Seminars and Colloquia by Series

Generating functions for induced characters of the hyperoctahedral group

Series
Algebra Seminar
Time
Monday, March 9, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mark SkanderaLehigh University

Merris and Watkins interpreted results of Littlewood to give generating functions for symmetric group characters induced from one-dimensional characters of Young subgroups.  Beginning with an n by n matrix X of formal variables, one obtains induced sign and trivial characters by expanding sums of products of certain determinants and permanents, respectively. We will look at a new analogous result which holds for hyperoctahedral group characters induced from the four one-dimensional characters of its Young subgroups.  This requires a 2n by 2n matrix of formal variables and four combinations of determinants and permanents.  This is joint work with Jongwon Kim.

Satellite operations and knot genera

Series
Geometry Topology Seminar
Time
Monday, March 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Allison MillerRice University

The satellite construction, which associates to a pattern knot P in a solid torus and a companion knot K in the 3-sphere the so-called satellite knot P(K), features prominently in knot theory and low-dimensional topology.  Besides the intuition that P(K) is “more complicated” than either P or K, one can attempt to quantify how the complexity of a knot changes under the satellite operation. In this talk, I’ll discuss how several notions of complexity based on the minimal genus of an embedded surface change under satelliting. In the case of the classical Seifert genus of a knot, Schubert gives an exact formula. In the 4-dimensional context the situation is more complicated, and depends on whether we work in the smooth or topological category: the smooth category is sometimes asymptotically similar to the classical setting, but our main results show that the topological category is much weirder.  This talk is based on joint work with Peter Feller and Juanita Pinzón-Caicedo.

Wobbling of pedestrian bridges

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 9, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guillermo GoldszteinGeorgia Tech

On June 10, 2000, the Millennium Bridge in London opened to the public. As people crossed the bridge, it wobbled. The sway of the bridge was large enough that prompted many on the bridge to hold on to the rails. Three days later, the bridge closed. It reopened only after modifications to prevent the wobbling were made, eighteen months later. We develop and study a model motivated by this event

Small torsion generating sets for mapping class groups

Series
Dissertation Defense
Time
Monday, March 9, 2020 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Justin LanierGeorgia Tech

A surface of genus $g$ has many symmetries. These form the surface’s mapping class group $Mod(S_g)$, which is finitely generated. The most commonly used generating sets for $Mod(S_g)$ are comprised of infinite order elements called Dehn twists; however, a number of authors have shown that torsion generating sets are also possible. For example, Brendle and Farb showed that $Mod(S_g)$ is generated by six involutions for $g \geq 3$. We will discuss our extension of these results to elements of arbitrary order: for $k > 5$ and $g$ sufficiently large, $Mod(S_g)$ is generated by three elements of order $k$. Generalizing this idea, in joint work with Margalit we showed that for $g \geq 3$ every nontrivial periodic element that is not a hyperelliptic involution normally generates $Mod(S_g)$. This result raises a question: does there exist an $N$, independent of $g$, so that if $f$ is a periodic normal generator of $Mod(S_g)$, then $Mod(S_g)$ is generated by $N$ conjugates of $f$? We show that in general there does not exist such an $N$, but that there does exist such a universal bound for the class of non-involution normal generators.

Graph fractal dimension and structure of fractal networks: a combinatorial perspective

Series
Combinatorics Seminar
Time
Friday, March 6, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pavel SkumsGeorgia State University

We study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product (or Prague or Nešetřil-Rödl) dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colorings and graph Kolmogorov complexity, and analyze the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework to evolutionary studies by revealing the growth of self-organization of heterogeneous viral populations over the course of their intra-host evolution, thus suggesting mechanisms of their gradual adaptation to the host's environment. As far as the authors know, the proposed approach is the first theoretical framework for study of network fractality within the combinatorial paradigm. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

Based on joint work with Leonid Bunimovich

Strong self concordance and sampling

Series
ACO Student Seminar
Time
Friday, March 6, 2020 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker

Motivated by the Dikin walk, we develop aspects of an interior-point

theory for sampling in high dimensions. Specifically, we introduce symmetric

and strong self-concordance. These properties imply that the corresponding

Dikin walk mixes in $\tilde{O}(n\bar{\nu})$ steps from a warm start

in a convex body in $\mathbb{R}^{n}$ using a strongly self-concordant barrier

with symmetric self-concordance parameter $\bar{\nu}$. For many natural

barriers, $\bar{\nu}$ is roughly bounded by $\nu$, the standard

self-concordance parameter. We show that this property and strong

self-concordance hold for the Lee-Sidford barrier. As a consequence,

we obtain the first walk to mix in $\tilde{O}(n^{2})$ steps for an

arbitrary polytope in $\mathbb{R}^{n}$. Strong self-concordance for other

barriers leads to an interesting (and unexpected) connection ---

for the universal and entropic barriers, it is implied by the KLS

conjecture.

Resonant tori of arbitrary codimension for quasi-periodically forced systems

Series
Math Physics Seminar
Time
Thursday, March 5, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guido GentileUniversita&#039; di Roma 3

Consider a system of rotators subject to a small quasi-periodic forcing which (1) is analytic, (2) satisfies a time-reversibility property, and (3) has a Bryuno frequency vector. Without imposing any non-degeneracy condition, we prove that there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases. This is a joint work with Livia Corsi.

Chromatic number of graphs with no indeuced 3-matching

Series
Graph Theory Working Seminar
Time
Thursday, March 5, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Joshua SchroederGeorgia Tech
In 1980, Wagon showed that $\chi(G) \leq f_m(n)$ for any $\{mK_2, K_n\}$-free graph $G$, where $f_m$ is a polynomial and $mK_2$ is an induced matching of size $m$. However this bound is not known to be sharp. Recently, Gaspers and Huang helped sharpen this bound by showing for any $\{2K_2, K_4\}$-free graph $G$, that $\chi(G) \leq 4$. This resolves the question raised by Wagon for $m=2$, $n=4$. For the case where $m = 3$, it was shown by Brandt in 2002 that $(K_3, 3K_2)$-free graphs are 4-colorable.  In this talk, I will provide the outline for an alternate proof of this fact, as a byproduct of my research project.

Martingales and descents

Series
Stochastics Seminar
Time
Thursday, March 5, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alperen OzdemirUniversity of Southern California

We provide a martingale proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order n^{-1/2}. The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics include inversions, descents in signed permutations, descents in Stirling permutations, the length of the longest alternating subsequences, descents in matchings and two-sided Eulerian numbers.