Wednesday January 24, 2018 1:00 PM  2:00 PM Math Tower P131
 Xujia Chen, Stony Brook University
Belyi's theorem and dessin d'enfantsEvery 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 onetoone 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 P131
 Saman Habibi Esfahani, Stony Brook University
Combinatorial Knot Floer HomologyKnot Floer homology is an invariant for knots and links in 3manifolds. 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 P131
 Aleksandar Milivojevic, Stony Brook University
Computations in Cartande Rham homotopy theoryThe 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 P131
 Nathan Chen, Stony Brook University
Elliptic Curves and the Monster groupThere 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 P131
 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 P131
 JeanFrancois Arbour, Stony Brook University
The fractal nature of isometry classes of Riemannian metricsAlthough 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 P131
 Hang Yuan, Stony Brook University
Morse theory and A infinity categoriesWe will first review the classical Morse complex associated to a MorseSmale 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 socalled Morse category, which turns out to be an A infinity (pre)category.

Wednesday March 21, 2018 1:00 PM  2:00 PM Math Tower P131
 Jiahao Hu, Stony Brook University
TBATBA

