symmetry, and the study of invariant and equivariant objects, is a deep and unifying principle underlying a variety of mathematical fields. in particular, geometric mechanics is characterized by the application of symmetry and differential geometric techniques to lagrangian and hamiltonian mechanics, and geometric integration is concerned with the construction of numerical methods with geometric invariant and equivariant properties. computational geometric mechanics blends these fields, and uses a self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes.

in this talk, we will introduce a systematic method of constructing geometric integrators based on a discrete hamilton's variational principle. this involves the construction of discrete lagrangians that approximate jacobi's solution to the hamilton-jacobi equation. jacobi's solution can be characterized either in terms of a boundary-value problem or variationally, and these lead to shooting-based variational integrators and galerkin variational integrators, respectively. we prove that the resulting variational integrator is order-optimal, and when spectral basis elements are used in the galerkin formulation, one obtains geometrically convergent variational integrators.

we will also introduce the notion of a boundary lagrangian, which is analogue of jacobi's solution in the setting of lagrangian pdes. this provides the basis for developing a theory of variational error analysis for multisymplectic discretizations of lagrangian pdes. equivariant approximation spaces will play an important role in the construction of geometric integrators that exhibit multimomentum conservation properties, and we will describe two approaches based on spacetime generalizations of finite-element exterior calculus, and geodesic finite-elements on the space of lorentzian metrics.

originally, groups emerged in mathematics as structures describing the symmetries of all kinds of objects. nowadays we of course also have an abstract definition of a group. it is this abstract description that recently has been exploited to generalize things to the notion of so-called quantum groups. once a good definition of a quantum group was established, people tried to reverse the process and see quantum groups as describing “symmetries”of various objects.

in this talk we will 1) try to understand the definition of a (compact) quantum group, 2) discuss several examples, and 3) see how we can define quantum symmetry groups via actions of quantum groups on various spaces.

everybody is welcome! necessary background beyond undergraduate analysis and algebra will be provided during the talk.

perfectoid spaces were introduced to provide a geometric framework to the field of norms isomorphism from $p$-adic hodge theory, however, have proven their value well beyond this old result. for this reason, the geometry of perfectoid spaces is worth
studying, for its own sake. in this talk, we explicitly compute the picard group for the projectivoid line.

with the desire to generalize this result to other perfectoid spaces, as well as to classify higher-rank vector bundles on these $p$-adic analytic spaces, we ask whether an appropriate gaga theorem holds for perfections of proper schemes over a base nonarchimedean field of characteristic $p >0$.

the log-sobolev inequality (lsi) is a very useful tool for analyzing high-dimensional situations. for example, the lsi can be used for deriving hydrodynamic limits, for estimating the error in stochastic homogenization, for deducing upper bounds on the mixing times of markov chains, and even in the proof of the poincaré conjecture by perelman. for most applications, it is crucial that the constant in the lsi is uniform in the size of the underlying system. in this talk, we discuss when to expect a uniform lsi in the setting of unbounded spin systems.

accurate simulations of elasticity properties can be constructed by solving second order elliptic boundary value problems which have been approximated using finite elements. this talk will examine the process of converting the given pde into a weaker form and applying the galerkin method. in addition, novel matlab programs will be introduced, which will display a visual depiction of an object after force is applied, given a subdivision of the shape into regular or irregular triangles, a dirichlet boundary condition, and a two dimensional force function.

while the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of
consecutive primes among the permissible pairs of reduced residue classes (mod q) is surprisingly erratic. we propose a conjectural explanation for this phenomenon, based on the hardy-littlewood conjectures, which fits the observed data very well. we also study the distribution of the terms predicted by the conjecture, which proves to be surprisingly subtle. this is joint work with kannan soundararajan.

a tensor is a vector that lies in a tensor product of vector spaces. the rank of a tensor t is the smallest integer r such that t can be written as a sum of r pure tensors. finding such low rank decompositions of a tensor t is known as the parafac model. this model has many applications: algebraic complexity theory, chemometrics, neuroscience, signal processing to name a few. an alternative is the convex decomposition (code) model. it uses the nuclear norm of tensors and is more numerically stable. we will discuss upper and lower bound for the rank and nuclear norm of some tensors of interest, and some applications.

supercurves are the simplest class of complex supermanifolds, “one-dimensional” in some sense and thus analogs of riemann surfaces. i will describe a remarkable duality between pairs of supercurves that generalizes serre duality for riemann surfaces. self-dual supercurves are precisely the “super riemann surfaces” introduced by physicists in connection with string theory. i’ll suggest connections between this duality and the classical duality between points and hyperplanes in projective space. no prior knowledge of supergeometry is required.
(this is a continuation of the talk i gave earlier this quarter, and will begin with a review of that talk.)