we will talk about kronheimer and mrowka's knot concordance
invariant, $s^\sharp$. we compute the invariant for various knots. our
computations reveal some unexpected phenomena, including that
$s^\sharp$ differs from rasmussen's invariant $s$, and that it is not
additive under connected sums. we also generalize the definition of
$s^\sharp$ to links by giving a new characterization of the invariant
in terms of immersed cobordisms.

we will discuss central sequence algebras of von neumann algebras, and provide a class of ii$_1$ factors whose central sequence algebra is not the ``tail'' algebra associated to any decreasing sequence of von neumann subalgebras. this settles a question of mcduff from 1969. we will also discuss an application of these techniques to the notion of tracial stability for groups. this is based on joint work with adrian ioana.

the a-function on a coxeter group w is a function a : w $\rightarrow$ n defined by lusztig which is intimately related to the partition of w into kazhdan--lusztig cells and to the representation theory of the hecke algebra of w. it is known that the identity element of w is the only element with a-value 0, while a non-identity element has a-value 1 if and only if it has a unique reduced word. however, as its definition relies on the kazhdan--lusztig basis of the hecke algebra, thea-function is often difficult to compute for general elements.
in this talk we will focus on elements of a-value 2, or a-2 elements. we show that a-2 elements arefully commutative in the sense of stembridge, which allows us to associate to them certain posets called heaps and, in many cases, certain generalized temperley--lieb diagrams. using heaps and temperley--lieb diagrams, we conjecture a combinatorial characterization of a-2 elements, classify all coxeter groups with finitely many a-2 elements, and enumerate a-2 elements for all groups from the classification. joint with richard green.

we will talk about several methods of proving non-asymptotic bounds on
the spectral norm of random matrices including epsilon-net argument and
slepian's lemma. the main idea is to relate the spectral norm of a
random matrix to the supremum of a random process.

in his thesis, m. bhargava proved parameterizations and identified local
conditions which he used to give asymptotic counts for $s_4$ quartic and
quintic number fields, ordered by discriminant. this talk will discuss
results in an ongoing project to add detail to bhargava's work by
considering in addition to the field discriminant, the lattice shape of
the ring of integers in the canonical embedding, and by giving strong
rates with lower order terms in the asymptotics. these results build on
earlier work of taniguchi-thorne, bhargava-shankar-tsimerman and
bhargava-harron.

i will discuss certain methods used to find so called ``cyclotomic
solutions'' to a system of polynomial equations. these are solutions in
which each coordinate is a root of unity. i will then give a simple real
world example in which these methods come in handy, coming from joint work
with christian woll.

belyi's theorem says that on one hand, a curve over a field of
characteristic 0 that admits a finite map to $\mathbf{p}^1$ ramified
over at most three points must descend to a subfield algebraic over
$\mathbb{q}$, and on the other hand any curve over such a subfield does
indeed admit such a morphism (without any further base extension). one
might ask whether a similar statement holds over a field of
characteristic $p$, replacing $\mathbb{q}$ with $\mathbb{f}_p$. for
general morphisms this is false, but it becomes true if we restrict to
tamely ramified morphisms to $\mathbf{p}^1$. such a statement was
originally given by saidi, in which the ``other hand'' assertion was made
conditional on the existence of some tamely ramified morphism from the
given curve to $\mathbf{p}^1$.

in the pre-talk, we will discuss how to establish existence of a tamely
ramified morphism in characteristic \mbox{$p\>2$}. this is ``classical'' over an
infinite algebraic extension of $\mathbb{f}_p$; to do it over a fixed
finite field requires a density statement in the style of poonen's
finite field bertini theorem.

in the talk proper, we will discuss work of sugiyama-yasuda that
establishes the existence of a tamely ramified morphism when the base
field is algebraically closed of characteristic 2. the case where the
base field is finite of characteristic 2 requires a further geometric
reinterpretation of the key construction of sugiyama-yasuda; this is
joint work with daniel litt and jakub witaszek.