have you always wondered how the banach-tarski paradox - namely cutting a ball into pieces and assembling those pieces into two balls - actually works? or what the mathematical reason behind it is? in this talk we will answer those questions by 1) reformulating the banach-tarski paradox, 2) explaining what paradoxical decompositions of various objects are, 3) constructing one for a ball, and 4) motivated by all those things introducing the concept of amenability, which has its origin in the aforementioned paradox and is ubiquitous in many areas of mathematics nowadays.

i will talk about the problem of separating multiple signals from each
other when we only have access to a few linear (or non-linear)
combinations of them. an example of this type of problem is at a
cocktail party when you are trying to have a conversation with a
friend but there are several conversations happening around you. your
ears provide you with a superposition of all the voices, and your
brain does remarkably well at focusing on your friend's voice and
drowning out all the others. we will talk about one computer algorithm
(or time permitting, more) that does such a task (reasonably)
successfully. along the way, we will talk about important tools in
mathematical signal processing, including the fourier transform and
sparsity.

a ``higgs bundle" is essentially a family of matrices, and if you
try to diagonalize one you get a "spectral cover." they arise in the
study of moduli of vector bundles. i will introduce higgs bundles and
spectral covers, and then i will describe their cohomological
invariants.

we shall discuss the recent work of brendle, choi and daskalopoulos on the classification via a pogorelov's type estimate.

fusion categories appear in many areas of mathematics. they are
realized by topological quantum field theories, representations of
finite groups and hopf algebras, and invariants for knots and
murray-von neumann subfactors. an important numerical invariant of
these categories are the frobenius-schur indicators, which are
generalized versions of those for finite group representations. using
these categorical indicators we are able to distinguish near-group
fusion categories, that is those fusion categories with one
non-invertible object, and obtain some realizations of their tensor
equivalence classes.

we present a general framework for approximating the component structure of random combinatorial assemblies when both the size $n$ and the number of components $k$ is specified. the approach is an extension of the usual saddle point approximation, and we demonstrate near-universal behavior when the rank $r := n-k$ is small relative to $n$ (hence the name `low rank’).

in particular, for $\ell = 1, 2, \ldots$, when $r \asymp n^\alpha$, for $\alpha \in \left(\frac{\ell}{\ell+1}, \frac{\ell+1}{\ell+2}\right)$, the size~$l_1$ of the largest component converges in probability to $\ell+2$. when $r \sim t\, n^{\ell/(\ell+1)}$ for any $t>0$ and any positive integer $\ell$, we have $p(l_1 \in \{\ell+1, \ell+2\}) \to 1$. we also obtain as a corollary bounds on the number of such combinatorial assemblies, which in the special case of set partitions fills in a countable number of gaps in the asymptotic analysis of louchard for stirling numbers of the second kind.

this is joint work with richard arratia

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convex cones over a real unital algebra, which in addition are closed under
inversion, may seem peculiar. however, convex invertible cones (cics)
naturally appear in stability analysis of continuous-time physical systems.
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with this motivation, in this talk we explore examples
of cics over some algebras and establish interconnections among them.
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this indicates at the importance of the study of
rational functions, of non-commuting variables, with
certain positivity properties.
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this talk is based on an ongoing research for many years.
some of it in collaboration with daniel alpay, chapman university,
california,
nir cohen, natal, brazil and the late leiba rodman, from the college
of william and mary, virginia.

given a rectangular board b with any set of forbidden states, the rook
number $r_b(k)$ is the number of ways of placing k non-attacking rooks
on board $b$ which avoid the set of forbidden states. when non-attacking
only means no two rooks lie in the same column, the number of such
configurations is called the file number, $f_b(k)$. when the board $b$ is
the staircase board, with row lengths $n-1, n-2, ..., 1$, then the rook
number coincides with the stirling numbers of the second kind, and the
file number coincides with the unsigned stirling numbers of the first
kind.

we will demonstrate, using poisson approximation and an explicit
coupling, how one can obtain quantitative bounds on rook and file
numbers under certain conditions. in the case of stirling numbers, our
results fill a gap in a recent asymptotic expansion which uses
explicitly defined parameters due to louchard.
this is joint work with richard arratia.