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# Discrete Mathematics (Leader: Prof. Greg McColm <mccolm (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'mccolm' + '&#64;' + 'usf.edu' + '">Prof. Greg McColm</a>');)

Title
Speaker
Time
Place

TBA
Daviel Leyva
2:00pm-2:50pm
CMC 108

Abstract

TBA

Title
Speaker
Time
Place

TBA
TBA
2:00pm-2:50pm
CMC 108

Abstract

TBA

## November 12, 2018

Veteran's Day Holiday -- no seminar this week.

Title
Speaker
Time
Place

TBA
TBA
2:00pm-2:50pm
CMC 108

Abstract

TBA

Title
Speaker
Time
Place

TBA
TBA
2:00pm-2:50pm
CMC 108

Abstract

TBA

Title
Speaker
Time
Place

TBA
TBA
2:00pm-2:50pm
CMC 108

Abstract

TBA

## Monday, October 15, 2018

Title
Speaker
Time
Place

Spaces of Geometric Graphs
Greg McColm
2:00pm-2:50pm
CMC 108

Abstract

A geometric graph is a graph embedded in some (nice) geometric space; usually a Euclidean space. A geometric graph may be generated from a putative quotient graph with some matrix groups assigned to the quotient graph's vertices and vectors (and perhaps matrices) assigned to the edges. If we treat the matrices as constants and the vectors as variables, we obtain an ensemble of geometric graphs parametrized by a vector space of input values.

## Monday, October 8, 2018

Title
Speaker
Time
Place

Covering and Navigating Geometric Graphs
Greg McColm
2:00pm-2:50pm
CMC 108

Abstract

A geometric graph is a graph embedded in some (nice) geometric space; usually a Euclidean space. Given a group acting on that Euclidean space, the symmetry group $$S$$ of a graph $$\Gamma$$ embedded in that space is the group of automorphisms of $$\Gamma$$ induced by the group acting on the underlying space. This symmetry group does not necessarily act freely on $$\Gamma$$, and yet we would like to “lift&rdqup; $$\Gamma$$ from $$\Gamma/S$$. We describe a method for doing so, and en route, we obtain a method for parametrizing classes of periodic (and other very regular) geometric graphs.

## Monday, October 1, 2018

Title
Speaker
Time
Place

Graph Covers, Quotients, Lifts, and Immersion
Greg McColm
2:00pm-2:50pm
CMC 108

Abstract

Given a graph $$\Delta$$, a covering graph of $$\Delta$$ is a graph $$\Gamma$$ such that there is a homomorphism $$\phi$$ from $$\Gamma$$ to $$\Delta$$ that is bijective from neighborhood to neighborhood. Connected covering graphs of connected graphs can be determined up to isomorphism by “lifting” walks on the covered walk up to vertices of the covering graph. We adapt this theory to the problem of generating a graph from its quotient graph and generators of its group.

## Monday, September 24, 2018

Title
Speaker
Time
Place

Greg McColm
2:00pm-2:50pm
CMC 108

Abstract

Metric spaces of finite and infinite geometric graphs of high symmetry can be described using linear forms. Given an abstract graph composed of forms, we obtain an ensemble of homomorphic images of that graph embedded in Euclidean space. We look at some fundamentals concerning these abstract graphs and the spaces of graphs that they characterize.

## Monda, September 17, 2018

Title
Speaker

Time
Place

Gromov-Monge Quasimetrics and Distance Distributions
Tom Needham
Ohio State University
2:00pm-2:50pm
CMC 108

Abstract

In applications in computer graphics and computational anatomy, one seeks measure-preserving maps between shapes which preserve geometry as much as possible. Inspired by this, we define a distance between arbitrary compact metric measure spaces by blending the Monge formulation of optimal transport with the Gromov-Hausdorff construction. We show that the resulting distance is an extended quasi-metric on the space of compact mm-spaces, which has convenient lower bounds defined in terms of distance distributions. We provide rigorous results on the effectiveness of these lower bounds when restricted to simple classes of $$mm$$-spaces such as metric graphs or plane curves. This is joint work with Facundo Mémoli.

## Monday, September 10, 2018

Topic
Speaker

Time
Place

Quasigroups and their application in several applied areas
Danilo Gligoroski
Norwegian University of Science and Technology
Trondheim, Norway
2:00pm-2:50pm
CMC 108

Abstract

In the talk, I will briefly define the algebraic structure quasigroup and quasigroup string transformations and their related combinatorial structures Latin Squares. Then I will show several applications of quasigroups in Cryptography, Coding Theory, Computer Science and 5G. In cryptography quasigroups have been used in primitives such as block ciphers, stream ciphers, hash functions and authenticated ciphers. Popular modes of operations of block ciphers such as CBC, OFB and CTR are actually quasigroup string transformations. I will show how can we use Latin rectangles to define balanced matrices that can be used in coding theory to define efficient erasure codes. Orthogonal matrices over finite fields are very useful mathematical objects in different areas of computer science since by their use we do not need to spend expensive time to compute their inverses, nor to spend space to store their inverses. I will show how can we use Latin rectangles to efficiently define orthogonal matrices over finite fields. Finally I will show how can we use Latin squares and partial Latin squares for one emerging scientific area: Network slicing in the upcoming 5G networks.