Seminars & Events for 2015-2016
Rational curves on elliptic surfaces
Given a non-isotrivial elliptic curve E over K=Fq(t), there is always a finite extension L of K which is itself a rational function field such that E(L) has large rank.
Yang-Mills gradient flow
We shall discuss long time existence and convergence properties of Yang-Mills gradient flow over closed, four-dimensional manifolds and applications to Morse theory for the quotient space of connections modulo gauge transformations.
Volumes of minimal hypersurfaces and stationary geodesic nets
We will prove an upper bound for the volume of a minimal hypersurface in a closed Riemannian manifold conformally equivalent to a manifold with Ric > -(n-1). In the second part of the talk we will construct a sweepout of a closed 3-manifold with positive Ricci curvature by 1-cycles of controlled length and prove an upper bound for the length of a stationary geodesic net.
Tokyo-Princeton algebraic geometry conference
Tokyo-Princeton algebraic geometry conference
Tokyo-Princeton algebraic geometry conference
Optimal detection of weak principal components in high-dimensional data
Please note special day (Monday). Principal component analysis is a widely used method for dimension reduction. In high dimensional data, the ``signal'' eigenvalues corresponding to weak principal components (PCs) do not necessarily separate from the bulk of the ``noise'' eigenvalues.
Stability and instability results for scalar waves on general asymptotically flat spacetimes
In the first part of this talk, we will prove a logarithmic decay result for solutions to the scalar wave equation $\square_{g}\psi=0$ on general asymptotically flat spacetimes $(\mathcal{M},g)$, possibly bounded by an event horizon with positive surface gravity and having a small ergosphere, provided a uniform energy boundedness estimate holds on $(\mathcal{M},g)$.
Tokyo-Princeton algebraic geometry conference
Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
Please note special day (Tuesday). Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière is an equidistribution result of eigenfunctions of the Laplacian in large frequency limit on a Riemannian manifold with an ergodic geodesic flow.
On the Hilbert Property and the fundamental group of algebraic varieties
Please note special day (Tuesday). This concerns recent work with P. Corvaja in which we relate the Hilbert Property for an algebraic variety (a kind of axiom linked with Hilbert Irreducibility, relevant e.g. for the Inverse Galois Problem) with the fundamental group of the variety. In particular, this leads to new examples (of surfaces) of failure of the Hilbert Property.
On the Solution of Elliptic Partial Differential Equations on Regions with Corners
The solution of elliptic partial differential equations on regions with non-smooth boundaries (edges, corners, etc.) is a notoriously refractory problem.
Ratner's theorems and applications
In the early 1980s, Marina Ratner published three papers about horocycle flows on the unit tangent bundle of a surface of constant negative curvature with finite volume. In the early 1990s, by expanding the ideas from her study of horocycle flows, Ratner proved a series of beautiful results about unipotent flows on homogeneous spaces.
Women and Math - Princeton Day
10:30—11:30 AM: |
Speaker: Zoltán Szabó, Professor of Mathematics, Princeton University Title: “Floer homology invariants for knots.” |
Randomized Algorithms for Matrix Decomposition
Matrix decompositions, and especially SVD, are very important tools in data analysis. When big data is processed, the computation of matrix decompositions becomes expensive and impractical. In recent years, several algorithms, which approximate matrix decomposition, have been developed. These algorithms are based on metric conservation features for linear spaces of random projections.
On p-torsion in class groups of number fields
Gauss famously investigated class numbers of quadratic fields, in particular characterizing the 2-divisibility of the class number for such fields. In general, it is expected that for a number field of any degree, and any rational prime p, the p-torsion part of the class group should be arbitrarily small, in a suitable sense, relative to the absolute discriminant of the field.
The Dirichlet problem for the constant mean curvature equation in Sol$_3$
Please note different day (Thursday). The Dirichlet problem for CMC surfaces with infinite boundary data was first studied for minimal graphs in $\mathbb{R}^3$ by Jenkins and Serrin and then by Sprück for CMC $H$ surfaces.
Uniqueness of immersed spheres in three-manifolds. Proof of a conjecture by Alexandrov
In this talk we generalize Hopf's famous classification of constant mean curvature spheres in R^3 to the general situation of classes of surfaces modeled by arbitrary elliptic PDEs in arbitrary three-manifolds, with the only hypothesis of the existence of a family of "candidate surfaces". In this way, we prove that any immersed sphere in such a class of surfaces is a candidate sphere.
Min-max theory in Gaussian space and Entropy Conjecture
Please note additional talk and special time (4:15). Minimal surfaces are critical points of the area functional. The min-max theory is a variational theory for constructing saddle point type, unstable minimal surfaces. In this talk, we will introduce a min-max theory in a specific space--the Gaussian probability space.
Divisibility of coefficients of modular forms
I will explain two recent results concerning non-zero coefficients of modular forms modulo a prime p. The first result, a joint work with K. Soundararajan, gives an asymptotic equivalent for the number of such coefficients. The second is concerned with the set of primes which are indices of non-zero coefficients.