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Series: Analysis Seminar

A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.

Series: Algebra Seminar

We determine the average size of the $\phi$-Selmer group in any quadratic twist family of abelian varieties having an isogeny $\phi$ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over $\mathbb{Q}$, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that $E/F$ is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if $F$ is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have $3$-Selmer rank 1. We also obtain consequences for Tate-Shafarevich groups of quadratic twists of a given elliptic curve. This is joint work with Manjul Bhargava, Zev Klagsbrun, and Ari Shnidman.

Series: Geometry Topology Seminar

The immersed Seifert genus of a knot $K$ in $S^3$ can be defined as the minimal genus of an orientable immersed surface $F$ with $\partial F = K$. By a result of Gabai, this value is always equal to the (embedded) Seifert genus of $K$. In this talk I will discuss the embedded and immersed cross-cap numbers of a knot, which are the non-orientable versions of these invariants. Unlike their orientable counterparts these values do not always coincide, and can in fact differ by an arbitrarily large amount. In further contrast to the orientable case, there are families of knots with arbitrarily high embedded 4-ball cross-cap numbers, but which are easily seen to have immersed cross-cap number 1. After describing these examples I will discuss a classification of knots with immersed cross-cap number 1. This is joint work with Seungwon Kim.

Series: CDSNS Colloquium

If h is a homeomorphism on a compact manifold which is chain-recurrent, we will try to understand when the lift of h to an abelian cover is also chain-recurrent. This has consequences on closed geodesics in manifold of negative curvature.

Series: GT-MAP Seminars

Data assimilation is a powerful tool for combining mathematical models
with real-world data to make better predictions and estimate the state
and/or parameters of dynamical systems. In this talk I will give an
overview of some work on models for predicting urban crime patterns,
ranging from stochastic models to differential equations. I will then
present some work on data assimilation techniques that have been
developed and applied for this problem, so that these models can be
joined with real data for purposes of model fitting and crime
forecasting.

Friday, October 20, 2017 - 13:55 ,
Location: Skiles 006 ,
John Etnyre ,
Georgia Tech ,
Organizer: John Etnyre

Note this talk is only 1 hour (to allow for the GT MAP seminar at 3.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we will continue studying branched covers of surfaces. Among other things we should be able to see how to use branched covers to see some relations in the mapping class group of surfaces.

Series: Combinatorics Seminar

We prove that every triangle-free graph with maximum degree $D$ has list chromatic number at most $(1+o(1))\frac{D}{\ln D}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that for any $r \geq 4$ every $K_r$-free graph has list-chromatic number at most $200r\frac{D\ln\ln D}{\ln D}$.

Friday, October 20, 2017 - 10:00 ,
Location: Skiles 114 ,
Kisun Lee ,
Georgia Institute of Technology ,
Organizer: Timothy Duff

We will introduce a class of
nonnegative real matrices which are called slack matrices. Slack
matrices provide the distance from equality of a vertex and a facet. We
go over concepts of polytopes and polyhedrons briefly,
and define slack matrices using those objects. Also, we will
give several necessary and sufficient conditions for slack matrices of
polyhedrons. We will also restrict our conditions for slack matrices for
polytopes. Finally, we introduce the polyhedral verification
problem, and some combinatorial characterizations of slack matrices.

Series: Stochastics Seminar

We present a unified framework for sequential low-rank matrix completion and estimation, address the joint goals of uncertainty quantification (UQ) and statistical design. The first goal of UQ aims to provide a measure of uncertainty of estimated entries in the unknown low-rank matrix X, while the second goal of statistical design provides an informed sampling or measurement scheme for observing the entries in X. For UQ, we adopt a Bayesian approach and assume a singular matrix-variate Gaussian prior the low-rank matrix X which enjoys conjugacy. For design, we explore deterministic design from information-theoretic coding theory. The effectiveness of our proposed methodology is then illustrated on applications to collaborative filtering.

Wednesday, October 18, 2017 - 13:55 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Jennifer Hom

I will talk about the Berge conjecture, and Josh Greene's resolution of a related problem, about which lens spaces can be obtained by integer surgery on a knot in S^3.