Abstract: In cryptography, a key exchange system is a protocol that allows users to agree on a common key to encrypt and decrypt their messages. This poster will be about key exchange systems designed to work over non-commutative structures. Such schemes particularly over braid groups have been proposed earlier. We will present these schemes as well as a new scheme that seems to work over matrices.

Abstract:The study of constraint satisfaction problems definable in various fragments of Datalog has recently gained considerable importance. We consider constraint satisfaction problems that are definable in the smallest natural recursive fragment of Datalog - monadic linear Datalog with at most one EDB per rule. We give combinatorial and algebraic characterisations of such problems, in terms of caterpillar dualities and lattice operations, respectively. We then apply our results to study graph H-colouring problems.

Abstract: Laver proved that there exists a division form for the terms of the free left distributive algebra on one generator, A. From the existence of this division form, we prove structure results on left divisors; it is hoped that these results will be helpful in answering the question of whether the set of left divisors of q \in A is well-ordered for every q. Though the existence of an analogous division form for the terms of A_\kappa, the free left distributive algebra on \kappa generators, remains open, we prove similar structure theorems for A_\kappa under the hypothesis that such a division form does exist.

Abstract: A finite group is cyclically generated if it has an automorphism that cycles through a generating set for the group. We present examples of cyclically generated groups and prove that every finite abelian group is cyclically generated.
The question of whether a finite abelian group has a minimal cyclic generating set is also addressed. As every symmetric generating set is a cyclic generating set, we can use results about symmetric generating sets for abelian groups to find examples of abelian groups that have cyclic generating sets of minimum size. We exploit connections between automorphisms, matrices, and roots of certain polynomials over rings of prime power order to construct suitable automorphisms or prove that no such automorphism exists. Necessary and sufficient conditions are given for finite abelian groups of rank 3 and 4 to have minimal cyclic generating sets. Partial results are given for abelian groups of rank 5 and higher prime and square free rank.

Abstract:In this poster we present the properties of the dynamic plane of functions in the family S_\lambda which are the meromorphic functions with two asymptotic values where one of the asymptotic values is also a pole. We show that each component of the Fatou set of S_\lambda is simply connected, and that there is at most one completely invariant domain of the Fatou set. We also prove that these results can be generalized to functions with finitely many singular values with certain restrictions. We also study the parametric representation of the family. We study the relationships between S_\lambda and the tangent family, and between S_\lambda and the exponential family.

Abstract: We can find a harmonic function on a Riemann surface with punctures by assigning boundary conditions. The harmonic function determines a graph on the surface. This graph contains combinatorial information, as well as extra structure, sufficient to reconstruct the surface. We examine properties of graphs arising in this way. A description of the surface-graph relationship will be presented with examples, as will consequences for cell decompositions of moduli spaces of such surfaces. This description of moduli spaces gives connections with string topology operations.

Abstract: Marcia Groszek, Rebecca Weber and Pete Winker defined D-reduction procedures to be Turing reduction procedures that are total on all c.e. oracles. This notion of reducibility is clearly implied by truth-table reducibility. They showed that this is not a transitive notion of reducibility and that it is not implied by wtt-reducibility and does not imply bs-reducibility. I will give necessary and "almost" sufficient conditions that make this reducibility transitive and show that we can separate the complete sets with respect to this reducibility from complete sets with respect to other notions of reducibility. The results in this poster are from my thesis.

Abstract: The degree spectrum of an additional relation on a computable structure is the collection of Turing degrees of that relation in computable copies of that structure. We examine the degree spectra of the successor relation in certain classes of computable linear orderings and show that it is closed upward in the c.e. Turing degrees.

Abstract: In mathematics, one often tries to classify some collection of objects, up to isomorphism. In mathematical logic, we can compare the complexity of the classification problem for classes of countable structures. There are two notions of "effective embedding" that each give rise to a partial ordering on classes of structures, and these partial orderings allow us to order the complexity of the classification problem for those classes. I will compare the orderings that arise from these two notions of effective embedding, and describe some general results that characterize when a class will embed into a given class of structures.

Abstract: Epistemic logics are modal systems modeling the acquisition knowledge amongst agents in a either a static or evolving environment. The natural addition of a temporal component allows for a much richer investigation. Here we contrast one traditional account of TEL, with relative-time modal operators for "Tomorrow" and "Yesterday", with one in which knowledge modalities are indexed by time-stamps (John McCarthy) where "K_i^t P" is read "agent i knows P at time t." The latter embeds into the former in many situations. When augmented by the justified common knowledge modality J (Artemov), the time-stamped systems provide a rather straightforward format for interpreting classic epistemic puzzles such as Wise Men (Women, in our case). The embedding naturally yields a traditional TEL solution which seems to be missing from the literature.

Abstract: Reverse mathematics and computable model theory both attempt to measure the strength of classical mathematical principles, the former from a proof-theoretic perspective and the latter from an effective one. Let HMT stand for the classical model theory result “If a set T of types satisfies the amalgamation and extension closure properties, there exists a homogeneous model that realizes exactly the types in T .” I will present computability-theoretic results on the strength of HMT and discuss how these degree-theoretic results provide insight into the reverse mathematical strength of HMT. In particular, I will discuss how HMT compares with the Atomic Model Theorem studied by Hirschfeldt, Shore, and Slaman.

Abstract: As we know, the fundamental group of the mapping torus of a pseudo-Anosov homeomorphism of a closed surface of genus greater than one is hyperbolic. Now we consider a graph of surfaces where the vertex surfaces and the edge surfaces are different closed surfaces, then we ask under which conditions the fundamental groups of this kind of graph of surfaces are hyperbolic. A particular case considered here is: for a graph of spaces GS, with underlying graph as a single loop, with edge and vertex space S and F respectively, S and F are closed surfaces with genus greater or equal to 2, p and q are covering maps from S to F, cut the edge open and reglue it with a pseudo-Anosov homeomorphism f of S. Let l^s and l^u be the stable and unstable geodesic laminations of f, let S' and F' be the universal covering spaces of S and F. For the lifted lamination L^s and L^u, there are degree(p) different images of L^s in F', and there are degree(q) different images of L^u in F'. Our hypothesis is that among all these degree(p)+degree(q) images in F', for any two of them their sets of leaf endpoints are disjoint subsets of the circle at infinity of F'. Our conclusion is the fundamental group of the graph of spaces is hyperbolic as long as m is sufficiently large.
And we give an example: for GS as the above, where S, F are genus 3 and 2 tori, f is any pseudo-Anosov homeomorphism whose virtual centralizer has trivial intersection with the deck transformation groups of p and q, m is a sufficiently large positive integer. Let a be a specific simple closed curve on F, and c is a simple closed curve on S such that the preimage of a under p is c, c in one componet of the preimages a under q. We claim that the fundamental group of GS is a hyperbolic group. And we construct a pseudo-Anosov homeomorphism f such that the fundamental group of GS is not commensurate to any surface-by-free group, even more it is not quasi-isometric to any surface-by-free groups.

Abstract: This project is about Karen Smith (1965-), Professor of Mathematics at the University of Michigan. Karen is a world leader in Commutative Algebra and Algebraic Geometry, she has been the recepient of very prestigious awards, and she is a devoted mother of three children. We will explore her path towards a career in mathematics, her outstanding acheivements, and her extremely positive influence on the numerous undergraduate students, graduate students and postdocs that she has mentored. We will also briefly discuss the role of Commutative Algebra and Algebraic Geometry in modern mathematics, to better understand Karen's contributions to these fields.

Abstract: Dr. Martha Euphemia Lofton Haynes is the first African American Woman to receive her Ph.D. in Mathematics. She received her Ph.D. from the Catholic University of America in 1943. The title of her thesis is “Determination of Sets of Independent Conditions Characterizing Certain Special Cases of Symmetric Correspondences.” Not only was Dr. Haynes well involved in the academic community, but she was also involved in many non-academic community related activities. She was a teacher, leader, philanthropist and mathematician. During her lifetime, Dr. Haynes has accomplished much and through her example, other women mathematicians can do the same.

Abstract: In 1976, the American mathematician Julia Robinson (1919-1985) became the first woman to be elected to the National Academy of Sciences. Robinson's work on existential definability and exponential Diophantine equations provided much of the basis for the solution of the famous Hilbert's Tenth Problem. Even though it was Yuri Matiyasevich who proved the negative solution to the Tenth Problem and not Robinson, his solution built fundamentally on Robinson’s joint work with Hilary Putnam and Martin Davis. Robinson made major contributions to the field of mathematical logic and became an inspiration to aspiring women mathematicians worldwide. Our poster will take a journey through Julia Robinson's life and her accomplishments as well as give an overview of Hilbert’s Tenth Problem and the Matiyasevich, Robinson, Davis, Putnam (MRDP) solution.

Abstract: Dr. Fiona Murnaghan is a Professor of Mathematics at the University of Toronto. Since earning her PhD from the University of Chicago in 1987, she has made many important contributions to the field of Representation theory of p-adic groups. We examined her rise to success from grade school, through graduate school and finally as a Professor of Mathematics. Next, we studied the general area in which Dr. Murnaghan’s research lies, group theory. After learning the basic results of group theory, we examined one of its classical results: A description of the symmetry group of each platonic solid.

Abstract: This project is about the British Mathematician, Sarah Rees (1957-), Professor of Mathematics at the University of New Castle in England. Professor Rees is a world leader in research at the junction of Group theory, geometry and combinatorics. Of her most recent projects, one examines connections between group theory and formal language theory, the other studies quantum computation from a group theoretic perspective. She has been internationally known for her work, and she is a devoted mother of a child. In this project I explore the decision problem known as word problem and its hardness using an article of Sarah Rees.

Abstract: The American mathematician Joan Birman was born in 1927. She is a leading expert in topology and one of the foremost experts in braids and knot theory. Her book Braids, Links, and Mapping Class Groups, which is based on a series of lectures she gave during her stay at Princeton in the spring 1987, has become a standard text in the subject. She has been influential in theoretical mathematics and has contributed to fundamental developments in topology. She has received numerous awards and recognitions for her work. Dr. Birman is currently a Research Professor Emerita at Barnard College, Columbia University, where she has been since 1973.