## Fall 2019 Undergraduate Friday Seminars

**Speaker: **Anush Tserunyan; Assistant Professor

**Title: **Perfect Set Property

**Abstract: **A subset of the reals has the perfect set property if it is either countable or contains a copy of Cantor space --- the space of all infinite sequences of 0s and 1s. This is a strong form of the so-called Continuum Hypothesis and the Cantor--Bendixon theorem states that this property holds for all closed subsets of reals. We will discuss this representative theorem of the subject called Descriptive Set Theory and its connection with infinite games, if time permits.

**Speaker:** Alice Chudnovsky; Undergraduate

**Title: **Classifying Homomorphisms from the Braid and Symmetric Groups

**Abstract:** One goal of group theory is to understand all possible homomorphisms between two groups. My group characterized maps from the symmetric group and braid group to solvable groups, abelian groups, dihedral groups, free groups, and other groups. This process was facilitated by the concept of Totally Symmetric Sets (TSS), a “basis” of sorts for a group, developed by Margalit, Kordek and Chen. TSS induces a correspondence: given a group homomorphism φ: G → H and a totally symmetric set S of size n in G, φ(S) is either a totally symmetric set of size n in H or of size 1. By determining two groups’ totally symmetric sets, we were able to prove for many cases that all possible homomorphisms factor through a cyclic group. Moreover, for those groups, we drew conclusions on the structure of their totally symmetric sets, and gave upper bounds for their possible size.

**Speaker:** Aubrey Laskowski; Undergraduate

**Title:** Error Correcting Codes

**Abstract:** Error correcting codes (ECCs) are some of the core connecting components between coding theory and information theory, and much of modern communication involves some aspect of an ECC - from CDs to Amazon Web Services and even to space missions. ECCs provide the ability to decode information from noise channels by providing a small amount of redundant data. I will discuss the history of ECCs, as well as specific codes such as Hamming codes and Reed Solomon codes. Important parameters of ECCs as a historical perspective will be discussed and, time permitting, I will introduce how ECCs connect to machine learning. For the history, no specific background is assumed, but linear and abstract algebra will likely be usful for other aspects of the talk.

**Speaker: **Jinghui Yang; Undergraduate

**Title:** Kuratowski 14-Sets Theorem and Related Classifications of Topological Spaces

**Abstract:** One very interesting theorem you may encounter when you first study topology is known as the famous Kuratowski 14-sets theorem: Given a topology space (X,T) with T is its topology, for any subset A of X, at most 14 sets can be obtained from A by taking closures and complements. This is really a shocking fact, but even more surprisingly, the “tool” to prove it is rather easy using a strong taste of algebra. Furthermore, this “tool” can be useful to do some basic classification of topology spaces; namely, you can decide the type of some topology spaces completely determined by its “K-Number”. This talk is based on the paper The Kuratowski Closure-Complement Theorem by B.J. Gardner and M. Jackson (2007), published in New Zealand Journal of Mathematics, Vol.38(2008), 9-44. Some basic concepts of topology will be reviewed.

**Speaker: **Alexi Block Gorman; PhD Student

**Title: **Where Automata Theory Meets Metric Geometry

**Abstract: **The results in this talk illustrate and expand on connections between automata theory and metric geometry. We will begin by defining automata, Buchi automata, fractals, and iterated function systems. We say that a function is regular if there is a Buchi automaton that accepts precisely the set of base n representations of points in the graph of the function. We show that a continuous regular function (with closed and bounded domain) "looks linear" almost everywhere, if you zoom in enough. As a result, we show that every differentiable regular function is a shift of linear function (or hyperplane, in higher dimensions).

**Speaker: **Colleen Robichaux; PhD Student

**Title: **Schubert Polynomials and Computational Complexity

**Abstract: **In this talk I will discuss recent work in Algebraic Combinatorics and how it relates to Computational Complexity. How do results in Computational Complexity influence work in Algebraic Combinatorics and vice versa? I will give introductions to both topics and discuss how they came together in joint work with Anshul Adve and Alexander Yong.

**Speaker:** Jenna Zomback; PhD Student

**Title: **Games That Take Forever

**Abstract:** Do you have a winning strategy for playing tic tac toe? What about chess? In a two player game (with no ties), we say that Player 1 has a winning strategy if she can always make sure that she wins the game, regardless of what Player 2 does. In this talk, we will make this definition a bit more formal, and we'll prove that in any game that ends after finitely many steps, one of the players has a winning strategy. We'll also discuss infinite games (that end after infinitely many steps), and what it means to have a winning strategy in these games. If time allows, we'll prove that in special types of infinite games, one of the players has a winning strategy.

**Speakers:** Anna Chlopecki & Jackie Oh

**Title:** EG Tableaux and Complexity

**Abstract:** The purpose of this talk is to examine a counting problem in algebraic combinatorics. We will be discussing connections between reduced words, Young tableaux, and the Lascoux- M.-P. Schützenberger transition algorithm in hopes to provide an intuition for proving that counting the number of Edelman Greene tableaux for a given permutation w and partition λ is in #P.

**Speaker: **Joseph Rogge

**Title: **Classifying Permutations Under Context-Directed Swaps

**Abstract:** In 2003, Prescott hypothesized a special sorting operation performed on the genomic material of ciliates. This operation, called cds, involves block interchanges of permuted lists. Christie (1996) discovered that among those permutations which are sortable by cds, cds sorts them using the fewest possible block interchanges of any kind. Adamyk et al. (2013) discovered an efficient way of quantifying the non-cds-sortability of a permutation called the strategic pile. My group partially characterized permutations with maximal strategic pile (in some sense, the permutations for which applying cds is the most unpredictable), completing this characterization when the number of available cds moves is minimal and when it is nearly maximal. We discover a ZnｘZn-action on permutations in S(n+1) that preserves the number of available cds moves. This group action, defined on permutations, has a very nice visual interpretation via chord diagrams (circles with intersecting chords). We characterize the stabilizer of this group action, and are thus able to use Lagrange’s Theorem to count the size of orbits of permutations. This allows us to count the number of permutations with a given number of cds moves.