in their 1967 seminal paper, foias and prodi captured precisely a notion of finitely many degrees of freedom in the context of the two-dimensional (2d) incompressible navier-stokes equations (nse). in particular, they proved that if a sufficiently large spectral projection of the difference of two solutions converge to 0 asymptotically in time, then the corresponding complementary projection of their difference must also converge to 0 in the infinite-time limit. in other words, the high modes are ``eventually enslaved`` by the low modes. one could thus define the number of degrees of freedom of the flow to be the smallest number of modes needed to guarantee this convergence for a given flow, insofar as it is represented by a solution to the nse. this property has since led to several developments in the understanding and application of the long-time behavior of solutions to the nse, for instance, in the context of turbulence, data assimilation, and the existence of determining forms. in this talk, we will discuss this asymptotic enslavement property as it regards the issue of uniqueness of invariant measures for stochastically forced equations, particularly the damped, stochastic kdv equation.

a new path-following primal-dual augmented lagrangian method is proposed
for solving nonlinear equality constrained optimization problems (nep).
at each iteration, a newton-like method is used to solve a perturbed
optimality condition that defines a penalty trajectory parameterized by
both the penalty parameter and the estimated lagrange multipliers.
we show that this method is globally convergent and has a quadratic
convergence rate in the limit. finally, numerical experiments on
problems from the cutest test collection are are used to support the
theoretical analysis.

the unit distance problem asks: given $n$ points in $\mathbb r^d$, how many pairs of points can have distance one? this problem can be re-phrased as a question in incidence geometry, and standard machinery from that field leads to certain non-optimal bounds. i will discuss some recent progress on the unit distance problem in three dimensions that goes beyond the standard tools of incidence geometry.

amphiphilic structures such as cell membranes and lipid vesicles play essential roles in biological applications. in this talk, we will first introduce the functionalized cahn-hilliard (fch) model for the free energy of amphiphilic mixtures. the fch model admits local minimizers corresponding to amphiphilic bilayers, filaments, micelles, and other defect structures. we will describe the geometric motion of bilayers, filaments, and their competition. to capture the coexistence and localization of amphiphilic structures, we introduce a degenerate fch functional as a modified model for the free energy of amphiphilic mixtures. we prove that the degenerate fch functional admits geometrically localized minimizers, which correspond to localized amphiphilic structures. specifically, we identify the leading order profile of bilayers under the assumption that the geometrically localized minimizers have bounded variations along the tangential directions.

for each $-1 < q < 1$, bozejko and speicher's q-gaussian functor is a natural and important generalization of voiculescu's free gaussian functor. we study a class of subalgebras of the corresponding q-gaussian von neumann subalgebras. we construct a riesz basis in the spirit of radulescu in the q-fock space. then we use this basis and follow popa's approach to show that when $|q| < 1/9$ , the generator subalgebras are maximal amenable inside those q-gaussian algebras. this is joint work with sandeepan parekh and koichi shimada.

in his influential paper ``modular curves and the eisenstein ideal,''
barry mazur studied congruences modulo p between cusp forms and the
eisenstein series of weight 2 and prime level n. in particular, he
defined the eisenstein ideal in the relevant hecke algebra, and showed
that it is locally principal. we'll discuss the analogous situation for
certain squarefree levels n, and show that, while the eisenstein ideal
may not be locally principal, we can count the minimal number of
generators and explain the arithmetic significance of this number. this
is joint work with carl wang-erickson.

the theta graph $\theta_{k, t}$ consists of two vertices with $t$ internally disjoint paths of length $k$ between them. since $\theta_{k,2}$ is a cycle of length $2k$, the study of the tur\'an number of this graph includes the notorious even-cycle problem. we show that for fixed $k$ and large $t$, a graph without a $\theta_{k,t}$ contains at most $c_k t^{1-1/k}n^{1+1/k}$ edges, and we use graphs constructed via random polynomials to show that this is sharp up to the constant $c_k$ when $k$ is odd.

in the 1990's, vafa-witten tested s-duality of n=4 susy yang-mills theory on a complex algebraic surface x by studying modularity of a certain partition function. in 2017, tanaka-thomas defined vafa-witten invariants by constructing a symmetric perfect obstruction theory on the moduli space of higgs pairs (e,$\phi$) on x. the instanton contribution ($\phi=0$) to these invariants is the virtual euler number of moduli space of sheaves. i outline a method to calculate this contribution, when x is of general type, by reducing to descendent donaldson invariants. for rank 2, this leads to verifications of a formula from vafa-witten. the method can be ``refined'' to virtual $\chi_y$ genus, elliptic genus, and cobordism class, which involves weak jacobi forms and borcherds lifts thereof. i also give a new formula for rank 3 vw invariants on general type surfaces, correcting an error in the physics literature. joint with g$\ddot{\text{o}}$
ttsche.