# Research

## Discrete Mathematics

### Thursday, December 5, 2013

Title: The Processes Inspired by the Functioning of Living Cells: Natural Computing Approach, Part III
Speaker: Grzegorz Rozenberg, Leiden University
The Netherlands
Time: 2:00pm‐3:00pm
Place: CMC 108

### Tuesday, December 3, 2013

Title: The Processes Inspired by the Functioning of Living Cells: Natural Computing Approach, Part II
Speaker: Grzegorz Rozenberg, Leiden University
The Netherlands
Time: 2:00pm‐3:00pm
Place: CMC 108

### Monday, December 2, 2013

Title: The Processes Inspired by the Functioning of Living Cells: Natural Computing Approach, Part I
Speaker: Grzegorz Rozenberg, Leiden University
The Netherlands
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Natural Computing is an interdisciplinary research field that investigates human-designed computing inspired by nature as well as computation taking place in nature, i.e., it investigates models, computational techniques, and computational technologies inspired by nature as well as it investigates phenomena/processes taking place in nature in terms of information processing.

One of the research areas from the second strand of research is a computational understanding of the functioning of the living cell. We view this functioning in terms of formal processes resulting from interactions between (a huge number of) individual reactions. These interactions are driven by two mechanisms, facilitation and inhibition: reactions may (through their products) facilitate or inhibit each other.

We present a formal framework for the investigation of these interactions. We motivate this framework by explicitly stating a number of assumptions that hold for processes resulting from these interactions, and we point out that these assumptions are very different from the ones underlying traditional models of computation. We discuss some basic properties of these processes, and demonstrate how to capture and analyse, in our formal framework, some notions related to cell biology and biochemistry.

Research topics in this framework are motivated by biological considerations as well as by the need to understand the underlying computations. The models we discuss turned out to be novel and attractive from the theory of computation point-of-view.

The lectures are of interest to mathematicians and computer scientists interested in formal models of computation as well as to bioinformaticians, biochemists, and biologists interested in foundational/formal understanding of biological processes. They are of a tutorial style and self-contained. In particular, no prior knowledge of biochemistry or cell biology is required.

The presented framework was developed jointly with A. Ehrenfeucht from University of Colorado at Boulder.

### Monday, November 25, 2013

Title: A Survey of a Periodic Placement Lemma
Speaker: Joy D'Andrea
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

The program Systre is designed to analyze periodic nets as they arise in the study of (extended or non-molecular) crystal structures. The acronym stands for “Symmetry, Structure (Recognition) and Refinement”. Systre uses a method called barycentric placement to determine the ideal (i.e., maximal embeddable) symmetry of a crystal net and to analyze its topological structure. In the course of designing systre, Olaf Delgado-Friedrichs wrote many papers explaining, examining, and describing the mathematics behind the program. In this talk, we will concentrate on one of the lemmas that was from one of three papers involving the design of systre. We will go through some background material, give a few brief visual examples, state the lemma, prove the lemma, and then close with questions from the audience.

### Monday, November 18, 2013

Title: Thompson's group $$F$$
Speaker: Suzana Milea
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

The presentation is from an algebraic perspective and is mainly based on the article Cannon, J. W.; Floyd, W. J.; Parry, W. R. (1996), “Introductory notes on Richard Thompson's groups”, L' Enseignement Mathématique. Revue Internationale. IIe Série 42 (3): 215–256. First, I will introduce the elements of the group $$F$$ as piecewise linear homeomorphisms from $$[0,1]$$ to itself and describe them using tree diagrams. In the rest of the talk I will go through the known properties of the group and prove a few basic results regarding its algebraic structure.

### Monday, November 11, 2013

Veteran's Day Holiday

### Monday, November 4, 2013

Title: An Introduction in Fuzzy Topology
Speaker: Jeremy Kerr
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

In this talk, I will discuss the notion of a fuzzy set and its roots in fuzzy logic. Then, I will go into the mathematics of fuzzy sets, in which I will be coving various topological properties in a fuzzy sense. Including the definition of a fuzzy set and topology, fuzzy continuous, fuzzy compactness, and the idea of a fuzzy Tychonoff theorem. This talk will by adapted from the publication “The Quest for a Fuzzy Tychonoff Theorem” by Stephan Carlson.

### Monday, October 28, 2013

Title: Hilbert's Third Problem
Speaker: Brian Tuesink
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

A brief discussion of the history of Hilbert's third problem. Definition, of the Dehn invariant, Lemma proving it is additive and corollary that if two polyhedrons are equdissectible then their Dehn invariants must be equal. followed by a proof that a cube and a tetra-hedron are not (the original proof by M. Dehn further simplified by Hadwiger, and Boltyanski).

### Monday, October 21, 2013

Title: Tile Self-Assembly Models: aTAM, $$2$$HAM, and Signals
Speaker: Daria Karpenko
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

The abstract tile assembly model (aTAM) was introduced by Erik Winfree in 1998 to describe the self-assembly of structures by sets of DNA-based tiles each of which behaves as a square with different “glues” on each side. Specifically, the aTAM models self-assembly that proceeds by single tile attachment of tiles from a specified set to a given “seed” tile. The two-handed assembly model ($$2$$HAM) is a variation of the aTAM where the seed requirement is removed and any two multi-tile structures are allowed to bind together as long as there is a sufficient number of matching glues between them. Motivated by recent advances in DNA nanotech, both the aTAM and the $$2$$HAM can be extended to allow glues on a particular tile to begin in an inactive state and later be activated by signals sent by adjacent tiles, enforcing a particular sequence of binding events. We'll look at what these models are known to be able to do, known to not be able to do, and not known to be able to do (that is, at their power and limitations, as well as open questions).

### Monday, October 14, 2013

Title: Point-free $$P_k$$-spaces
Speaker: Joseph Van Name
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

If $$k$$ is an infinite cardinal, then a $$P_k$$-space is a completely regular space where the intersection of less than $$k$$ many open sets is open, and a $$P$$-space is a completely regular space where the intersection of countably many open sets is open. In this talk, we shall first give an introduction to some notions from point-free topology. We shall then look at various ways that the notion of a $$P$$-space and a $$P_k$$-space can be generalized to point-free topology.

### Monday, October 7, 2013

Title: Cohomology of Groups and Crystallography
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

We will review the notion of a Crystal that is — “a triple” — (point group $$H$$, lattice $$M$$, action of $$H$$ on $$M$$), consider an equivalence relation on the set of Crystals (arithmetic equivalence), and introduce the necessary ingredient from Group Cohomology to state the main result (of Howard Hiller) relating arithmetic crystal classes and cohomology of groups.

### Monday, September 30, 2013

Title: Polynomial Term Sieving and Quadratic Diophantine Equations
Speaker: Jonathan Burns
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

We identify a recursive structure among the integer factorizations of polynomial terms into two factors. For quadratics, this recursive structure is shown to be equivalent to a non-recursive Diophantine identity. In the special cases where the function is one of the prime-producing polynomials of the form $$n^2-n+\varepsilon$$ or $$n^2+\lambda$$ this structure gives a complete characterization of the integer factorization for the terms into two factors. Finally we transform the quadratic identity into geometric terms, showing that each integer factorization of the structure corresponds to a lattice point of a conic. This final context leads to a new identity and some solutions to the Pell equation.

### Monday, September 23, 2013

Title: Self-similar groups acting essentially freely on the boundary of a rooted tree
Speaker: Dima Savchuk
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

In this talk I will discuss self-similar groups that act essentially freely on the boundary of rooted trees. I will start from a motivation and explain how these groups could be used to construct new interesting counterexamples. The main result of our study is a complete classification of groups generated by 3-state automata over 2-letter alphabet that act essentially freely on the boundary of the binary tree. Among all these groups I will concentrate on one most interesting new example that has not been studied before. It is an extension of the rank 2 lamplighter group by the group of order 2. This group rather surprisingly has a subgroup of infinite index, whose closure has index 2 in the closure of the whole group. This is a joint work with Rostislav Grigorchuk.

### Monday, September 16, 2013

Title: Crystallography and Cohomology of Groups, Part II
Speaker: Greg McColm
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

We will continue the presentation of Howard Hiller's “Crystallography and Cohomology of Groups”, in the American Mathematical Monthly, 93:10 (1986), pp. 765-779. This paper is available via the USF Library via its JSTOR subscription to the Monthly. This presentation will be on the Bieberbach theorems.

### Monday, September 9, 2013

Title: Crystallography and Cohomology of Groups
Speaker: Milé Krajčevski
Time: 3:05pm‐3:55pm
Place: CMC 120

#### Abstract

We will begin a presentation of Howard Hiller's “Crystallography and Cohomology of Groups”, in the American Mathematical Monthly, 93:10 (1986), pp. 765–779. This paper is available via the USF Library via its JSTOR subscription to the Monthly.