The Junior Colloquium is a colloquium series presented primarily by graduate students. Any graduate student or faculty member is welcome to present research or other mathematical topics of interest. Talks are intended to be accessible to all graduate students, though everyone is welcome to attend.
Talks are generally given on Thursday from 3:00pm to 3:50 in Minard 212. Coffee and treats are served at each meeting. If you are interested in giving a talk please contact Jonathan Totushek at jonathan.totushek@ndsu.edu or Faraad Armwood at faraad.armwood@ndsu.edu.
Knots are obtained by embedding a circle in $\mathbb{R}^3$. Pictures and handicrafts of knots have appeared in ancient Chinese art several hundred years B.C. The first systematic study of knots though started in the 19th century by a Scottish physicist Peter Guthrie Tait who classified all knots to 10 crossings and made several important conjectures about knots (the most recent of these conjectures is proven in 1991).
Nowadays, knots are interesting objects not just to mathematitions, but also to physicists, biologists, chemists, etc. In my talk, I will recall some elementary notions of knots (such as prime knots, torus knots, fibered knots, pretzel knots, alternating knots, satellite knots, etc.). Many, many examples and pictures will be provided. Then, I'll briefly mention several well known knot invariants, but concentrate mostly on computing the Alexander polynomial of knots.
Many fractals can be defined by an iteration function system (IFS) consisting of finitely many maps. We can associate a letter to each map and use a symbolic space to describe the fractal. One way to define a sub-fracta is to relate the IFS to a sub-shift of finite type (SFT) on the symbolic space. In this talk, we will formally describe the connection between such sub-fractals and SFTs. We will also discuss some techniques for calculating the Hausdorff dimension of these sets.
Short exact sequences (a.k.a., "extensions") are fundamental in algebra. For instance, one uses these to prove the existence of rational canonical forms in linear algebra. In this talk I will discuss how the cohomological construction "Ext" controls the behavior of extensions in surprisingly deep ways.
In this talk we are going to discuss the pricing of one of the most popular fincancial derivatives which is called "swap". Swap is a financial derivatie in which two counter parties exchange financial instruments such as cash flow. We use non-Gaussian Ornstein-Uhlenbeck process driven by Levy subordinator to model the dynamics of stock price (BN-S model) and use it to value the price of variance, volatility, and covariance swap. Nowadays investment banks or other sectors are actively quoting such swaps. If you have some cash and don't know what to do with it, this talk might be the right place to attend.
Construction of a projective resolution of a module is the process of iteratively approximating the module by projective modules. It is natural to ask when this process terminates. I will discuss work of Barbara Osofsky showing that the answer depends on whether or not you assume that the Continuum Hypothesis holds.
Let $K$ and $L$ be two convex bodies containing the origin. If for each hyperplane passing through the origin, the projection of $K$ can be rotated around the origin to fit inside the projection of $L$. Can $K$ be rotated to fin into $L$? We will present counterexamples to this question, and additional conditions on $K$ and $L$ that make the result true.
Properties of a monomial ideal can be seen in a lattice diagram representing a monomial ideal $I$ in a polynomial ring $R$. Such properties include whether $I$ is m-primary. If it is, we can relatively eassily calculate the length of modules of the form $R/I^n$ and $I^{n-1}/I^n$, integral closure, the Hilber-Samuel multiplicity, and obtain information about reductions. If $I$ is not m-primary, we can calculate the j-multiplicity.
The Riemann zeta-function $\zeta(s)$ and its generalizations, $L$-functions, are ubiquitous yet mysterious functions in number theory. These functions can be defined in association with a plethora of mathematical objects, including Dirichlet characters, number fields, and modular forms. We will begin with an introduction to the Riemann zeta-function and motivate the study of its moments. We will then consider general $L$-functions, specifically arbitrary products of $L$-functions attached to irreducible cuspidal automorphic representations of GL$(m)$ over $\mathbb{Q}$. The Langlands program suggests essentially all $L$-functions are of this form. Assuming some standard conjectures, I will discuss how to estimate two types of moments: the continuous moment of an arbitrary product of primitive $L$-functions and the discrete moment (taken over fundamental discriminants) of an arbitrary product of primitive $L$-functions twisted by quadratic Dirichlet characters. This is a generalization of results of K. Soundararajan and is inspired by the work of V. Chandee.
Projective and injective modules are of key importance in algebra, in part because of their useful homological properties. The notion of $C$-projective and $C$-injective modules generalizes these constructions. In particular, these modules may be used to construct resolutions and define related homological dimensions in a natural way. When $C$ is a semidualizing module, the $C$-projective and $C$-injective modules have particularly useful homological properties. Further, one may combine projective and $C$-projective resolutions to construct complete $PC$-resolutions (and, dually, complete $IC$-resolutions) that yield other modules with nice homological properties. We will survey these constructions. No prior knowledge of homological algebra is assumed.
In 2000, Viswanath published a paper that used technology to give a specific example of a general result due to Furstenberg and Kesten (1960). The specific example was that if one took the standard Fibonacci sequence, where the nth term was obtained by addition of the previous two terms (1,1,2,3,5,8...), and instead of adding each time, flipped a fair coin that decided whether one adds or subtracts. It is a well known fact that the quotient of consecutive Fibonacci numbers converges to the Golden Ratio as n approaches infinity, and Furstenberg and Kesten proved that the random Fibonacci sequence would almost always have a similar convergence property. Viswanath gave the initial digits of the new mathematical constant, 1.13....
In this talk, we will discuss a subset of the truly random Fibonacci sequences: the periodic Fibonacci sequences. The subset has measure 0, but there are still many interesting properties that they possess. If time permits, we will also discuss similar properties for the tribonacci sequence, and the n-nacci sequence in general.
In this talk, I'll tell you about my thesis work in narrative form, including the stories behind the various results. I'll also include some tips for getting through grad school and learning to do research.
We will consider a particualr operator algebra, called a semicrossed product, generated by a symbol and continuous functions on a compact Hausdorff space. We will then define point derivations and discuss their basic properties and what it means for a point derivation to be inner. Finally, we will focus on a particular representation of a semicrossed product into an upper triangular matrix, and we will show that every point derivation associated with this representation is inner. A passing acquaintance with Math 750 and matrices is all that is required.
A conncected sum is a construction that can be used to construct Gorenstein rings. I will discuss a characterization of connected sums of $k$-algebras, and explore some relations between connected sums and associated graded rings, based on joint work with H. Ananth and E. Celikbas.
In this talk, we will discuss the construction of tree fractals and canopies. The fractal dimension is a useful way to distinguis two fractals. Weill focus on the Hausdorff dimension and will discuss different methods for calculation. We will focus on one calculation method, which involves defining a topological pressure and finding a Borel probability measure that is supported by the canopy. We will use this method to find the Hausdorff dimension of some well known fractals.
This will be a two-part talk; the first talk is meant to be a refresher on (or brief introduction to) the necessary basic operator theory and $C^{*}$-algebras. This is meant for those who have not seen these befoe or for whom it has been a while since working in these topics. The second talk will be an introduction to Graph Algebras; this is a study of certain families of operators on (generally finite) directed graphs, and the $C^{*}$-algebras these families generate. We will explore the basic properties of these, as well as some interesting known rewsults about them.
The branch of mathematics known as Ramsey Theory is the study of minimal conditions for a particular structure to appear. On the integers, we consider the following problem: Given a linear equation, inequality, or system of equations/inequalities $L$, find the least integer $n$ such that when we color the ordered set $(1,2,\dots,n)$ using $t$ colors, we are guaranteed a solution to $L$ in $(1,2,\dots,n)$ in one color. In this talk, we will look at this problem using various choices for $L$. This talk will be accessible to both graduate and undergraduate students as no prior knowledge is needed.
Differential equations are sometimes considered (especially within the US system of higher education) as something which is taught right after the calculus, with a clear engineering taste and a lot of tedious and uninsightful calculations. Upper lever courses in differential equations often have at least some applied flavor. These facts notwithstanding, the mathematical theory of differential equations is an extremely interesting and rich area on its own, which gave birth to such subjects as topology and Lie group theory, to mention just a few, and which is in the center of current mathematical activity. The modern theory of differential equations can be roughly divided into two parts: Qualitative theory of differential equations and Dynamical systems theory. The former deals mostly with analytical (or even polynomial) systems on the plane and can be described as relatively "orderly world," whereas the latter deals with flows in dimensions greater than two, and can be called "the realm of chaos." In this presentation I would like to talk about the qualitative theory of differential equations, which centered during approximately last hundred years around the 16th Hilbert's problem. This problem (to be precise, the second part of it) can be stated as: What can be said about the number and location of limit cycles of a planar polynomial vector field of degree n? This problem is still open in its full generality, and I will discuss what is actually known today about it.
The Beurling-Selberg Extremal Problem looks at finding optimal entire functions of a given exponential type that majorize or minorize a real valued function while minimizing the $L^1$-norm of the difference. Often the existence of such functions lead to optimal bounds in certain inequalities. In this talk, I will describe some classical applications of such problems and then present some results and generalizations for the Poisson and conjugate Poisson kernels. This talk is intended to be accessible to all graduate students and advanced undergraduates as I will describe any needed results from Complex and Fourier Analysis.
This is going to be a nontechnical talk, aimed to a general audience. This talk can be considered as an introduction to my graduate course in ODE scheduled for Fall 2014.
Given an infinite, directed, graded graph $G$, an adic transformation is a dynamical system on the space of infinite paths of $G$. We describe some well-known dynamical systems and give their equivalent adic transformations. In particular, we discuss some dynamical properties of an adic transformation whose underlying graph is related to Pascal's triangle.
Recall than an element $x$ in a ring $R$ is idempotent if $x^2=x$. We'll use idempotents in a discussion about clean rings and neat rings. We'll also see that PIDs are neat!
Let $R$ be a commutative, noetherian ring with identity. An $R$-module $C$ is semidualzing if $C$ is finitely generated over $R$, the homothety map $\chi^R_C:R\to\operatorname{Hom}_R(C,C)$ is an isomorphism, and $\operatorname{Ext}^i_R(C,C)=0$ for all $i>0$. We discuss the existence of nontrivial semidualizing modules (and complexes) over tensor products.
First, we will define what it means for a module to be Artinian and under what conditions the module $R/I$ ($R$ a commutative ring with identity and $I$ an ideal) has finite length. The associated graded ring and the Hilbert function will be defined and we will prove that the Hilbert function eventually becomes a polynomial. From this Hilbert polynomial, we will define the multiplicity $e(I)$ and mention a few theorems regarding it. Lastly, we will generalize these ideas to all ideals and mention several open questions.
When $E$ is a semidualizing module over a commutative ring, Holm and Jørgensen studied some connections between $E$-Gorenstein injectivity/projectivity and Nagata's trivial extension. We generalized some of those results to other general constructions including D'Anna and Fontana's amalgamated duplication of a ring along an ideal and Esescu's Pseudocanonical covers. We also identified some key properties of those general constructions that enable their connections to Gorenstein injectivity and projectivity.
The Enigma cipher machine was used extensively by German military forces during the second World War to send and receive encrypted messages. Though it was thought to be unbreakable, several flaws in the machine's operation led to breakthroughs by Polish and British cryptanalysts that allowed Allied forces to decrypt messages and shortened the length of the war by several years. In the first part of this talk, we will investigate the Enigma machine's operation, make working paper simulators, and discuss the combinatorics that gave the machine its reputation as an "unbreakable" cipher. In the second part, we will develop the mathematical tools used by Polish mathematicians to break the cipher, and work through a complete message crack. If time permits, we will also discuss the British role in the Enigma crack, paying special attention to the role of Alan Turing in ending the war. This talk requires only a basic understanding of combinatorics, permutation theory, and the operation of scissors and tape, and will be accessible to all graduate students and advanced undergraduates.
Public-key cryptography provides a unique and fascinating solution to problems involving authentication and privacy of information. The idea is to use a pair of encryption keys instead of a single key used by all parties. We'll explore the underlying mathematics and applications.
I am going to talk about very basic models in mathematical epidemiology, mathematical insights that can be obtained from these simple models, shortcomings of these models, and about the connections with random network theory, which has recently become a very powerful tool in mathematical epidemiology (and in various other fields of current research). In a more general context, my talk can be considered as a promotion of Fall 2013 graduate course Math 767: Topics in Applied Mathematics: Mathematics of Networks; I will try to present a wide picture of what I am going to discuss in this course, and what kind of mathematics can be learned.
Let $R$ be a commutative noetherian ring. A finitely generated $R$-module $C$ is said to be semidualizing if $\text{Ext}_R^i(C,C) = 0$ for all $i > 0$ and $R \cong \text{Hom}_R(C,C)$. When $R$ is local, an artinian $R$-module $T$ is said to be quasidualizing if $\text{Ext}_R^i(T,T) = 0$ for all $i > 0$ and $\widehat{R}\cong \text{Hom}_R(T,T)$. Using the notion of $I$-cofiniteness, we introduce a unifying notion that recovers each of the above notions as special cases. (Joint work with Sean Sather-Wagstaff)
Let $R$ be a local noetherian ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Auslander, Buchsbaum, and Serre proved that $R$ is a regular local ring if and only if $\text{pd}_R(M)$ for every $R$-module $M$. Similar results characterize complete intersection rings with the complete intersection projective dimension $\text{CI}$-$\text{pd}_R(M)$. This talk will explore other types of homological dimensions and what types of information we can get from them.
Every module $M$ over a commutative ring $R$ is a homomorphic image of a free $R$-module. Iterating this procedure, one sees that $M$ has a free resolution, that is, there is an exact sequence \[ \cdots \to F_2 \to F_1 \to F_0 \to M \to 0 \] such that each $F_i$ is free. This is the starting point, in some sense, for homological algebra. One goal in this area is to give explicit algorithms for describing such resolutions. In general, this problem is intractable. In this talk, after providing some some background information, I will describe some situations where this problem can be solved.
How would you teach a computer to play games against a human player? Using move trees and the minimax algorithm, we can develop techniques for simple game strategies. We will discuss how the algorithm works and use it to play simple games. We will also discuss evaluation functions and ways they apply to more complicated games like chess. This talk will be very informal and should be accessible to everyone.
In my talk I will give an introduction to the mathematical approaches to model biological evolution. In particular, I will formulate the celebrated Eigen's quasispecies model and state the basic results concerning the quasispecies concept and the error threshold. Connections with the statistical physics will be presented. I will conclude my presentation with possible research directions. (slides)
A recurring question in convex analysis is the determination of a convex body $K$ from a given set of lower dimensional information about it, such as the volume of its sections or its projections. In this talk we will review several problems about the determination of $K$ from its hyperplane sections. Different problems arise depending on the type of sections considered (central sections, maximal sections, sections tangent to a prescribed ball inside of $K$, etc.) In all cases, the main questions are the following:
And the answers may be very different (and surprising) for each type of problem.
A large part of commutative algebra research is focused on the study of modules over commutative rings. However, some results about modules can only be proved by considering a larger class of objects: chain complexes. I will discuss some aspects of this area, giving lots of background. In particular, I will only assume familiarity with ideas from Math 721.
Factorization usually investigates how to break an element in an integral domain into a product of irreducibles. This usually isn't possible in the exotic world of purgatory and antimatter domains. In this talk we will look into some natural ways to peel the layers of the factorization onion by examining the group of divisibility of an integral domain, and we will develop some structure theorems about groups of divisibility.