Speaker: Evan Gawlik (UCSD)

Title: Numerical Methods for Partial Differential Equations on Evolving Domains

Abstract: Many important and challenging problems in computational science and
engineering involve partial differential equations with a high level
of geometric complexity. Examples include moving-boundary problems,
where the domain on which a PDE is posed evolves with time in a
prescribed fashion; free-boundary problems, where the domain is one of
the unknowns in and of itself; and geometric evolution equations,
where the domain is an evolving Riemannian manifold. Such problems are
inherently challenging to solve numerically, owing not only to the
difficulty of discretizing functions defined on evolving geometries,
but also to the coupling, if any, between the geometry’s evolution and
the underlying PDE. Similar difficulties, which are in some sense dual
to those just mentioned, are faced when the goal is to numerically
approximate functions taking values in a manifold. This talk will
focus on tackling these unique challenges that lie at the intersection
of numerical analysis, PDEs, and geometry.

Speaker: Tam Nguyen Phan (Binghamton U.)

Title: Examples of negatively curved and nonpositively curved manifolds

Abstract: Let M be a noncompact, complete, Riemannian manifold. Gromov proved that if the sectional curvature of M negative and bounded, and if the volume of M is finite, then M is homeomorphic to the interior of a compact manifold overline{M} with boundary B. In other words, M has finitely many ends, and each end of M is topologically a product of a closed manifold C with a ray. A natural question is how the geometry (i.e. in terms of the curvature) of M controls the topology of C. The same question is interesting in nonpositive curvature settings. I will discuss what topological restrictions there are on each end and give old and new constructions of such manifolds.

Title: Matrix algebra dimensions

Abstract: What is the dimension of a triply generated commutative matrix algebra? It seems that not much is known, but we’ll discuss some relevant ideas. For example, an old result, often called Gerstenhaber’s Theorem, states that the algebra of polynomials in two commuting nxn matrices has dimension at most n. Here we discuss the possibility of extending this result to algebras generated by three commuting matrices. Related questions concern the reducibility of the variety of commuting triples and the question of “approximate simultaneous diagonalizability”. We present some experimental results based on the Weyr canonical form (an under–appreciated alternative to the JCF).

Speaker: Farbod Shokrieh (Cornell U.)

Title:
Metric graphs, potential theory, and algebraic geometry

Abstract:
A metric graph can be viewed, in many respects, as an analogue of an
algebraic curve. For example, there is a notion of “Jacobian” for
graphs.
More classically, metric graphs can be viewed as electrical networks.
I will discuss the interplay between these two points of view, as well
as some recent applications to problems in algebraic geometry.

The Logic Seminar will meet again this Friday. The speaker will be Bjørn Kjos-Hanssen.

Title:
Superposition as memory: unlocking quantum automatic complexity

Time:
Friday March 17, 2:30-3:20

Place: Keller 404 (Note: this might change)

Abstract:
Imagine a lock with two states, “locked” and “unlocked”, which may be manipulated using two operations, called 0 and 1. Moreover, the only way to (with certainty) unlock using four operations is to do them in the sequence 0011, i.e., $0^n1^n$ where $n=2$. In this scenario one might think that the lock needs to be in certain further states after each operation, so that there is some memory of what has been done so far. Here we show that this memory can be entirely encoded in superpositions of the two basic states “locked” and “unlocked”, where, as dictated by quantum mechanics, the operations are given by unitary matrices. Moreover, we show using the Jordan–Schur lemma that a similar lock is not possible for $n=60$.

Speaker: Ben Hutz (Saint Louis U.)

Title: A bound on the periodic part of the forward orbit of a projective subvariety

Abstract: Let $f:mathbb{P}^N to mathbb{P}^N$ be a morphism and $X subseteq mathbb{P}^N$ a (projective) variety. We obtain a bound on the periodic part of the forward orbit of $X$ by examining the orbit modulo a prime of good reduction. This bound depends only on the degree of the map, the degree of the subvariety, the dimension of the projective space, the degree of the number field, and the prime of good reduction.

This talk will be split into two different cases. First, a review of the case when $dim(X) = 0$, i.e., iteration of points. For the projective line, a series of results due to Li, Morton-Silverman, Narkiewicz, Pezda, and Zieve provide the main theorem. This was generalized by the speaker in 2009 to any $N$. Second, the main part of the talk describes the speaker’s new result for the case $dim(X) > 0$.

The Logic Seminar will meet again this Friday, usual place: Keller Hall 404. The speaker will be Achilles Beros.

Title: Teachers, Learners and Oracles

Abstract: When identifying r.e. sets from enumeration, a teacher is a
computational aide that pre-processes the data and only passes the
“useful” examples to the learner. Access to a teacher does not affect
the learnability of a family of r.e. sets, but it can affect the speed
with which learning is accomplished. Another computational aide is the
membership oracle. We consider four different forms of polynomial
bounds on learning and compare the performance of learners equipped with
teachers and learners equipped with oracles. We find that in most cases
neither strategy is uniformly superior. In this talk I will survey the
results and show a proof that utilizes a strategy analogous to integrity
checks in TCP (Transmission Control Protocol). The paper presented is
joint work with Colin de la Higuera.