in this talk, we explore the relationship between symplecticity and the
preservation of quadratic invariants. symplectic runge-kutta methods,
the bread and butter of numerical geometric integration, are exactly
the ones that preserve all quadratic first integrals of a system. but
when we expand our focus to larger classes of methods, we will find a
more nuanced connection. it is also known that for symplectic rk
methods, the action of discretization commutes with forming variational
equations, and we will discuss the expansion of this result to a larger
class of methods.

we generalize the notion of a de bruijn sequence to a ``multi de
bruijn sequence'': a cyclic or linear sequence that contains every
$k$-mer over an alphabet of size $q$ exactly $m$ times. for example,
over the binary alphabet $\{0,1\}$, the cyclic sequence $(00010111)$
and the linear sequence $000101110$ each contain two instances of each
$2$-mer $00,01,10,11$. we derive formulas for the number of such
sequences. the formulas and derivation generalize classical de bruijn
sequences (the case $m=1$). we also determine the number of multisets
of aperiodic cyclic sequences containing every $k$-mer exactly $m$
times; for example, the pair of cyclic sequences $(00011)(011)$
contains two instances of each $2$-mer listed above. this uses an
extension of the burrows-wheeler transform due to mantaci et al., and
generalizes a result by higgins for the case $m=1$.

in their celebrated work, p. li and s.-t. yau proved the famous li-yau gradient bound for positive solutions of the heat equation on manifolds with ricci curvature bounded from below. since then, li-yau type gradient bounds has been widely used in geometric analysis, and become a powerful tool in deriving geometric and topological properties of manifolds.

in this talk, we will present our recent works on li-yau type gradient bounds for positive solutions of the heat equation on complete manifolds with certain integral curvature bounds, namely, $\vert ric_\vert$ in $l^p$ for some $p>n/2$ or certain kato type of norm of $\vert ric_\vert$ being bounded together with a gaussian upper bound of the heat kernel. these assumptions allow the lower bound of the ricci curvature to tend to negative infinity, which is weaker than the assumptions in the known results on li-yau bounds. these are joint works with qi s. zhang.

i present a survey on the use of polynomial optimization in game theory. in recent years, there has been increased interest in applying methods from polynomial optimization to game theory. this research often studies games where individuals have polynomial utility functions or polynomial approximations to continuous utility functions. the presentation will include a discussion of research on zero sum games, generalized nash equilibrium, principal-agent problems, and potential games.

we investigate in this talk the average root number (i.e. sign of the functional equation) of
non-isotrivial one-parameter families of elliptic curves (i.e elliptic curves over q(t), or elliptic
surfaces over q). for most one-parameter families of elliptic curves, the average root number
is predicted to be 0. helfgott showed that under chowla's conjecture and the square-free
conjecture, the average root number is 0 unless the curve has no place of multiplicative
reduction over q(t). we then build non-isotrivial families of elliptic curves with no place
of multiplicative reduction, and compute the average root number of the families. some
families have periodic root number, giving a rational average, and some other families have
an average root number which is expressed as an innite euler product.
we then prove several density results for the average root number of non-isotrivial families
of elliptic curves, over z and over q (the previous density results found in the literature were
for isotrivial families). we also exhibit some surprising examples, for example, non-isotrivial
families of elliptic curves with rank r over q(t) and average root number $-(-1)^r$, which
were not found in previous literature.