Department of Mathematics

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Abstracts

Week of October 9, 2022

October 10, 2022
Applied Mathematics	Completing large low rank matrices with only few observed entries: A one-line algorithm with provable guarantees	3:00pm - AKW 200	Abstract: Suppose you observe very few entries from a large matrix. Can we predict the missing entries, say assuming the matrix is (approximately) low rank? We describe a very simple method to solve this matrix completion problem. We show our method is able to recover matrices from very few entries and/or with ill conditioned matrices, where many other popular methods fail. Furthermore, due to its simplicity, it is easy to extend our method to incorporate additional knowledge on the underlying matrix, for example to solve the inductive matrix completion problem. On the theoretical front, we prove that our method enjoys some of the strongest available theoretical recovery guarantees. Finally, for inductive matrix completion, we prove that under suitable conditions the problem has a benign optimization landscape with no bad local minima. Joint work with Pini Zilber. Bio: Boaz Nadler received his Ph.D. at Tel Aviv University. After 3 years as a Gibbs instructor/assistant professor at Yale University, he joined the faculty at the department of computer science and applied mathematics at the Weizmann Institute of Science. His research interests span mathematical statistics, machine learning, applied mathematics, and various applications, including optics and signal processing.
Geometry, Symmetry and Physics	Rational Hodge isometries of hyper-Kahler varieties of K3^[n] and generalized Kummer type are algebraic	4:30pm - LOM214	Let X and Y be compact hyper-Kahler manifolds deformation equivalent to the Hilbert scheme of n points on a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subalgebra generated by Chern classes of coherent analytic sheaves. Let f be a Hodge isometry of the second rational cohomologies of X and Y with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence F between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When X and Y are projective the correspondences f and F are algebraic. The case n=1 (when X and Y are K3 surfaces) was known as the Shafarevich conjecture and was solved by Nikolay Buskin in 2015 using work of Mukai. The higher dimensional case is based on recent results on equivalences of derived categories of hyperkahler varieties due to Taelman. We will also comment on the analogous result for hyperkahler manifolds of generalized Kummer deformation type (work in progress).

October 13, 2022
Algebra and Geometry lecture series	Cohomology of Hitchin moduli spaces and the P=W conjecture.	4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214

October 14, 2022
Friday Morning Seminar	Friday Morning Seminar	10:30am - LOM 205	A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!
Graduate Student Seminar	Geodetic Graphs: Geometry, Diameter and the Problem of Classification	12:00pm - LOM	A graph is called geodetic if there is a unique shortest path between any two vertices (i.e - a geodesic). Ore (’62) sought to characterize such graphs, and despite a considerable body of work on this problem - a characterization is surprisingly elusive. In this talk we introduce the idea of viewing graphs from a geometric perspective and survey its connections to the problem of classification. In particular, we review some known classifications and constructions of geodetic graphs, and show that such graphs must be of relatively high connectivity. This talk is based on joint work with Nati Linial.
Geometric Analysis and Application	Equivariant min-max theory to construct free boundary minimal surfaces in the unit ball	2:00pm - LOM 215	Abstract: A free boundary minimal surface (FBMS) in the three-dimensional Euclidean unit ball is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ball (i.e., the unit sphere). A very natural question is whether there are FBMS in the unit ball of any given topological type. In this talk, we will present the construction of a family of FBMS with connected boundary and arbitrary genus, via an equivariant version of Almgren-Pitts min-max theory à la Simon-Smith. We will see how this method allows us to control the topology of the resulting surface and also to obtain information on its index.