suppose $\omega$ is a nonempty open set with lipschitz continuous boundary in $\mathbb{r}^n$. there are certain exponents $e\in \mathbb{r}$ and $q\in (1, infty)$ for which $\displaystyle \frac{\partial}{\partial x^j}: w^{e,q}(\omega) \rightarrow w^{e-1,q}(\omega)$ is not a well-defined continuous operator. now suppose $m$ is a compact smooth manifold. in this talk we will try to discuss the
following questions:
\begin{enumerate}
\item how are sobolev spaces of sections of vector bundles on $m$ defined?
\item is it possible to extend $d: c^\infty(m)\rightarrow c^\infty(t^{*}m)$ to a continuous linear map from $w^{e,q}(m)$ to $w^{e-1,q}(t^{*}m)$ for all $e\in \mathbb{r}$ and $q\in (1,\infty)$?
\item why are we interested in the above question?
\end{enumerate}

in recent years there has been growing interest in foliations
in complex algebraic geometry, both as a tool to study algebraic varieties
and as objects in their own right.
i will describe some recent work on applying the ideas of mori theory to
foliations, and in particular, work on developing a foliated minimal model
program (mmp)
and a foliated kodaira-enriques classification.

we investigate riemannian (non-kahler) ricci flow solutions that develop finite-time singularities
with the property that parabolic rescalings at the singularities converge to singularity
models that are shrinking kahler--ricci solitons, specifically, the conjecturally stable
``blowdown soliton'' discovereed by feldman, ilmanen and knopf. this is a joint work with isenberg and knopf.

i will first talk about a recent project focusing on regulation of cell aging:

cell aging is a universal biological phenomenon, but mechanisms that regulate aging remain unclear. using novel microfluidic technologies, we tracked the replicative aging of single yeast cells and found a dramatic loss of heterochromatin silencing leading to cell aging and death. the dynamics of silencing loss during aging can be dissected into an early phase with sporadic silencing loss, followed by sustained silencing loss preceding cell death. although the length of the later phase is relatively constant, the length of the early phase is highly variable among cells and largely determines lifespan. the intermittent silencing dynamics during the early phase depends on a conserved histone deacetylase sir2 and is important for longevity, whereas either sustained silencing or loss of silencing shortens lifespan. these findings reveal that the temporal dynamics of key molecular processes can directly influence cell aging.

i will then show a couple of examples of projects, in which we are facing challenges in single-cell imaging analysis.

one important technical ingredient in many arithmetic statistics papers is upper bounds for finite exponential sums which arise as fourier transforms of characteristic functions of orbits. this is typical in results obtaining power saving error terms, treating ``local conditions'', and/or applying any sort of sieve. in my talk i will explain what these exponential sums are, how they arise, and what their relevance is.

i will outline a new method for explicitly and easily evaluating them, and describe some pleasant surprises in our end results. i will also outline a new sieve method for efficiently exploiting these results, involving poisson summation and the bhargava-ekedahl geometric sieve. for example, we have proved that there are ``many'' quartic field discriminants with at most eight prime factors. this is joint work with takashi taniguchi.

abstract: turán and szemerédi-type theorems prove the existence of certain well-behaved patterns in dense graphs and hypergraphs. here we show much stronger/larger patterns exist for graphs and hypergraphs that arise from geometry or algebra. the talk is based on recent joint works with jacob fox, jános pach, adam sheffer, and josh zahl.

guided by the framework of conformal geometry in four dimensions, we consider several global invariants for cr geometry in 3-d that has implications for the embedding problem as well as the rigidity questions for cr structures.

i will talk about gromov-witten theory of calabi-yau one-folds
and fan-jarvis-ruan-witten theory of counterpart landau-ginzburg models.
the gw invariants and fjrw invariants are enumerative counting of stable
maps and sections of certain orbifold line bundles. we prove these
invariants are coefficients of expansions of appropriate quasi-modular
forms at different points, thus can be related by cayley transformations.