it has been a puzzling question why several organisms reproduce sexually. fisher and muller hypothesized that reproducing by sex can speed up the evolution. they explained that in the sexual reproduction, recombination can combine beneficial alleles that lie on different chromosomes, which speeds up the time that those beneficial alleles spread to the entire population. we consider a population model of fixed size $n$, in which we will focus on two loci on a chromosome. each allele at each locus can mutate into a beneficial allele at rate $\mu_n$. the individuals with 0, 1, and 2 beneficial alleles die at rates $1, 1-s_n$ and $1-2s_n$ respectively. when an individual dies, with probability $1-r_n$, the new individual inherits both alleles from one parent, chosen at random from the population, while with probability $r_n$, recombination occurs, and the new individual receives its two alleles from different parents. under certain assumptions on the parameters $n, \mu_n, s_n$ and $r_n$, we obtain an asymptotic approximation for the time that both beneficial alleles spread to the entire population. when the recombination probability is small, we show that recombination does not speed up the time that the two beneficial alleles spread to the entire population, while when the recombination probability is large, we show that recombination decreases the time, which agrees with fisher-muller hypothesis, and confirms the advantage of reproducing by sex.

a new primal-dual path-following shifted penalty-barrier method will be described for solving nonlinear inequality constrained optimization problems (nip). the proposed method has a bi-level structure in which a trajectory parameterized by the penalty and barrier parameters and lagrangian multipliers estimates is closely followed towards a constrained local minimizer of nip. this method inherits some features of the primal-dual augmented lagrangian method for solving nonlinear equality constraint problems (nep) but has been extended to handle inequality constraints. global and local convergence results will be presented. finally, numerical results from the cutest test collection will be provided to support the robustness of the proposed algorithm.

a simple character with delusions of grad school applies his abstraction-through-incomprehension style of independent squalorship to banalities which appear, from an obtuse angle, to concern foundational questions in mathematics, providing an abject lesson in how common misconceptions may be avowed.

let $k$ be a global field, that is, a number field of finite
degree
over $\bbb q$ or the function field of a smooth projective curve $c$
over a finite field $f$. let $x$ be a smooth projective variety over $k$,
and let $k$
be the maximal cyclotomic extension of $k$, obtained by adjoining
all roots of unity. if $x$ is an abelian variety, a famous theorem,
due to ribet in the number field case and lang-neron in the
function field case when $x$ has trace zero over the constant subfield
of $k$, asserts that the torsion subgroup of the mordell-weil group of $x$
over $k$ is finite. denoting by $k^{sep}$ a separable closure of $k$,
this result is equivalent to finiteness of the fixed part
by $g=gal(k^{sep}/k)$ of the etale cohomology group
$h^1(x_{k^{sep}},\bbb q/\bbb z)$,
where we ignore the $p$-part in positive characteristic $p$. in a recent
paper, roessler-szamuely generalize this result to all odd cohomology groups.
the trace zero assumption in the function field case is replaced by
a ''large variation'' assumption on the characteristic polynomials of
frobenius acting on the cohomology of the fibres of a
morphism $f: \mathcal{x}\to c$ from a smooth projective variety
$\mathcal{x}$
over a finite field to $c$ with generic fibre $x$.
in this talk, i will discuss the case of even degree, proving some
positive results in the number field case and negative results in the
function field case.

non-abelian $c^\ast$-algebras can be understood better from the examination of their maximal abelian subalgebras. in particular, renault showed that in the presence of a cartan subalgebra, a $c^\ast$-algebra can be associated in a canonical way with a topological twisted groupoid.

in joint work with jon brown, adam fuller, and david pitts, we extend renault's result by identifying cartan pairs revealed by gradings by a group.

the shape of a degree $n$ number field is a $n-1$-variable real quadratic form
(up to equivalence and scaling) which keeps track of the lattice shape of its ring
of integers relative to $\mathbb{z}$. for number fields of small degree, in previous
joint work with bhargava, we showed that shapes of $s_n$-number fields are
equidistributed, when ordered by absolute discriminant. the proof relies heavily
on bhargava's parametrizations which introduces but ultimately ignores the notion
of resolvent rings. this talk discusses work in progress, joint with christelle vincent,
in which we define the joint shape of a ring and its resolvent ring in order to prove
equidistribution of joint shapes of quartic fields and their cubic resolvent fields.

almost everything we encounter in our 3-dimensional world is a surface - the outside of a solid object. comparing the shapes of surfaces is, not surprisingly, a fundamental problem in both theoretical and applied mathematics. deep mathematical results are now being used to study objects such as bones, brain cortices, proteins and biomolecules. this talk will discuss recent joint work with patrice koehl that introduces a new metric on the space of genus-zero surfaces and applies it in this context.