I will present work that aim at understanding the failure of analytic hypoellipticity of special differential operators, namely sums of squares of (analytic) vector fields. According to Hörmander, given such an operator P that satisfies the ``bracket condition'' and given a smooth function f, if u is distribution solution to Pu=f then u is also a smooth function. P is then said to be hypoelliptic. But if f is real analytic, then u need not to be analytic but merely smooth Gevrey for some indices, usually to be guessed. Examples were built by Metivier, Matsuzawa, Bove, Baouendi-Goulaouic..., using real variables methods. In this joint work with Paulo Cordaro, by using methods of complex analysis, we show that this failure of analytic hypoellipticity due to the presence of irregular singularity of some holomorphic ODEs the analysis of which defines the best Gevrey indices to be expected. The theory of summability of formal solutions of holomorphic ODEs as developped by Ramis, Malgrange, Braaksma is a fundemental tool here.

In this talk, I will describe a new algebraic characterization of sectional curvature bounds that only involves curvature terms in the Weitzenboeck formulae for symmetric tensors. This characterization is further clarified by means of a symmetric analogue of the Kulkarni-Nomizu product, which renders it computationally amenable. Furthermore, a related application of the Bochner technique to closed 4-manifolds with indefinite intersection form and positive or nonnegative sectional curvature will be discussed, yielding some new insight about the Hopf Conjecture. This is based on joint work with R. Mendes (Univ. zu K$\ddot{\text{o}}$ln, Germany).

The Gaussian entropy, introduced by Colding and Minicozzi, is a rigid motion and scaling invariant functional which measures the complexity of hypersurfaces of the Euclidean space. It is defined to be the supremal Gaussian area of all dilations and translations of the hypeprsurface, and as such, is well adapted to be studied by mean curvature flow. In the case of the n-th sphere in $R^{n+1}$, the entropy can be computed explicitly, and is decreasing as a function of the dimension n. A few years ago, Colding Ilmanen Minicozzi and White proved that all closed, smooth self-shrinking solutions of the MCF have larger entropy than the entropy of the n-th sphere. In this talk, I will describe a generalization of this result, which derives better (sharp) entropy bounds under topological constraints. More precisely, we show that if M is any closed self-shrinker in $R^{n+1}$ with a non-vanishing k-th homotopy group (with $k\leq n$), then its entropy is higher than the entropy of the k-th sphere in $R^{k+1}$. This is a joint work with Brian White.

Study of algebraic equations is one of the oldest and most celebrated themes in mathematics. It was understood that finite systems of equations are decidable in the fields of complex and real numbers. The celebrated Hilbert tenth problem stated in 1900 asks for a procedure which, in a finite number of steps, can determine whether a polynomial equation (in several variables) with integer coefficients has or does not have integer solutions. In 1970 Matiyasevich, following the work of Davis, Putnam and Robinson, solved this problem in the negative. Similar questions can be asked for arbitrary rings. We will give a survey of results on the Diophantine problem in different rings and prove the undecidability of equations (the undecidability of the Diophantine problem) for free Lie algebras of rank at least 3 over an arbitrary field. These are joint results with A. Miasnikov.

Many scientific datasets are of high dimension, and the analysis usually requires retaining the most important structures of data. Many existing methods work only for data with structures that are mathematically formulated by curves, which is quite restrictive for real applications. To get more general graph structures, we develop a novel graph structure learning framework that captures the local information of the underlying graph structure based on reversed graph embedding. A new learning algorithm is developed that learns a set of principal points and a graph structure from data, simultaneously. Experimental results on various synthetic and real world datasets show that the proposed method can uncover the underlying structure correctly.

The class number formula connects the residue of the Dedekind zeta
function at s=1 to the regulator, which measures the covolume of the
lattice generated by logarithms of units. Beilinson defined a vast
generalization of the regulator morphism and conjectured a class number
formula associated to the cohomology of any smooth proper variety over a
number field. His formula provides arithmetic meaning for the orders of
the so-called trivial zeros of L-functions at integer points as well as
the value of the first nonzero derivative at these points.

We study this conjecture for the middle degree cohomology of
compactified Shimura varieties associated to unitary groups of signature
(2,1) and (2,2) over Q. We construct explicit Beilinson-Flach elements
in the motivic cohomology of these varieties and compute their
regulator. This is joint work with Aaron Pollack.