In recent work with Guy David we introduce the notion of almost
minimizer for a series of functionals previously studied by Alt-Caffarelli
and Alt-Caffarelli-Friedman.
We prove regularity results for these almost minimizers and explore the
structure of the corresponding free boundary. A key ingredient in the study
of the 2-phase problem is the existence of almost monotone quantities. The
goal of this talk is to present these results in a self-contained manner,
emphasizing both the similarities and differences between minimizers and
almost minimizers.
A theorem of Eichler and Shimura says that the space of cusp forms with complex coefficients appears as a direct summand of the cohomology of the compactified modular curve. Ohta has proven an analog of this theorem for the space of ordinary p-adic cusp forms with integral coefficients. Ohta's result has important applications in the Iwasawa theory of cyclotomic fields.
We discuss a new proof of Ohta's result using the geometry of Hecke correspondences at places of bad reduction.
The study of isoperimetric inequalities has a long history,
it's humble beginnings in Ancient Greek mathematics belying a deep and
rich theory. A major tool in the study of the isoperimetric profile is
the Calculus of Variations. Variational arguments lead to weak
differential inequalities for the isoperimetric profile, which allows
analytical tools such as the maximum principle to be employed. Of
central importance here is the connection with curvature, which is
intimately connected with the topology of isoperimetric regions. I
will survey some of the results in this direction, paying particular
attention to the interplay of the isoperimetric profile and curvature
flows which is the focus of my current research.
Combinatorial Hodge Theory recently arises with a growing number of
applications in statistical ranking, game theory, and computer vision etc.
As classical Hodge Theory is a milestone connecting topology and geometry,
such a combinatorial version brings computational topology and geometry into
these fields. Here we particularly focus on statistical ranking via crowdsourcing,
where Hodge theory leads to an orthogonal decomposition of pairwise ranking
data on a graph into gradient flow, harmonic flow, and curl flow. It enables us
to pursue both global ranking and the intrinsic inconsistency as cyclic rankings
which plays a central role in Arrow's impossibility theorem in social choice theory.
Several developments with applications are introduced for such a decomposition,
which shows elements of applied Hodge theory in connection to computer science
and statistics where paired comparison data grow rapidly with crowdsourcing technology.
For any irregular prime p, one has a Hida family of cuspidal eigenforms of level 1 whose residual Galois representations are all reducible. This family has already played a starring role in Wiles’ proof of Iwasawa’s main conjecture for totally real fields. In this talk, we instead focus on the Iwasawa theory of these modular forms in their own right. We will discuss new phenomena that occur in this residually reducible case including the fact that analytic mu-invariants are unbounded in this family and directly related to the p-adic zeta-function. This is a joint work with Joel Bellaiche.
In this talk I shall provide a survey of my recent works and their environs on differential geometry of Lagrangian submanifolds in specific K\"ahler manifolds, such as complex projective spaces, complex space forms, Hermitian symmetric spaces and so on. I shall emphasis on the relationship between certain minimal Lagrangian submanifold in complex hyperquadrics and isoparametric hypersurfaces in spheres. This talk is mainly based on my joint work with Associate Professor Hui Ma (Tsinghua University, Beijing).
An isometric action of a connected Lie group on a Riemannian manifold is called polar if there exists a connected closed submanifold that meets each orbit of the action and intersects it orthogonally. Dadok established in 1985 a remarkable, and mysterious, relation between polar actions on Euclidean spaces and Riemannian symmetric spaces. Soon afterwards an attempt was made to classify polar actions on symmetric spaces. For irreducible symmetric spaces of compact type the final step of the classification has just been achieved by Kollross and Lytchak. In the talk I want to focus on symmetric spaces of noncompact type. For actions of reductive groups one can use the concept of duality between symmetric spaces of compact type and of noncompact type. However, new examples and phenomena arise from the geometry induced by actions of parabolic subgroups, for which there is no analogon in the compact case. I plan to discuss the main difficulties one encounters here and some partial solutions.