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Series: Job Candidate Talk

This talk will be about random lozenge tilings of a class of
planar domains which I like to call "sawtooth domains." The basic
question is: what does a uniformly random lozenge tiling of a large
sawtooth domain look like? At the first order of randomness, a remarkable
form of the law of large numbers emerges: the height function of the tiling
converges to a deterministic "limit shape." My talk is about the next
order of randomness, where one wants to analyze the fluctuations of tiles
around their eventual positions in the limit shape. Quite remarkably, this
analytic problem can be solved in an essentially combinatorial way, using a
desymmetrized version of the double Hurwitz numbers from enumerative
algebraic geometry.

Series: Job Candidate Talk

Under the operation of connected sum, the set of knots in the 3-sphere forms a monoid. Modulo an equivalence relation called concordance, this monoid becomes a group called the knot concordance group. We will consider various algebraic methods -- both classical and modern -- for better understanding the structure of this group.

Series: Job Candidate Talk

In first-passage percolation (FPP), one places random non-negative
weights on the edges of a graph and considers the induced weighted
graph metric. Of particular interest is the case where the graph is
Z^d, the standard d-dimensional cubic lattice, and many of the
questions involve a comparison between the asymptotics of the random
metric and the standard Euclidean one. In this talk, I will survey
some of my recent work on the order of fluctuations of the metric,
focusing on (a) lower bounds for the expected distance and (b) our
recent sublinear bound for the variance for edge-weight distributions
that have 2+log moments, with corresponding concentration results.
This second work addresses a question posed by Benjamini-Kalai-Schramm
in their celebrated 2003 paper, where such a bound was proved for only
Bernoulli weights using hypercontractivity. Our techniques draw
heavily on entropy methods from concentration of measure.

Series: Job Candidate Talk

It is known that certain medium, for example electromagnetic
field and Bose Einstein condensate, has positive speed of sound. It
is observed that if the medium is in its equilibrium state, then an
invading subsonic particle will slow down due to friction; and the
speed of a supersonic particle will slow down to the speed of sound
and the medium will radiate. This is called Cherenkov radiation. It
has been widely discussed in physical literature, and applied in
experiments. In this talk I will
present some rigorous mathematical results. Joint works with Juerg
Froehlich, Israel Michael Sigal, Avy Soffer, Daniel Egli, Arick Shao.

Series: Job Candidate Talk

Spin glasses are disordered spin systems originated from the
desire of understanding the strange magnetic behaviors of certain alloys in
physics. As mathematical objects, they are often cited as examples of
complex systems and have provided several fascinating structures and
conjectures. This talk will be focused on one of the famous mean-field spin
glasses, the Sherrington-Kirkpatrick model. We will present results on the
conjectured properties of the Parisi measure including its uniqueness and
quantitative behaviors. This is based on joint works with A. Auffinger.

Series: Job Candidate Talk

The Hales--Jewett theorem is one of the pillars of Ramsey theory, from
which many other results follow.
A celebrated result of Shelah from 1988 gives a significantly improved
bound for this theorem. A key tool used in his proof, now known as the cube
lemma, has become famous in its own right. Hoping to further improve
Shelah's result, more than twenty years ago, Graham, Rothschild and Spencer
asked whether there exists a polynoimal bound for this lemma. In this talk,
we present the answer to their question and discuss numerous connections of
the cube lemma with other problems in Ramsey theory.
Joint work with David Conlon (Oxford), Jacob Fox (MIT), and Benny Sudakov
(ETH Zurich).

Series: Job Candidate Talk

Experimentalists observed that microscopically disordered systems exhibit
homogeneous geometry on a macroscopic scale. In the last decades elegant
tools were created to mathematically assert such phenomenon. The classical
geometric results, such as asymptotic graph distance and isoperimetry of
large sets, are restricted to i.i.d. Bernoulli percolation. There are many
interesting models in statistical physics and probability theory, that
exhibit long range correlation.
In this talk I will survey the theory, and discuss a new result proving,
for a general class of correlated percolation models, that a random walk on
almost every configuration, scales diffusively to Brownian motion with
non-degenerate diffusion matrix. As a corollary we obtain new results for
the Gaussian free field, Random Interlacements and the vacant set of Random
Interlacements. In the heart of the proof is a new isoperimetry result for
correlated models.

Series: Job Candidate Talk

Anna Vershynina is a job candidate. She is a Mathematical Physicist working on the rigorous mathematical theory of N-body problem and its relation with quantum information.

Entanglement is one of the crucial phenomena in quantum theory. The existence of entanglement between two parties allows for notorious protocols, like quantum teleportation and super dense coding. Finding a running time for many quantum algorithms depends on how fast a system can generate entanglement. This raises the following question: given some Hamiltonian and dissipative interactions between two or more subsystems, what is the maximal rate at which an ancilla-assisted entanglement can be generated in time. I will review a recent progress on bounding the entangling rate in a closed bipartite system. Then I will generalize the problem first to open system and then to a higher multipartite system, presenting the most recent results in both cases.

Series: Job Candidate Talk

The perimeter of a convex set in R^n with respect to a given measure is the measure's density averaged against the surface measure of the set. It was proved by Ball in 1993 that the perimeter of a convex set in R^n with respect to the standard Gaussian measure is asymptotically bounded from above by n^{1/4}. Nazarov in 2003 showed the sharpness of this bound. We are going to discuss the question of maximizing the perimeter of a convex set in R^n with respect to any log-concave rotation invariant probability measure. The latter asymptotic maximum is expressed in terms of the measure's natural parameters: the expectation and the variance of the absolute value of the random vector distributed with respect to the measure. We are also going to discuss some related questions on the geometry and isoperimetric properties of log-concave measures.

Series: Job Candidate Talk

Orthonormal bases (ONB) are used throughout mathematics and its
applications. However, in many settings such bases are not easy to come by.
For example, it is known that even the union of as few as two intervals may
not admit an ONB of exponentials. In cases where there is no ONB, the next
best option is a Riesz basis (i.e. the image of an ONB under a bounded
invertible operator).
In this talk I will discuss the following question: Does every finite union
of rectangles in R^d, with edges parallel to the axes, admit a Riesz basis
of exponentials? In particular, does every finite union of intervals in R
admit such a basis? (This is joint work with Gady Kozma).