-called Morse category, which turns out to be an A infinity (pre-)category.


Wednesday
March 28, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Jiahao Hu, Stony Brook University
Introduction to Morse theory

A typical differentiable function on a manifold will reflect the topology quite directly. Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. In this talk, I will use basic linear algebra and calculus to establish classical Morse theory, and then introduce a generalization—Morse-Bott theory. In application, we will compute the cohomology group of sphere, complex projective space and complex grassmannian if time permits.


Wednesday
April 04, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Aaron Ackbarali, Stony Brook University
Higher Topos Theory

Who, what when, where, and why?


Wednesday
April 11, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Ben Wu, Stony Brook University
Hirzebruch Surfaces

Hirzebruch surfaces are projectivizations of rank 2 vector bundles over the projective line. For each nonnegative integer, $n$, there is a corresponding Hirzebruch surface $\mathbb{F}_n$. It can be shown that any minimal rational surface is isomorphic to either $\mathbb{P}^2$ or $\mathbb{F}_n$ for $n \neq 1$. In this talk, we will show that for $n >0$, there is a unique irreducible curve on $\mathbb{F}_n$ of negative self intersection. In particular, we will realize this curve as the image of a section of the projective bundle $\mathbb{F}_n$. We will begin with a brief discussion on the intersection theory of surfaces in order to understand the notion of self intersection.


Wednesday
April 18, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Jordan Rainone, Stony Brook University
Black Holes: Mathematically and Physically

You probably have a picture of what a black hole is in your head. I'm going to give you several more pictures which we can use to explore what happens at the event horizon and inside of an ideal black hole (spherically symmetric, static, vacuum solution, ie Schwarzschild). You'll also get the "technical definition" of a "black hole" and discussion of some big name theorems which require these definitions. Lastly, if time permits, we'll discuss progressively weirder back holes possibly ended at the Black Ring.


Wednesday
April 25, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Paul Frigge, Stony Brook University
Dynamics of root-finding algorithms

Iterative algorithms for determining the roots of a polynomial, such as Newton's method, can be viewed as a dynamical system. In this talk, the focus will be on the secant root finding algorithm, which can be viewed as a two-dimensional rational map. The main example will be the secant maps associated with quadratic polynomials, which can (almost) all be conjugated to the monomial map (x,y) -> (y,xy) over an appropriate compactification of C^2. I will describe the dynamics of this function and if time permits, I will explain how the secant map can also find the critical points of higher degree polynomials.


Wednesday
May 02, 2018

1:00 PM - 2:00 PM
Math Tower P-131
Ying Hong Tham, Stony Brook University
Knots, Categories, and Quantum Groups

In 1985, Vaughan Jones defined his polynomial invariant of links in S^3 via Von Neumann algebras. We will take a different path, through representations of quantum groups, and see how this leads to the Reshetikhin-Turaev invariant of links in arbitrary 3-manifolds.


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars