Thursday January 25, 2018 4:00 PM  5:00 PM Math Tower P131
 Frank Thorne, University of South Carolina
An Analytic View of Arithmetic Statistics"Arithmetic statistics" is all about counting arithmetic objects: number fields, ideal class groups, Selmer groups of ellitpic curves, and so on. One common approach is to parametrize such objects in terms of lattice points in a vector space, up to an equivalence relation given by the action of a suitable group. One then counts the lattice points.
I will give an overview of some work on this subject, from Gauss to contemporary work of Manjul Bhargava, myself, my host Bob Hough, and many others. I will concentrate on the analytic and quantitative side of the subject, and explain how refinements to our lattice point counting methods lead to improved results.

Thursday February 01, 2018 4:00 PM  5:00 PM Math Tower P131
 Xiuxiong Chen, Stony Brook University
On constant scalar curvature Kaehler metricsInspired by the celebrated $C^0, C^2 $ and $C^3$ a priori estimates of Calabi, Yau and others on Kaehler Einstein metrics, we present a report on a priori estimates on constant scalar curvature Kaehler metrics. With this estimate, we prove the Donaldson conjecture on geodesic stability and the well known properness conjecture on the Mabuchi energy functional. This is a joint work with Cheng JingRui from University of Wisconsin at Madison.
This lecture is intended to be expository. This will be followed by a more
technical lecture by Dr. Cheng Jingrui on Geometry/topology seminar (Feb. 6th).

Thursday February 15, 2018 4:00 PM  5:00 PM Math Tower P131
 Leonid Kovalev, Syracuse University
Lipschitz geometry of finite subset spacesLet $X(n)$ be the set of all nonempty subsets of a metric space X with cardinality at most n. Equipped with the Hausdorff metric, $X(n)$ becomes a metric space of its own, a finite subset space of X. Such spaces have been studied since the 1931 paper of Borsuk and Ulam, mostly from the topological perspective. The investigation of their metric structure is more recent.
The finite subset spaces form a natural chain of isometric embeddings: $X = X(1) ⊂ X(2) ⊂ X(3) ⊂ ...$ For example, when X is a circle, this chain describes a trefoil knot bounding a Möbius strip in the 3sphere. For some classes of spaces these embedding split, even in the Lipschitz category. The existence of Lipschitz retractions $X(n) → X(n1)$ generally corresponds to X being nonpositively curved, but the exact relationship is yet to be understood.

Thursday February 22, 2018 4:00 PM  5:00 PM Math Tower P131
 Nets Katz, Caltech
Semialgebraic sets and the Kakeya ProblemIn joint work with Keith Rogers, we study the connection between the Kakeya problem and the highlights of the theory of semialgebraic sets such as Tarski's projection theorem and Gromov's algebraic lemma.

Thursday March 01, 2018 4:00 PM  5:00 PM Math Tower P131
 Dzmitry Dudko,
On the problem of local connectivity of the Mandelbrot set.The Mandelbrot set encodes how the dynamics of $z^2+c$ depends on $c$. In the 1980s A. Douady and J. Hubbard conjectured that the Mandelbrot set is locally connected  the MLC conjecture. This conjecture would result in a simple abstract ``pinched disk'' model for the Mandelbrot set. Since the 1990s, local connectivity was established for a large class of parameters, but the full conjecture is still open. In the talk we will discuss recent developments in the area. Our main tool is ``Pacman renormalization'' responsible for selfsimilarity of the Mandelbrot set near its main cardioid.
Based on a joint work with Misha Lyubich.

Thursday April 05, 2018 4:00 PM  5:00 PM SCGP Auditorium 103
 John Milnor, Stony Brook University
Points on a Circle: from Pappus to ThurstonA soft introduction to the easiest examples of DeligneMumford compactifications.

Thursday April 12, 2018 4:00 PM  5:00 PM SCGP102
 Piotr Sulkowski, University of Warsaw (Poland) & Caltech
Categorification in mathematical physics workshop Colloquium: Knotsquivers correspondenceI will present a surprising relation between knot invariants and quiver representation theory, motivated by various string theory constructions. Consequences of this relation include the proof of one of the famous integrality conjectures of BPS invariants (i.e. LabastidaMarinoOoguriVafa conjecture for symmetric representations), explicit (and unknown before) formulas for colored HOMFLYPT polynomials for various knots, new viewpoint on knot homologies, a novel type of categorification, new dualities between quivers, solutions to certain combinatorial problems, and many others.

Thursday April 19, 2018 4:00 PM  5:00 PM Math Tower P131
 Percy Deift, NYU Courant
Universality in numerical computation with random data.
Case studies and analytical results.The speaker will discuss various universality aspects of numerical computations using standard algorithms.
These aspects include empirical observations, rigorous results, and some speculations about computation in a broader sense.
Joint with C.Pfrang, G.Menon, S.Olver and Thomas Trogdon.

Thursday April 26, 2018 4:00 PM  5:00 PM Math Tower P131
 Mohammed Abouzaid, Columbia University
Mirror symmetry, loop spaces, and immersed LagrangiansA compelling approach to mirror symmetry, initiated by
StromingerYauZaslow and Fukaya, is to associate to a symplectic manifold with a Lagrangian torus fibration a mirror space consisting of the space of objects of the Fukaya category supported on the fibres with multiplicity one. In the simplest setting of the cotangent bundle of the torus, the mirror correspondence thus amounts to the natural identification between the group ring of the fundamental group and Laurent polynomials. I will begin by explaining how one can globalise this argument to yield a proof of homological mirror symmetry for manifolds admitting nonsingular Lagrangian fibrations. Then I will explain how the main obstruction to generalise this proof is to formulate an appropriate notion of loop space for singular Lagrangians. Finally, I will sketch a construction that plays the role of the loop space for immersed Lagrangians, and indicate how this should suffice to prove mirror symmetry for examples arising from toric degenerations.

Thursday May 03, 2018 2:30 PM  3:30 PM Math Tower P131
 Chris Bishop, Stony Brook University
Snowflakes and TreesThis is a short survey of various results involving (mostly planar) harmonic measure, ranging roughly from Makarov's theorems of the 1980's to the present, but focusing mostly on topics related to "balancing" harmonic measure on two sides of a common boundary. We touch on topics like Brownian motion, conformal maps, circle homeomorphisms, dessins d'enfants, transcendental dynamics, wandering domains and European history.

