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Series: Geometry Topology Seminar

Series: Other Talks

Lunch will be provided. The talk will be the first 25 minutes of the hour and then will be followed by discussion.

In a recent article to appear in the American Mathematical Mothly next year, we use the Lambert series generating function for Euler’s totient function to introduce a new identity for the number of 1’s in the partitions of n. New expansions for Euler’s partition function p(n) are derived in this context. These surprising new results connect the famous classical totient function from multiplicative number theory to the additive theory of partitions. We will define partitions and several variants of Euler's partition function in the talk to state our new results.

Series: Analysis Seminar

Dynamical sampling is the problem of recovering an
unknown function from a set of space-time samples. This problem has many
connections to problems in frame theory, operator theory and functional
analysis. In this talk, we will state the problem and discuss its
relations to various areas of functional analysis and operator theory,
and we will give a brief review of previous results and present several
new ones.

Wednesday, October 11, 2017 - 13:55 ,
Location: Skiles 006 ,
Justin Lanier ,
Georgia Tech ,
Organizer: Jennifer Hom

We will discuss the mapping class groupoid, how it is generated by handle slides, and how factoring in the mapping class groupoid can be used to compute Heegaard Floer homology. This talk is based on work by Lipshitz, Ozsvath, and Thurston.

Friday, October 13, 2017 - 10:00 ,
Location: Skiles 114 ,
Libby Taylor ,
GA Tech ,
Organizer: Timothy Duff

We will give an overview of divisor theory on curves and give
definitions of the Picard group and the Jacobian of a compact Riemann
surface. We will use these notions to prove Plucker’s formula for the
genus of a smooth projective curve. In addition,
we will discuss the various ways of defining the Jacobian of a curve and
why these definitions are equivalent. We will also give an extension
of these notions to schemes, in which we define the Picard group of a
scheme in terms of the group of invertible sheaves
and in terms of sheaf cohomology.

Series: ACO Student Seminar

We show variants of spectral sparsification routines can preserve thetotal spanning tree counts of graphs, which by Kirchhoff's matrix-treetheorem, is equivalent to determinant of a graph Laplacian minor, orequivalently, of any SDDM matrix. Our analyses utilizes thiscombinatorial connection to bridge between statistical leverage scores/ effective resistances and the analysis of random graphs by [Janson,Combinatorics, Probability and Computing `94]. This leads to a routinethat in quadratic time, sparsifies a graph down to about $n^{1.5}$edges in ways that preserve both the determinant and the distributionof spanning trees (provided the sparsified graph is viewed as a randomobject). Extending this algorithm to work with Schur complements andapproximate Choleksy factorizations leads to algorithms for countingand sampling spanning trees which are nearly optimal for dense graphs.We give an algorithm that computes a $(1\pm \delta)$ approximation tothe determinant of any SDDM matrix with constant probability in about$n^2\delta^{−2}$ time. This is the first routine for graphs thatoutperforms general-purpose routines for computing determinants ofarbitrary matrices. We also give an algorithm that generates in about$n^2\delta^{−2}$ time a spanning tree of a weighted undirected graphfrom a distribution with total variation distance of $\delta$ fromthe w-uniform distribution.This is joint work with John Peebles, Richard Peng and Anup B. Rao.

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

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). Along the way several open problems will be discussed.

Series: Combinatorics Seminar

The original notion of poset dimension is due to Dushnik and Miller (1941). Last year, Uerckerdt (2016) proposed a variant, called local dimension, which has garnered considerable interest. A local realizer of a poset P is a collection of partial linear extensions of P that cover the comparabilities and incomparabilities of P. The local dimension of P is the minimum frequency of a local realizer where frequency is the maximum multiplicity of an element of P.
Hiraguchi (1955) proved that any poset with n points has dimension at most n/2, which is sharp. We prove that the local dimension of a poset with n points is O(n/log n). To show that this bound is best possible, we use probabilistic methods to prove the following stronger result which extends a theorem of Chung, Erdős, and Spencer (1983): There is an n-vertex bipartite graph in which each difference graph cover of the edges will cover one of the vertices Θ(n/log n) times. (This is joint work with Jinha Kim, Ryan R. Martin, Tomáš Masařı́k, Warren Shull, Andrew Uzzell, and Zhiyu Wang)

Friday, October 13, 2017 - 15:00 ,
Location: Skiles 154 ,
Bhanu Kumar ,
GT Math ,
Organizer: Jiaqi Yang

Birkhoff's Theorem is a result useful in characterizing the boundary of certain open sets U ⊂ T^1 x [0, inf) which are invariant under "vertical-tilting" homeomorphisms H. We present the method used by A. Fathi to prove Birkhoff's theorem, which develops a series of lemmas using topological arguments to prove that this boundary is a graph.