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 HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: Alex PerryHARVARD UNIVERSITY Categorical joins on Tuesday, October 13, 2015, at 3:00 PM in MIT E17-122 Homological projective duality is a powerful theory developed by Kuznetsov for studying the derived categories of varieties. It can be thought of as a categorification of classical projective duality. I will describe a categorical version of the classical join of two projective varieties, and its relation to homological projective duality. I will discuss some applications to the structure of the derived categories of Fano varieties and to derived equivalences of Calabi-Yau varieties. This is work in progress with Alexander Kuznetsov

 DIFFERENTIAL GEOMETRY SEMINAR: Adrian ZahariucHARVARD UNIVERSITY Specialization of Quintic Threefolds to the Chordal Variety on Tuesday, October 13, 2015, at 4:15 PM in Science Center 507 We consider a family of quintic threefolds specializing to a certain reducible threefold with two irreducible components. What are the flat limits of the rational curves in this degeneration? I will provide a “first approximation” to the answer by describing the space of genus zero stable morphisms to the central fiber (as defined by J. Li).

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS GEOMETRIC ANALYSIS SEMINAR: Pengfei GuanMcGILL UNIVERSITY Isometric embeddings of $(S^2,g)$ to general warped product space $(N^3,\bar g)$. on Tuesday, October 13, 2015, at 10:00 AM in Science Center 232 We discuss recent work on the Weyl problem of isometric embeddings of $(S^2, g)$ to general ambient space $(N^3, \bar g)$ with warped product structure. When $N^3$ is $R^3$, it is the classical Weyl's problem, which was solved by Nirenberg in 1950s (when $g$ is analytic, it was solved by H. Levy prior to Nirenberg's work). Pogorelov also considered problem when $N$ is the hyperbolic space $H^3$. Solutions to Weyl's problem in $R^3$ and $H^3$ have important applications in general relativity, e.g., the Brown-York and Liu-Yau quasi local masses. The recent work of Wang-Yau further indicates the importance of understanding isometric embedding problem. We establish curvature estimates for immersed surfaces in warped product space in general, discuss the existence of solutions of the problem.  In contrast to the space form cases, there is non-rigidity when $N^3$ is not a space form. Part of the work is jointly with Siyuan Lu. The openness and non-rigidity results are due to C. Li and Z. Wang.

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: Benjamin SchweinhartCENTER OF MATHEMATICAL SCIENCES & APPLICATIONS Universality Conjectures for Curvature Flow on Graphs on Wednesday, October 14, 2015, at 2:00 - 3:00 PM in Science Center 232 Curvature flow on graphs and cell complexes is an important model for grain growth in polycrystalline materials. I will introduce these concepts, and state several universality conjectures about their long term behavior. This talk is based on joint work with Robert MacPherson and Jeremy Mason.

 NUMBER THEORY SEMINAR: Wei ZhangCOLUMBIA UNIVERSITY Cycles on the moduli of Shtukas and Taylor coefficients of L-functions (II) **Note: This is a continuation of Zhiwei Yun's talk, on Oct 13, 4:30-5:30pm in MIT room E17-122. on Wednesday, October 14, 2015, at 3:00 - 4:00 PM in Science Center 507 In our joint work, we prove a generalization of Gross-Zagier formula in the function field setting. Our formula relates self-intersection of certain cycles on the moduli of Shtukas for GL(2) to higher derivatives of L-functions.

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: Boaz BarakHARVARD UNIVERSITY SEAS Convexity, Bayesianism, and the quest towards Optimal Algorithms on Wednesday, October 14, 2015, at 4:00 PM in Science Center 507 In this high level and accessible talk I will describe a recent line of works aimed at trying to understand the intrinsic complexity of computational problems by finding *optimal* algorithms for large classes of such problems. In particular, I will talk about efforts centered on convex programming as a source for such candidate algorithms. As we will see, a byproduct of this effort is a computational analog of Bayesian probability that is of its own interest. I will demonstrate the approach using the example of the hidden (or planted) clique problem - a central problem in average case complexity with connections to machine learning, community detection, compressed sensing, finding Nash equilibrium and more. While the complexity of this problem is still wide open, this line of works has led to interesting insights on it.

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS EVOLUTION EQUATIONS SEMINAR: Zhouping XinCHINESE UNIVERSITY OF HONG KONG On Global Well-Posedness of The Compressible Navier-Stokes Systems with Large Oscillations on Thursday, October 15, 2015, at 10:00 AM in Science Center 232 I this lecture, I will discuss some issues involving the global (in time) well-posedness of strong solutions to the multi-dimensional compressible Navier-Stokes systems with data which may have large oscillations and vacuum. Blow-up phenomena will be discussed and global existence of regular solutions with small energy will be proved. Though such solutions have small energies, yet they may have large oscillations and contain vacuum in both interior and far fields. In particular, The uniqueness and regularity of weak solutions by P. L. Lions will be shown provided that the weak solutions have small Initial energy. Some of the main ideas for the analysis will be discussed.

 GAUGE THEORY, TOPOLOGY & SYMPLECTIC GEOMETRY SEMINAR: Cheuk Yu MakMINNESOTA Divisorial caps, uniruled caps and Calabi-Yau caps on Friday, October 16, 2015, at 3:30 - 4:30 PM in Science Center 507 We illustrate how a nice symplectic cap captures properties of symplectic fillings of a contact 3-manfiold. Three kinds of symplectic caps are introduced. Divisorial caps are motivated from compactifying divisors in algebraic geometry. Uniruled caps give strong restriction to symplectic fillings for a class of contact 3-manifolds strictly larger than the planar ones. Calabi-Yau caps, in particular, can be used to derive uniform Betti numbers bounds on Stein fillings of the standard unit cotangent bundle of any hyperbolic surface. This is a joint work with Tian-Jun Li and Kouichi Yasui.

 BASIC NOTIONS SEMINAR: Wilfried SchmidHARVARD UNIVERSITY Automorphic distributions on Monday, October 19, 2015, at 3:00 - 4:00 PM in Science Center 507 Automorphic distributions determine, and are determined by, automorphic representations. I shall describe how they can be used to prove the analytic continuation and holomorphy of certain Langlands L-functions.

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: Nicholas CookUCLA Random regular digraphs: singularity and spectrum on Wednesday, October 21, 2015, at 2:00 - 3:00 PM in Science Center 232 We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with i.i.d. bounded edge weights. Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the graph has degree linear in the number of vertices. Towards establishing the same result for the unweighted adjacency matrix, we prove that it is invertible with high probability, even for sparse digraphs with degree growing only poly-logarithmically.

 NUMBER THEORY SEMINAR: David HansenCOLUMBIA UNIVERSITY Critical p-adic L-functions on Wednesday, October 21, 2015, at 3:00 - 4:00 PM in Science Center 507 I'll explain some new results on the critical p-adic L-functions associated with an elliptic curve over Q at a prime of good ordinary reduction. In particular, I'll explain why the two approaches to defining such a p-adic L-function (due to Kato and Pollack-Stevens) in fact yield the same function. This comparison explains a certain pattern (discovered numerically by Pollack-Stevens) in the zeros of the p-adic L-function; I'll describe this as well.

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: Zhouping XinCHINESE UNIVERSITY OF HONG KONG Entropy and Uniqueness of Weak Solutions to The Multi-Dimensional Compressible Euler Systems on Wednesday, October 21, 2015, at 4:00 PM in Science Center 507 For the ideal compressible Euler systems, which are fundamental in fluid-dynamics and pro-type examples of nonlinear hyperbolic systems, one of the main features is that the characteristic speeds of a wave propagation depend on the wave itself which leads to the finite-time formation of shocks in general. Thus one has to work with weak solutions globally. Yet the uniqueness of the "physical" solutions becomes a challenging issue. In the one- dimensinal case, various admissible criterion have been introduced to rule out the non-physical solutions. In particular, the physical entropy can guarantee the uniqueness of weak solutions at least in the case of small variations. However, in higher space dimensions, for some given initial data, there are infinitely many highly oscillatory solutions (wild solutions) which are bounded, measurable and satisfying the physical entropy. In talk, I will review some progress on the constructions of such "wild solutions" by a method of convex integration; present some results on the structure of such "wild solutions"; and investigate the effects of lower order dissipations. Some open problems will be discussed.1

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS EVOLUTION EQUATIONS SEMINAR: Xiangdi HuangCHINESE ACADEMY OF SCIENCES On Nash's problem for compressible flows on Thursday, October 22, 2015, at 10:00 AM in Science Center 232 We establish a unified blow-up criterion for strong solutions of various compressible models, including baratropic, fully compressible and heat-conducting magneto hydrodynamic flows, which is analogous to the Serrin's criterion of Navier-Stokes equations. It gives an affirmative answer to a problem proposed by J. Nash in 1958. As an application, we prove global-in-time classical solutions of compressible flows allowing vacuum and large fluctunation of initial data provided the initial energy is small. In the end, we prove a strong version of Nash’s problem based on new observations.

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS SPECIAL SEMINAR: Monther R. Alfuraidan & Mohamed Amine KhamsiKing Fahd University & University of Texas at El Paso Graphical approach to some fixed point problems on Wednesday, October 28, 2015, at 11:00 AM in Science Center 530 In this joint talk, we will discuss a new area that overlaps between metric fixed point theory and graph theory. This new area yields interesting generalizations of the Banach contraction principle in metric and modular spaces endowed with a graph. The bridge between both theories is motivated by the fact that they often arise in industrial fields such as image processing engineering, physics, computer science, economics, ladder networks, dynamic programming, control theory, stochastic filtering, statistics, telecommunication and many other applications.

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS EVOLUTION EQUATIONS SEMINAR: Pin YuTSINGHUA UNIVERSITY Shock formations for 3 dimensional wave equations on Thursday, October 29, 2015, at 10:00 AM in Science Center 232 We study a family of 3 dimensional quasi-linear wave equations derived from variational principle. We exhibit a family of smooth initial data and show that the foliation of the incoming characteristic hypersurfaces collapses. No symmetry condition is imposed on the initial datum. The proof is inpired by the techniques used in studying the formation of shocks for compressible Euler equations and the formation of black holes in general relativity.

 CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: Louise-Pierre ArguinCUNY The maximum of the characteristic polynomial of random unitary matrices on Wednesday, November 18, 2015, at 2:00 - 3:30 PM in Science Center 232 A recent conjecture of Fyodorov, Hiary & Keating (FHK) states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of a specific class of Gaussian fields, the so-called log-correlated Gaussian fields. These include important examples such as branching Brownian motion and the 2D Gaussian free field. In this talk, we will highlight the connections between the two problems. We will outline the proof of the conjecture for the leading order of the maximum. We will also discuss the connections with the FHK conjecture for the maximum of the Riemann zeta function on the critical line. This is based on joint works with D. Belius (NYU), P. Bourgade (NYU), and A. Harper (Cambridge).