The term "Gray code" usually refers to an ordering of the n-bit binary
strings in which successive strings differ in a single bit. For
example, 000, 001, 011, 010, 110, 111, 101, 100 suffices for $n=3$. The
term "de Bruijn sequence" usually refers to a circular string of $2^n$
bits in which each n-bit binary string appears once as a substring. For
example, 0000100110101111 suffices for $n=4$ since its substrings are
0000, 0001, 0010, ..., 1110, 1100, 1000 (where the last three substrings
wrap-around).

These concepts have natural generalizations to other combinatorial
objects. For example, 123, 132, 312, 321, 231, 213 is a Gray code for
the permutations of {1,2,3} in which successive permutations differ by
an adjacent-transposition. Simiarly, 321312 is a universal cycle for
the 2-permutations of {1,2,3}. (The term "universal cycle" is often
used for non-binary de Bruijn sequences.)

Gray codes and universal cycles are usually constructed 'globally',
meaning that the overall order is easy to describe. In this talk, we
instead consider 'local' constructions, where it is easy to describe the
next object in the order. With this change in focus, we are able to
unify a number of previous results, and solve several well-known open
problems. The talk will begin with an overview of the research area and
its applications, and should be of interest to computer scientists and
discrete mathematicians.

In addition to frequent single-nucleotide mutations, mammalian and
many other genomes undergo rare and dramatic changes called genome
rearrangements. These include inversions, fissions, fusions, and
translocations. Although analysis of genome rearrangements was
pioneered by Dobzhansky and Sturtevant in 1938, we still know very
little about the rearrangement events that produced the existing
varieties of genomic architectures. Recovery of mammalian
rearrangement history is a difficult combinatorial problem that I will
cover in this talk. Our data sets have included sequenced genomes
(human, mouse, rat, and others), as well as radiation hybrid maps of
additional mammals.

Kronecker coefficients count the multiplicities of
irreducible representations in the tensor product of two irreducible
representations of the symmetric group. While their study was initiated
almost 75 years, very little is still known about them, and one of the
major problems of algebraic combinatorics is to find a positive
combinatorial interpretation for the Kronecker coefficients. Recently
this problem found a new meaning in the field of Geometric Complexity
Theory, initiated by Mulmuley and Sohoni, where certain conjectures on
the complexity of computing and deciding positivity of Kronecker
coefficients are part of a program to prove the ``P vs NP'' Millennium
problem.

In this talk we will describe several problems and some results on
different aspects of the Kronecker coefficients. A conjecture of Jan
Saxl states that the tensor square $S_\rho$ $ \otimes$ $S_\rho$, where $\rho$ is
the staircase partition, contains every irreducible representation of
$S_n$. We will present results towards this conjecture, as well as a tool
for determining the positivity of certain Kronecker coefficients. We
will also explore the combinatorial aspect of the problem and show how
to prove certain unimodality results using Kronecker coefficients,
including Sylvester's theorem on the unimodality of $q$-binomial
coefficients (as polynomials in $q$) and some new extensions thereof, like
strict unimodality. The results are based on joint work with Igor Pak
and Ernesto Vallejo.

Shrinking Ricci solitons are generalizations of positive Einstein
manifolds which arise naturally in the analysis of singularities of
the Ricci flow. At present, all known complete noncompact examples
either split locally as products or possess conical structures at
infinity. I will describe recent joint work with Lu Wang in which we
prove that such conical structures admit little flexibility: if two
shrinking solitons are asymptotic along some end of each to the same
regular cone, then the solitons must actually be isometric on some
neighborhoods of infinity of these ends. As an application, we prove
that the only complete connected shrinking soliton asymptotic to a
rotationally symmetric cone is the Gaussian soliton.

We consider critical branching Brownian motion with absorption, in which
there is initially a single particle at $x > 0$, particles move according to
independent one-dimensional Brownian motions with the critical drift of
negative the square root of 2, and particles are absorbed when they reach
zero. Kesten (1978) showed that almost surely this process eventually
dies out. Here we obtain upper and lower bounds on the probability that
the process survives until some large time t. These bounds improve upon
results of Kesten (1978), and partially confirm nonrigorous predictions of
Derrida and Simon (2007). We will also discuss results concerning the
behavior of the process before the extinction time, as x tends to
infinity. We estimate the number of particles in the system at a given
time and the position of the right-most particle, and we obtain asymptotic
results for the configuration of particles at a typical time. This is
based on joint work with Julien Berestycki and Nathanael Berestycki.

Technological innovations have revolutionized the process of scienti�c research and knowledge discovery. Nowadays, massive abundance of data is not uncommon in many scientific areas ranging from genomics to economy. They give rise to many novel opportunities in knowledge discovery but come coupled with many unforeseen computational and statistical challenges, including scalability and storage bottleneck, noise accumulation, spurious correlation, incidental endogeneity, and measurement errors. These challenges are distinguished and require both new computational and new statistical paradigms. My current research lies at the frontier of such big data paradigm and focuses mostly on only one task of variable selection and feature extraction.
In this talk I will expose the challenges that arise in this one task and show how optimization, probability and matrix analysis interplay in the new statistical theory.

Let $\chi$ be a totally odd character of a totally real number
field. In 1981, B. Gross formulated a p-adic analogue of a conjecture of
Stark which expresses the leading term at s=0 of the p-adic L-function
attached to $\chi\omega$ as a product of a regulator and an algebraic
number. Recently, Dasgupta-Darmon-Pollack proved Gross' conjecture in the
rank one case under two assumptions: that Leopoldt's conjecture holds for F
and p, and a certain technical condition when there is a unique prime above
p in F. After giving some background and outlining their proof, I will
explain how to remove both conditions, thus giving an unconditional proof
of the conjecture. If there is extra time I will explain an application to
the Iwasawa Main Conjecture which comes out of the proof, and make a few
remarks on the higher rank case.

Many biochemical processes at the sub-cellular level involve a small number of molecules. The local numbers of these molecules vary in space and time, and exhibit random fluctuations that can only be captured with stochastic simulations. We present a novel stochastic operator-splitting algorithm to model such reaction-diffusion phenomena. The reaction and diffusion steps employ stochastic simulation algorithms and Brownian dynamics, respectively. Through theoretical analysis, we have developed an algorithm to identify if the system is reaction-controlled, diffusion-controlled or is in an intermediate regime. The time-step size is chosen accordingly at each step of the simulation. We have used three examples to demonstrate the accuracy and robustness of the proposed algorithm. The first example deals with diffusion of two chemical species undergoing an irreversible bimolecular reaction. It is used to validate our algorithm by comparing its results with the solution obtained from a corresponding deterministic partial differential equation at low and high number of molecules. In this example, we also compare the results from our method to those obtained using a Gillespie multi-particle (GMP) method. The second example, which models simplified RNA synthesis, is used to study the performance of our algorithm in reaction- and diffusion-controlled regimes and to investigate the effects of local inhomogeneity. The third example models reaction-diffusion of CheY molecules through the cytoplasm of Escherichia coli during chemotaxis. It is used to compare the algorithm’s performance against the GMP method. Our analysis demonstrates that the proposed algorithm enables accurate simulation of the kinetics of complex and spatially heterogeneous systems. It is also computationally more efficient than commonly used alternatives, such as the GMP method.