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 BASIC NOTIONS SEMINAR: Yaiza CanzaniHARVARD UNIVERSITY Local geometry of the nodal sets of Laplace eigenfunctions on Monday, March 10, 2014, at 3:00 PM in Science Center 507 Let $(M,g)$ be a compact real analytic surface with no boundary, and let $H$ denote a closed analytic curve in $M$. Write $\varphi_\lambda$ for the eigenfunctions of the Laplacian $\Delta_g$ with eigenvalue $\lambda^2$. When $M$ is the two-torus and $H$ has non-vanishing curvature (Burgain-Rudnick) or when $M$ is an arithmetic surface and $H$ is a geodesic circle (Jung), it has been shown that $H$ is a \emph{good} curve in the sense that $\| \varphi_{\lambda} \|_{L^2(H)} \geq e^{-C \lambda}$ for some $C>0$ and all $\lambda >\lambda_0$. In these cases it was proved that $$\# \{ \varphi_{\lambda}^{-1}(0) \cap H\}= O(\lambda). \quad \quad (*)$$ In this talk we show that the bound $(*)$ holds on the general surface $(M,g)$ provided $H$ is a good curve. This is joint work with John Toth.

 HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: Bangere PurnaprajnaUNIVERSITY OF KANSAS Geometry of surfaces of general type on Tuesday, March 11, 2014, at 4:00 PM in Science Center 507 In this talk I will survey my recent results with my coauthors on varieties of general type with particular emphasis on the case of algebraic surfaces. The first theme will relate the deformation of canonical maps to construction of simple canonical surfaces, addressing a question of Enriques. The framework we develop allows us to describe some components of infinitely many moduli spaces of surfaces of general type. The second theme is to explore a higher dimensional analogue of the uniformization theorem of Riemann and Kobe, the so-called holomorphic convexity of the universal cover of a projective variety, which goes under the name of Shafarevich conjecture. Until recently, this was not known in its full generality for even surfaces fibered by genus two curves. We prove some general statements about fundamental groups of surfaces fibered by hyperelliptic curves of arbitrary genus. Examples show that this is an optimal result. As a byproduct we prove, a stronger form of Shafarevich conjecture for these surfaces, and a very attractive conjecture of Nori on fundamental groups. This also yields statements on second homotopy groups of fibered surfaces.

 NUMBER THEORY SEMINAR: Adriana SalernoBATES COLLEGE Effective computations in arithmetic mirror symmetry on Wednesday, March 12, 2014, at 3:00 PM in Sci Center 507 In this talk, I will talk about computational approaches to the problem of arithmetic mirror symmetry. One of the biggest questions facing string theorists is the one of mirror symmetry. In arithmetic mirror symmetry, we approach the conjecture from a number theoretic point of view, namely by computing Zeta functions of mirror pairs. I will define all of these terms and then explain our work through a couple of examples of families of K3 surfaces. This is joint work with Xenia de la Ossa, Charles Doran, Tyler Kelly, Stephen Sperber, and Ursula Whitcher.

 INFORMAL DYNAMICS & GEOMETRY SEMINAR: Curtis McMullenHARVARD UNIVERSITY Graphs with expansion < 8 on Wednesday, March 12, 2014, at 4:00 PM in Science Center 530

 HARVARD LOGIC COLLOQUIUM: Paul LarsonMIAMI UNIVERSITY Forcing axioms in ℙmax extensions on Wednesday, March 12, 2014, at 4:00 - 5:00 PM in 2 Arrow Street, Room #420 Martin's Maximum, the maximal forcing axiom, was introduced in the 1980's by Foreman, Magidor and Shelah, who proved its consistency relative to a supercompact cardinal. In the 1990's, Woodin introduced ℙmax, a method for forcing over models of determinacy, and used it produce a new consistency proof for MM(𝔠), the restriction of Martin's Maximum to partial orders of cardinality the continuum. Later results of Sargsyan showed that the determinacy hypothesis used in Woodin's result has consistency strength below a Woodin limit of Woodin cardinals. We will discuss a recent attempt to extend Woodin's theorem to partial orders of cardinality 𝔠+. In particular, we will show that in the ℙmax extension of a suitable determinacy model, certain ◻ principles fail at both ω2 and ω3, significantly reducing the consistency strength upper bound for such a result. Part of the talk will focus on models of determinacy. This is joint work with Caicedo, Sargsyan, Schindler, Steel and Zeman.

 GAUGE THEORY & TOPOLOGY SEMINAR: Mohammed AbouzaidCOLUMBIA UNIVERSITY Formality and Symplectic Khovanov Homology on Friday, March 14, 2014, at 3:30 - 4:30 PM in Science Center 507 I will describe the first (completed) half of a project whose goal is to prove that Khovanov homology agrees with the symplectic analogue. The main result is the formality of the subcategory of the Fukaya category of the nilpotent slice studied by Seidel-Smith which corresponds to Khovanov's arc algebra. They key new ingredient is an abstract criterion for formality due to Seidel, and its implementation using counts of holomorphic curves in a partial compactification of these spaces. This is joint work with I. Smith.

 INFORMAL DYNAMICS & GEOMETRY SEMINAR: Tom ChurchSTANFORD & MIT Moebius functions and growth rates of groups on Wednesday, March 26, 2014, at 4:00 PM in Science Center 530

 BASIC NOTIONS SEMINAR: Joe HarrisHARVARD UNIVERSITY Uniformity of rational points and the Lang conjecture on Monday, March 31, 2014, at 3:00 PM in Science Center 507 This talk concerns joint work with Lucia Caporaso and Barry Mazur. The question we’ll address is very straightforward. Faltings’ proof of the Mordell conjecture asserts that every curve $C$ of genus $g \geq 2$ defined over a number field $K$ has only a finite number of rational points. The question is, given $g$ and $K$, is there an upper bound on these numbers?