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Mathematics & Statistics

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**Title**

**Speaker**

**Time**

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Simulating Substitution Tiling via Self-Assembly

Daniel Cruz

2:00pm-2:50pm

CMC 108

**Abstract**

Chaim Goodman-Strauss showed that every known substitution tiling of the Euclidean plane can be enforced with a finite set of matching rules. We study which substitution tilings can be simulated via self-assembly. We introduce a notion of hierarchical assembly within an existent tile assembly model to describe substitution tilings and generalize to non-square tiles. We develop a method for simulating substitution tilings via hierarchical assembly and give sufficient conditions for such simulation.

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Ehrenfeucht-Fraïssé Games: Examples and Applications

Daviel Leyva

2:00pm-2:50pm

CMC 108

**Abstract**

In this talk we shall use Ehrenfeucht-Fraïssé Games to examine through examples to what extent two structures are indistinguishable from one another in first-order logic, as well as show Connectivity is not expressible in first-order logic. We shall also connect Ehrenfeucht-Fraïssé Games with the notion of satisfaction via Hintikka sentences.

**Title**

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Group of intermediate growth, aperiodic order, and Schröedinger operators

Rostislav Grigorchuk

Texas A&M University

2:00pm-2:50pm

CMC 108

**Abstract**

I will explain how seemingly unrelated objects: the group \(G\) of intermediate growth constructed by the speaker in 1980, the aperioidc order, and the theory of (random) Schröedinger operators can meet together. The main result, to be discussed, is based on a joint work with D. Lenz and T. Nagnibeda. It shows that a random Markov operator on a family of Schreier graphs of \(G\) associated with the action on the boundary of a binary rooted tree has a Cantor spectrum of the Lebesgue measure zero. This will be used to gain some information about the spectrum of the Cayley graph. The main tool of investigation is given by a substitution, that, on the one hand, gives a presentation of \(G\) in terms of generators and relations, and, on the other hand, defines a minimal substitutional dynamical system which leads to the use of the theory of random Shröedinger operators.

No special knowledge is assumed, and the talk is supposed to be easily accessible for the audience.

**Note:** This is a joint Discrete Mathematics Seminar/Colloquium talk.

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Closed groups of rooted tree automorphisms, symbolic dynamics, and Rabin automata

Zoran Šunić

Hofstra University

2:00pm-2:50pm

CMC 108

**Abstract**

The set of automorphisms of a rooted tree can be considered in several settings: group theoretic, topological, or symbolic dynamics. We consider topologically closed, self-similar groups of tree automorphisms (thus, sets of tree automorphisms closed in each of the three settings). Prompted by a question by Nataša Jonoska, we provide a weak analog of a theorem of Kitchens on one-sided group shifts, which leads to interesting conclusions about the tree languages representing the closures of some well known self-similar groups. For instance, while the closures of many “unusual” self-similar groups (Grigorchuk group, Hanoi Towers group, ...) are finitely constrained groups (tree shifts of finite type) and, therefore, can be recognized by rather simple Rabin automata, the closure of the infinite cyclic group \(\mathbb{Z}\) acting as odometer on the tree is not even Rabin recognizable (the same conclusion is valid for many other level transitive, self-similar groups that satisfy an algebraic law). Joint work with Andrew Penland.

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Self-assembly of shapes by cotranscriptional folding

Shinnosuke Seki

University of Electro-Communications

Tokyo, Japan

2:00pm-2:50pm

CMC 108

**Abstract**

Transcription is a process in which an RNA sequence (of A, C, G, U) is synthesized out of its template DNA sequence (of A, C, G, T). The molecular Xerox called RNA polymerase attaches to the template DNA sequence, scans it through, and produces its RNA copy letter by letter according to the rule A \(\to\) U, C \(\to\) G, G \(\to\) C, and T \(\to\) A (e.g., from the DNA sequence AACGT, the polymerase produces the RNA sequence UUGCA). Much earlier than being fully transcribed, the elongating RNA sequence (transcript) starts folding upon itself into a complex conformation by forming hydrogen bonds in order to get stabilized. This phenomenon is called cotranscriptional folding. In nature, various information processing is done by cotranscriptional folding, in particular, of shapes. For instance, fluoride riboswitches in *Bacillus cereus* bacteria cotranscriptionally fold into a terminator stem hairpin or not, in order to regulate gene expression. In 2014, Geary, Rothemund, and Andersen proposed an architecture of a DNA sequence whose transcript highly probably folds into a unique rectangle cotranscriptionally.

Using a theoretical model called oritatami system, the study of self-assembly of shapes by cotranscriptional folding was initiated recently. In this talk, I will first formalize the problem of folding a shape cotranscriptionally by an oritatami system. I will then lead audiences through the design process of an OS that cotranscriptionally folds into an arbitrary finite portion of Heighway dragon fractal as well as its verification. Related results and open problems will be also discussed.

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Attack-Resilient Monitoring of Cloud Services in Cyber-Physical Infrastructures

Kaiqi Xiong

2:00pm-2:50pm

CMC 108

**Abstract**

Attack-resilient scalable monitoring of cloud services becomes critical in the steady-state operation of a cyber-physical infrastructure. In this talk, I will first introduce cloud services and then present the cloud status problem for the monitoring of cloud services. I will further discuss a series of my attack-resilient algorithms for solving the cloud status problem, give some properties of these algorithms through theoretical analysis, and evaluate these algorithms through experiments. Moreover, the control center of a cloud is highly vulnerable to Distributed Denial of Service (DDoS) attacks. I will briefly discuss my Software Defined Networking (SDN)-based approaches for DDoS attack mitigation and detection.

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Developing a Model to Predict Prevalence of Compulsive Behavior in Individuals with OCD

Lindsay Fields

2:00pm-2:50pm

CMC 108

**Abstract**

Obsessive-Compulsive Disorder (OCD) is a mental disorder which affects approximately 2.3% of people worldwide at some point in their lives. Although it is commonly accepted by mental health experts that obsessions and compulsions are self-perpetuating, there is little research to support this theory. As such, this research proposes that continued ritualistic behavior, as opposed to calming the OCD sufferer, actually serves to amplify anxiety, which in turn incites further rituals.

We have designed a model which simulates OCD behavior consistent with this theory for individuals with moderate to severe OCD. The model predicts, given a set of environmental parameters, the probability of an individual with OCD performing compulsive behavior and the prevalence of such behavior. By applying a simple signaling game to the neural system known as the worry circuit, a program was composed and simulations run which suggest that the likelihood of compulsive behavior can be predicted using a function of the number of compulsions performed previously. These results may be considered preliminary, given the sample size of the case study and the primitive nature of the model.

Spring Break — no seminar this week.

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A singular trip through the world of algebraic curves over finite fields

Annamaria Iezzi

2:00pm-2:50pm

CMC 108

**Abstract**

For an algebraic curve defined over a finite field Fq, the number of rational points, i.e., of points of the curve with coordinates in Fq, is always finite. Due to this discrete aspect, algebraic curves over finite fields have found, in the last 40 years, several applications in cryptography and coding theory.

In this talk, we will explore this problem of counting points, by showing the fundamental tools and results that appear in this theory. In particular, we will see how the case of smooth curves generalizes to the case of singular ones and we will present a construction of singular curves with many rational points.

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Design strategies for DNA tile assembly

Margherita Maria Ferrari

2:00pm-2:50pm

CMC 108

**Abstract**

New laboratory techniques have been developed using the Watson-Crick complementarity properties of DNA strands to achieve the self-assembly of nanostructures, particularly polyhedra. For all of these methods, an essential step is designing the component molecular building blocks. We present a mathematical model that captures the geometric constraints of rigid tiles, that are star-shaped molecules used as building blocks for tile-based DNA self-assembly. We illustrate the functionality of the model by providing DNA self-assembly strategies to effciently construct Platonic and Archimedean 3-regular polyhedral skeletons.

No seminar this week.

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Platform Color Designs for Interactive Molecular Arrangements

Jasper Braun

2:00pm-2:50pm

CMC 108

**Abstract**

It has been shown that alternating attachments of two types (species) of floating molecular (DNA based) tiles on a predesigned array that consists of communicating neighboring DNA tiles complementary to the floating tiles can dynamically simulate some types of cellular automata (CA). We address the question of which design of the platform array provides communication across the whole plane. We show that for square tiles only the checkerboard arrangement of the two species can provide communication between any two tiles of the plane. On the other hand, there are an uncountable number of arrangements of two colors of hexagonal tiles on the plane which provide communication between any two tiles.

No seminar this week.

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Too Many Periodic Graphs

Greg McColm

2:00pm-2:50pm

CMC 108

**Abstract**

A *periodic graph* is a graph embedded in Euclidean space, whose vertex set is discrete, and whose symmetry group has a free abelian subgroup of translations spanning the space. Our project is to catalog — or otherwise make accessible — all periodic graphs. We classify periodic graphs using syntactic graphs of linear forms, and for each graph of forms, we obtain a parametrization of the Euclidean graphs that are *realizations* of that graph of forms. The two primary cataloguing problems are: (1) even restricting our attention to very simple periodic graphs — say, edge transitive periodic graphs — there are a lot of graphs of linear forms, and worse, (2) some graphs of linear forms have infinitely many isomorphism classes of realizations. We describe one approach for finessing the latter nuisance.

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On Logic Definition and Comparison Games and First-Order Logic

Daviel Leyva

2:00pm-2:50pm

CMC 108

**Abstract**

During the Twentieth century, Leon Henkin, Jaakko Hintikka, and others connected game theory and logic. In this talk, we shall introduce the basic ideas connecting these two branches of mathematics as well as a method to obtain semantics from syntax via games developed by Hintikka. We shall also see how we can tell if two structures are “indistinguishable” (in a given logic) from one another as prescribed by Roland Fraïssé and Andrzej Ehrenfeucht.