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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).

Series: Job Candidate Talk

Experimental neuroscience is achieving rapid progress in the ability
to collect neural activity and connectivity data. This holds promise to
directly test many theoretical ideas, and thus advance our understanding
of "how the brain works." How to interpret this data, and what exactly
it can tell us about the structure of neural circuits, is still not
well-understood. A major obstacle is that these data often measure
quantities that are related to more "fundamental" variables by an
unknown nonlinear transformation. We find that combinatorial topology
can be used to obtain meaningful answers to questions about the
structure of neural activity.
In this talk I will first introduce a new method, using tools from
computational topology, for detecting structure in correlation matrices
that is obscured by an unknown nonlinear transformation. I will
illustrate its use by testing the "coding space" hypothesis on neural
data. In the second part of my talk I will attempt to answer a simple
question: given a complete set of binary response patterns of a network,
can we rule out that the network functions as a collection of
disconnected discriminators (perceptrons)? Mathematically this
translates into questions about the combinatorics of hyperplane
arrangements and convex sets.

Series: Job Candidate Talk

Synapses in many cortical areas of the brain are dominated by local, recurrent connections. It has long been suggested, therefore, that cortical networks may serve to restore a noisy or incomplete signal by evolving it towards a stored pattern of activity. These "preferred" activity patterns are constrained by the excitatory connections, and comprise the neural code of the recurrent network. In this talk I will briefly review the permitted and forbidden sets model for cortical networks, first introduced by Hahnloser et. al. (Nature, 2000), in which preferred activity patterns are modeled as "permitted sets" - that is, as subsets of neurons that co-fire at stable fixed points of the network dynamics. I will then present some recent results that provide a geometric handle on the relationship between permitted sets and network connectivity. This allows us to precisely characterize the structure of neural codes that arise from a simple learning rule. In particular, we find "natural codes" that can be learned from few examples, and that closely mimic receptive field codes that have been observed in the brain. Finally, we use our geometric description of permitted sets to prove that these networks can perform error correction and pattern completion for a wide range of connectivities.

Series: Job Candidate Talk

Driving nanomagnets by spin-polarized currents offers exciting
prospects in magnetoelectronics, but the response of the magnet to
such currents remains poorly understood. For a single domain
ferromagnet, I will show that an averaged equation describing the
diffusion of energy on a graph captures the low-damping dynamics of
these systems. In particular, I compute the mean times of thermally
assisted magnetization reversals in the finite temperature system,
giving explicit expressions for the effective energy barriers
conjectured to exist. I will then outline the problem of extending the
analysis to spatially non-uniform magnets, leading to a transition
state theory for infinite dimensional Hamiltonian systems.