An outstanding challenge in large scale computing is to enable the casual
application programmer to realize performance obtained by an expert.
The challenge has grown in recent years due to disruptive technological
changes, which are expected to continue. In HPC, performance programming
generally relies on a priori knowledge about the application. However,
it is important to avoid entangling application software with knowledge
about the hardware.

The HPC community relies heavily on libraries, which have helped insulate
application software against technological change. However, not all
change can be accommodated via libraries, and an alternative approach
is to restructure the source using a custom translator that incorporates
the required a priori knowledge.

I will describe custom source-to-source translators targeting different
performance programming problems arising in large scale computation.
The first translator, Saaz, reduces the overheads of abstraction by up
to an order of magnitude in application libraries used to construct tools
for data discovery in turbulent flow simulation. The second translator,
MATE, restructures MPI applications to tolerate significant amounts
of communication on distributed memory computers. The third translator,
Mint, transforms annotated C++ stencil codes into highly optimized CUDA
that comes close (80%) to the performance of carefully hand coded CUDA
running on GPUs.

Each translator incorporates application semantics into the optimization
process, which are unavailable through a traditional compiler working
with conventional language constructs. In effect, the translators treat
idiomatic constructs or library APIs as a domain specific language
embedded within a conventional programming language--in our case C or C++.

Domain specific translation is an effective means of managing development
costs, enabling the domain scientist to remain focused on the domain
science, while realizing performance usually attributed to expert coders.

I will conclude the talk with earlier work on run times, that led to the
research in domain specific translation.

In 1995, Gessel introduced a multivariate formal power series $G$ tracking the distribution of ascents and descents in labeled binary trees. In addition to showing the $G$ is a symmetric function, he conjectured that $G$ is Schur-positive. In this talk, we'll see how to expand $G$ positively in terms of ribbon Schur functions. Moreover, we'll see how certain specializations of $G$ relate to actions on hyperplane arrangements. As an application of our work, we get a proof of gamma-positivity of the distribution of right edges over the set of local binary search trees.

Classical multidimensional scaling is an important tool for data reduction in many applications. It takes in a distance matrix and outputs low-dimensional embedded samples such that the pairwise distances between the original data points can be preserved, when treating them as deterministic points. However, data are often noisy in practice. In such case, the quality of embedded samples produced by classical multidimensional scaling starts to break down, when either the ambient dimensionality or the noise variance gets larger. This motivates us to propose the modified multidimensional scaling procedure which applies a nonlinear shrinkage to the sample eigenvalues. The nonlinear transformation is determined by sample size, the ambient dimensionality, and moment of noise. As an application, we consider the problem of clustering high-dimensional noisy data. We show that modified multidimensional scaling followed by various clustering algorithms can achieve exact recovery, i.e., all the cluster labels can be recovered correctly with probability tending to one. Numerical studies lend strong support to our proposed methodology.

Dr. Qiang Sun is currently an Assistant Professor of Statistics at the University of Toronto within the Department of Statistical Sciences and Department of Computer and Mathematical Sciences. Previously, he worked at Princeton University as an associate research scholar. He earned his Ph.D. from the University of North Carolina Chapel Hill in 2014 and his B.S. from University of Science and Technology of China in 2010. His research interests span a broad spectrum, including hypothesis-driven imaging genetics, clustering, manifold learning, nonconvex optimization and robust statistics.

A semitoric system is a type of 4-dimensional integrable system which possesses a circular symmetry; semitoric systems are classified in terms of five invariants by a result of Pelayo-Vu Ngoc. In this talk we will introduce semitoric systems, discuss their classification, and discuss several recent results related to explicitly constructing such systems. The general strategy of such constructions is via a one-parameter family of systems, known as a semitoric family, which passes through certain degeneracies to transition into the desired system. Using these families, we find several new explicit semitoric systems which display various behavior and are of importance in the semitoric minimal models program. The work presented is joint with Y. Le Floch and S. Hohloch (and work joint with D. Kane and A. Pelayo will also be mentioned).

In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.

We use Erman and Wood's semiample extension of Poonen's closed
point sieve to compute the probability that a semiample complete
intersection over a finite field is smooth. This generalizes work of Bucur
and Kedlaya, who provided the analogous calculation in the ample setting.
We further extend the result by allowing the requirement that the complete
intersection meet a closed subscheme transversely, so long as the subscheme
satisfies a mild Altman Kleiman type condition. In both cases the
probability stabilizes to a product of local factors determined by the
semiample divisor in question.

We prove a gap theorem for asymptotic entropy on ancient Ricci flows. We also prove an assertion made by Perelman in his paper ``The entropy formula for the Ricci flow and its geometric applications'', saying that for an ancient solution with bounded nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales.

The Shuffle Theorem of Carlsson and Mellit gives a well-studied combinatorial expression for the bigraded Frobenius characteristic of the ring of diagonal harmonics. The Rational Shuffle Theorem of Mellit and the Delta Conjecture proposed by Haglund, Remmel and Wilson are two natural generalizations of the Shuffle Theorem. The Primary goal of this dissertation is to prove some special cases of the conjectures, and compute the Schur function expansion of the corresponding symmetric function expressions.