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Series: Math Physics Seminar

During the last few years there has been a systematic pursuit for sharp estimates of the energy components of atomic systems in terms of their single particle density. The common feature of these estimates is that they include corrections that depend on the gradient of the density. In this talk I will review these results. The most recent result is the sharp estimate of P.T. Nam on the kinetic energy. Towards the end of my talk I will present some recent results concerning geometric estimates for generalized Poincaré inequalities obtained in collaboration with C. Vallejos and H. Van Den Bosch. These geometric estimates are a useful tool to estimate the numerical value of the constant of Nam's gradient correction term.

Friday, October 27, 2017 - 15:00 ,
Location: Skiles 154 ,
Hassan Attarchi ,
Georgia Tech ,
Organizer:

This presentation is about the results of a paper by Y. Sinai in
1970. Here, I will talk about dynamical systems which resulting from the
motion of a material point in domains with strictly convex boundary,
that is, such that the operator of the second quadratic form is
negative-definite at each point of the boundary, where the boundary is
taken to be equipped with the field of inward normals. It was proved
that such systems are ergodic and are K-systems. The basic method of
investigation is the construction of transversal foliations for such
systems and the study of their properties.

Series: Combinatorics Seminar

Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed distribution. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk, generated by permutations with a binomially distributed number of two-cycles. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group.
Includes joint work with Nayantara Bhatnagar, Evita Nestoridi, and Igor Pak.

Friday, October 27, 2017 - 13:00 ,
Location: Skiles 006 ,
John Etnyre ,
Georgia Tech ,
Organizer: John Etnyre

Notice the seminar is back to 1.5 hours this week.

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 should be able to finish our discussion of branched covers of surfaces and transition to 3-manifolds.

Friday, October 27, 2017 - 10:00 ,
Location: Skiles 114 ,
Jaewoo Jung ,
GA Tech ,
Organizer: Timothy Duff

For any undirected graph, the Stanley-Reisner ideal is generated by monomials correspoding to the graph's "non-edges." It is of interest in algebraic geometry to study the free resolutions and Betti-tables
of these ideals (viewed as modules in the natural way.) We consider the relationship
between a graph and its induced Betti-table. As a first step, we look at how operations on graphs effect on the Betti-tables. In this
talk, I will provide a basic introduction, state our result about clique sums of graphs (with proof), and discuss the next things to do.

Series: Stochastics Seminar

When considering smooth functionals of dependent data, block bootstrap methods have enjoyed considerable success in theory and application. For nonsmooth functionals of dependent data, such as sample quantiles, the theory is less well-developed. In this talk, I will present a general theory of consistency and optimality, in terms of achieving the fastest convergence rate, for block bootstrap distribution estimation for sample quantiles under mild strong mixing assumptions. The case of density estimation will also be discussed. In contrast to existing results, we study the block bootstrap for varying numbers of blocks. This corresponds to a hybrid between the subsampling bootstrap and the moving block bootstrap (MBB). Examples of `time series’ models illustrate the benefits of optimally choosing the number of blocks. This is joint work with Stephen M.S. Lee (University of Hong Kong) and Alastair Young (Imperial College London).

Series: Geometry Topology Seminar

In a joint work with M. Yampolsky, we gave a classification of Thurston maps with parabolic orbifolds based on our previous results on characterization of canonical Thurston obstructions. The obtained results yield a solution to the problem of algorithmically checking combinatorial equivalence of two Thurston maps.

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.