The Department of Mathematics encourages faculty, staff and students to participate and attend conferences and colloquia. For more information, please contact Professor Gabriel Prajitura or Howard Skogman.

## Main Page Content

## Recent and Upcoming Talks

**Speaker: **Howard Skogman**Title: **Ozanam's Rule is False

**Wednesday, Feb. 21, 2:30 - 3:20**

Time:

Time:

**214 Albert Brown Building**

Location:

Location:

**Ozanam's rule is a test to determine whether a given integer is a polygonal number. It was originally stated in 1694 (by Jacques Ozanam) and generalized in 1778 (by Jean-Etienne Montucla) but in 2017 it was shown to be false in general by Adam Krause (Brockport Math MA student). We discuss the history of this rule and give a corrected version.**

Abstract:

Abstract:

**Speaker: **Jason Morris**Title: **Familiar (and less familiar) Spaces of Sequences

**Wednesday, Nov. 8, 2:45 - 3:45**

Time:

Time:

**104 Holmes**

Location:

Location:

**It’s a pleasant surprise when you find out that you can do arithmetic with sequences just as if they were vectors in 2D or 3D. But you can do more than just vector arithmetic. You can make sense of magnitude and inner products, and you can perform linear operations on sequences. You can even have a sequence of sequences that converges (to a sequence, of course)! The choice of how to measure magnitude has consequences, and different choices lead to classical sequence spaces, such as l^∞ , c, c_0 , l^1 , l^2 , and l^p . These are introduced, and some of their properties are considered. Then we can get to the “less familiar” part: does it make sense to have l(p_n) where p_n itself is a sequence?**

Abstract:

Abstract:

Notes:

1. If you studied infinite series in calculus 2, you’ll understand (most of) this talk.

2. If you have taken linear algebra or even real analysis, you’ll understand even more!

3. There may be opportunities for research projects based on this topic.

4. I’m working up to a future talk about the possibility of research in solving differential equations using Sobolev Spaces with variable exponent...but you don’t need to be interested in that to come to this talk about sequences.

**Speaker: **Trevor Jarvis (Brockport Undergraduate)**Title: **Introduction to Knot Theory

**Monday, Oct. 30th, 2:45 - 3:45**

Time:

Time:

**104 Holmes**

Location:

Location:

**In this talk, we will discuss some of the basics of knot theory, following Colin Adams’ The Knot Book. We will cover the first few chapters, some information related to the content of those chapters, as well as some applications of knot theory, essentially, why do we care about it. This talk will be accessible for all undergraduates**

Abstract:

Abstract:

**.**

**Speaker: **Melissa Dimarco**Title: **Quivers, an Introduction

**Wednesday, Oct. 25th, 2:45 - 3:45**

Time:

Time:

**104 Holmes**

Location:

Location:

**A quiver is a directed graph with an algebraic context. In this talk, we will introduce quivers and discuss some of their key properties. Hopefully by the end of the talk, you will be astonished by how worthwhile quivers can be in the study of algebra. This talk is accessible to all undergraduates, especially those familiar with vector spaces.**

Abstract:

Abstract:

**Speaker**: Nathan Reff**Title**: Quaternion Matrices and Gain Graphs**Time**: Wednesday, Oct. 18th, 2:45 - 3:45**Location**: 104 Holmes**Abstract**: Gain graphs are a special kind of graph where each orientation of an edge is given
a group value, which is the inverse of the group value assigned in the opposite orientation.
If these edge values are chosen to be unit quaternions, then we can define matrices
associated to these graphs so that their algebraic properties can be related to the
original graph. In particular, we can find the left and right-eigenvalues associated
to these matrices and bound them using graph parameters. The real challenge is that
quaternions are not commutative over multiplication. I will define all of these things.
Very little prior knowledge is necessary, except perhaps some linear algebra.

**Speaker: **Howard Skogman**Title: **Kernel (method) Panic

**Wednesday, Oct. 5th, 3:35 - 4:35**

Time:

Time:

**B1 Holmes**

Location:

Location:

**Abstract**: The "kernel method" is a technique for using generating functions to solve a wide variety of problems. However, each type of problem may require its own "tricks" to make the method work. We will describe one combinatorial problem where the kernel method seems to lead to incorrect or absurd results. More importantly, I have not discovered the trick I need for this problem (or I have and I do not see it). Note that no familiarity with any of the above terms will be assumed in this talk.

## Previous Semesters

Speaker: Gabriel Prajitura

Title: Problem Solving as a Path to Research

Time: Thursday, May 4, 4:30 - 5:45pm

Location: 106 Holmes

Abstract:

The math of pizzas, pancakes, potatoes, cookies, trees, and many other subjects.
Special appearances by Patrick and Sponge Bob.

Speaker: John Steiner

Title: Approximation of Fractals

Time: Monday, April 24, 3:35 - 4:35pm

Location: B2 Holmes

Abstract:

A fractal is a geometric object whose basic structure repeats at all scales of magnification.
We will explore some unique and interesting fractals, the construction and approximation
of these fractals, and how to calculate the dimensions. Using the prior results, we
will then explore approximating the construction of fractals with an irrational dimension.

Speaker: Tasneem Zaihra

Title: Workshop on R: Long and Short of R in an Hour

Time: Monday, Feb. 27, 3:35 - 4:35pm

Location: B2 Holmes

Abstract:

R is a freely available language and environment for statistical computing and graphics
that has become popular in academia and in many industries. This short workshop will
introduce participants to using R with the help of RStudio ( IDE) in an integrated
way. It is designed to be accessible to those with little or no experience with R,
and will provide participants with skills, examples, and resources that they can use
in their own teaching and research. Participants should bring a laptop to the session.

The presenter has been using R to teach statistics to undergraduates at all levels as well as for her research and will share her approach and some favorite examples. Topics will include, building comprehensive html/pdf documents, which can include mathematical formula (tex) as well as R code and it’s corresponding output all put together.

We will be using RStudio environment, which provides novices with a powerful but manageable set of tools, data visualization, basic statistical inference using R. Much of it will be facilitated using the mosaic package.

Optional things you can do prior to workshop:

Install or update R on your laptop. (Available at https://cran.r-project.org/)

Install or update RStudio on your laptop (available at https://www.rstudio.com/products/rstudio/download/)

Install the mosaic package and its dependencies on your laptop. (In RStudio, click
on the Packages tab, then on Install, then type mosaic in the entry area and R should
do its thing.)

Notes for Dr. Zaihra's talk may be found at http://rpubs.com/statooed/253536 & http://rpubs.com/statooed/254109.

**Speaker:** Gabriel Prajitura

**Title:** Numerical range and Aluthge transforms

**Time:** Monday, Dec. 5, 2016 3:35 - 4:35pm

**Location:** B2 Holmes

**Abstract:**

The numerical range of a matrix is a very sensitive set of values which encodes more
information than the eigenvalues. The Aluthge transform of a matrix is a certain permutation
of factors. Both are simple objects and lead to many open problems even for 3 by 3
matrices.

**Speaker:** Nathan Reff

**Title:** Open Problem: Oriented Hypergraphs and Associated Matrices

**Time:** Wednesday Nov. 9, 2016 3:35 - 4:35pm

**Location:** B2 Holmes

**Abstract:**

For a given hypergraph, an orientation can be assigned to each vertex-edge incidence.
These orientations are used to define the incidence, adjacency and Laplacian matrices.
The adjacency and Laplacian matrices are particularly nice because they are symmetric,
so their eigenvalues are real. A natural question arises: how are the eigenvalues
of the adjacency and Laplacian matrices related to the original oriented hypergraph?
In this talk, I will present a background on oriented hypergraphs, explain some of
the known eigenvalue relationships, and give some current open problems. If time permits,
I will show some applications of oriented hypergraphs to chemical reaction networks.

**Speaker:** Gabriel Prajitura

**Title:** Open Problem: Korenblum's Constant

**Time:** Wednesday Oct. 12, 2016 3:35 - 4:35pm

**Location:** B2 Holmes

**Abstract:**

The setting is any vector space of complex functions defined for numbers of absolute
value less than 1. In general, if the values of a function (in absolute value) are
less than the values of another function at every point then the magnitude of the
first function (considered as a vector) is less than the magnitude of the second function.
That is, there is a preservation of order when we move from local to global. If, instead
of considering the values everywhere in the disc we restrict to only the values in
an annulus , there is no reason to expect such a preservation of order. Nevertheless,
in many vector spaces of function this preservation of order is present. The phenomenon
was conjectured by Korenblum in 1995 and it was proved to be true in a large class
of spaces in 1999. This raised the question of finding the inner radius of the smallest
annulus for which this happens. The value of this radius is called Korenblum's constant.
The exact value of it is not known. What we know today is that it is something between
0.14 and 0.67, with the last improvement dating from 2008. In 2014 Chakraborty and
Solynin proved that if we restrict to polynomials of degree at most n then there is
a particular Korenblum's constant only for this kind of functions. As with the general
case, the values of these polynomial constants are not known.

**Speaker:** Howard Skogman

**Title:** Open Problem: Maximal Moment Distributions of Character Sums

**Time:** Wednesday Oct. 5, 2016 3:35 - 4:35pm

**Location:** B2 Holmes

**Abstract:**

Given a set of numbers $S = \{a_1, a_2, a_3, ...\}$ and an integers $k$, we define
the $k$-th moment of the set to be the sum of the $k$-th powers of its elements, that
is $$ m_k(S) = \sum_{i} a_i^k $$

In this talk we fix a positive integer $m$ and a prime number $p$ and consider the distribution of the largest moment of a set of $m$, distinct $p$-th complex roots of unity. For the case of $m = 2$ (that is a set of two distinct $p$-th roots of unity), we show that in fact there is only one value. However, if $m > 2$ the number of possible values in the distribution increases. Some natural (and completely open) questions that arise are: (a) Given $m$ and $p$ how many possible values are in the distribution? (b) What are those values? (c) Can we give any lower bounds on these values?

**Speaker:** Gabriel Prajitura

**Title:** Open Problem: Gauss - Lucas Theorem Generalizations

**Time:** Monday, Sept. 19, 2016 3:35 - 4:35pm

**Location:** B2 Holmes

**Abstract:**

For real functions, Rolle's theorem tells us that in between two zeros of a function
there is always a zero of the derivative. Gauss Lucas theorem shows a property of
the same nature in the case of a complex polynomial: the zeros of the derivative are
inside the smallest convex polygon containing the zeros of the polynomial. For example,
for a polynomial of degree three, the zeros are the vertices of a triangle. The theorem
says that the two zeros of the derivative are inside that triangle.

For a polynomial of degree 4, if the zero form a convex quadrilateral, then the zeros of the derivative are inside it. However, there is the possibility that the 4 zeros of the polynomial do not for a convex quadrilateral. For example three of them can be the vertices of a triangle while the fourth is inside that triangle. The theorem still says that the three zeros of the derivative are inside the triangle. But where?

Connecting the fourth point with the other 3, the triangle is divided into 3 smaller triangles. It was prove that only 2 of the three can have zeros of the derivative inside. It is not known if all three zeros of the derivative can be inside only one of the smaller triangle. Or how many of them can be on the sides.

As the degree of the polynomial increases even less is known.

**Speaker:** Howard Skogman

**Title:** Open Problem: General Incomplete Gauss Sums and Lehmer Spirals

**Time:** Monday, Sept. 12, 2016 3:35 - 4:35pm

**Location:** B2 Holmes

**Abstract:**

Gauss sums are certain sums of complex n-th roots of unity. In 1973 D. H. Lehmer
investigated incomplete versions and noticed that the incomplete sums spiral towards
particular values. In fact he showed that almost all of the incomplete sums are located
close to one of two points. In this talk we consider more general incomplete Gauss
sums and consider similar graphs of the incomplete sums.

**Speaker:** Howard Skogman (College at Brockport)

**Title:** Introduction to Modular Forms

**Time:** Wednesday, Oct. 21, 2015 3:35-4:35pm (note was rescheduled)

**Location:** B2 Holmes

**Abstract:**

Modular forms (or Automorphic Forms) have become one of the most important tools
in modern mathematics. They played essential roles in the resolution of Fermat's Last
Theorem in 1995 as well as in the " Monstrous Moonshine Conjectures" also in what
is known as the "Langland;'s Program", in addition they also appear in "String; Theory"
from theoretical physics. In this talk we will give a classical introduction to Modular
Functions and Forms, we will discuss their definition, some motivation and some standard
examples. This talk will be accessible to undergraduate and graduate students.

**Speaker:** Daniel Gaile (University of Buffalo)

**Title:** The Parametric t-test's Latent Weakness

**Time:** Monday Nov. 9, 2015 3:35-4:35pm

**Location:** B2 Holmes

**Abstract:**

When a latent class structure is present, parametric t-tests conducted on the observed
continuous variable can be anti-conservative. This problem is exacerbated by: A) test
multiplicity across large numbers of manifest assays, each with a latent structure,
and B) increased accuracy of the manifest assays to discriminate underlying latent
structures. While it is not surprising that violations of the parametric t-test's
underlying assumptions can impact its performance, we demonstrate that latent state
conditions can lead to profound overstatements of statistical significance and profound
loss of error control. For example, we provide a motivating 'toy' data-set for which
the parametric t-test quantifies the evidence against the null hypothesis as approximately
12.5 million to 1 when it should be quantified as approximately 250 to 1. This result
is relevant in many modern experimental settings, such as pilot array / next-generation
sequencing studies, where an underlying latent structure is either known to be true
(e.g., methylation and array comparative genomic hybridization) or plausible (e.g.,
down/up-regulated gene networks). Our findings are also applicable to small animal
studies (e.g., mouse and rat studies), for which latent state biological mechanisms
are often plausible and the parametric t-test is often applied. Time permitting, we
will briefly discuss the effect of latent structure on common distance estimators
and present some methylation array results.

**Speaker:** Dr. Rebecca Smith (College at Brockport)

**Title:** Sorting things out with stacks

**Time:** Wednesday Sept. 30, 3:35-4:35pm

**Location:** B2 Holmes

**Abstract:**

A stack is a restricted queue where entries may only enter and exit via the top.
As such, a stack sorts using a "last in, first out" process. We shall use this machine
with permutations. Mostly, we will be concerned with permutations simply as an arrangement
of the numbers 1,2,3,...,n. However, there are some uses for more algebraic properties
of permutations. Generally, one either tries to sort a permutation, meaning using
a stack to obtain the identity permutation 123...n or one starts with a permutation
(generally the identity) and determines which permutations can be generated using
a stack. We will start with the problem of sorting permutations using a single stack.
We will then talk about different restrictions or freedoms that can be applied as
well as ways to string together more than one stack to create a larger machine.

**Speaker:** Dr. Patrick Papadopulos (University of Rochester)

**Title:** Math in a Factory: Algebraic Topology and Configuration Spaces

**Time:** Tuesday, March 10, 2015 2:00pm-3:00pm

**Location:** B6 Holmes

**Abstract:**

In this talk, we will give an overview of some of the main components of Algebraic
topology. In particular, we will introduce common objects and spaces that are studied.
Our main goal, however, is to introduce the concept of a configuration space and focus
on applications to real world problems. This talk will be aimed at undergraduate students
with a basic understanding of point set topology and will include a number of pictures.

**Speaker:** Dr. Nathan Reff and Dr. Howard Skogman (College at Brockport)

**Title:** Hadamard Matrices and Oriented Hypergraphs II

**Time:** Thursday February 26, 11am-12pm

**Location:** B6 Holmes

**Abstract:**

This talk will be a continuation of the previous talk. In particular Nathan will
present the proof of the equivalency between Hadamard matrices and certain oriented
hypergraphs. Howard will then review known constructions, along with (time permitting)
equivalent problems, potential other constructions, and applications.

**Speaker:** Dr. Nathan Reff (College at Brockport)

**Title:** Hadamard Matrices and Oriented Hypergraphs

**Time:** Thursday February 19, 11am-12pm

**Location:** B6 Holmes

**Abstract:**

An nxn matrix with all entries +1 or -1 whose rows are mutually orthogonal is called
a Hadamard matrix. It is known that n must be 1,2 or a multiple of 4 for all Hadamard
matrices, however it is not known whether Hadamard matrices exist for every n which
is a multiple of 4. This problem of existence is well over a century old and is known
as the Hadamard conjecture.

A signed graph is a graph with edges labelled either +1 or -1. In this talk I will present a problem equivalent to the Hadamard conjecture which involves signed graphs and their associated matrices. Hadamard matrices enjoy a wide range of applications including a direct connection to error-correcting codes. If time permits, I will mention some known constructions.

**Speaker:** Dr. Jason Morris (College at Brockport)

**Title:** Does an infinite matrix have eigenvalues?

**Time:** Thursday October 30, 11am-12pm

**Location:** B6 Holmes

**Abstract:**

For finite square matrices, we are usually taught to use the determinant to find
eigenvalues. There is no easy analog of determinant in the case of infinite matrices,
so we explain how to use the insolvability of Av=cv to characterize the so-called
"spectral" values of A. It turns out that there are several types of insolvability,
leading to a situation where some spectral values could be eigenvalues, and some not.
This is the first of possibly several talks that will serve as an overview of spectral
theory. The goal is to set a foundation to enable the study of relationships between
infinite graphs and some matrices that represent them. Most of the topics should be
accessible to students with some background in linear algebra (and in convergence
of infinite series).

**Speaker:** Dr. Howard Skogman (College at Brockport)

**Title:** Covering Graphs, Part II

**Time:** Thursday October 9, 11am-12pm

**Location:** B6 Holmes

**Abstract:**

We will go through more examples of the normal (or Galois) cover construction, along
with related results. If there is time we will discuss block-diagonalizing the adjacency
matrix or the universal cover of a graph.

**Speaker:** Dr. Howard Skogman (College at Brockport)

**Title:** Covering Graphs, Part I

**Time:** Thursday October 2, 11am-12pm

**Location:** B6 Holmes

**Abstract:**

We will define a covering graph, prove some results about their structure, discuss
general constructions as well as normal (or Galois) covers and non-normal covering
graphs.

**Speaker:** Dr. Nathan Reff (College at Brockport)

**Title:** Oriented gain graphs, oriented hypergraphs, line graphs and eigenvalues

**Time:** Thursday September 25, 11am-12pm

**Location:** B6 Holmes

**Abstract:**

We define line graphs of gain graphs and study matrix properties of complex unit
gain graphs. As with graphs and signed graphs, there is a relationship between the
incidence matrix of a complex unit gain graph and the adjacency matrix of the line
graph. The line graph of a gain graph is defined using oriented gain graphs, a new
structure that generalizes Zaslavsky's oriented signed graphs and their line graphs.
The line graph of an oriented hypergraph is similarly defined and will also be discussed.

**Speaker:** Dr. Nathan Reff (College at Brockport)

**Title:** Gain Graphs, Oriented Hypergraphs and Matrices Part II

**Time:** Thursday September 18, 11am-12pm

**Location:** B6 Holmes

**Abstract:**

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given
a label of +1 or -1. Recently, there has been an interest in finding a suitable way
to study matrices associated to hypergraphs. I will show how oriented hypergraphs
provide a very natural setting to study matrices associated to hypergraphs, and introduce
some eigenvalue properties.

If time permits I will mention some potential projects related to these structures and their applications.

**Speaker:** Dr. Nathan Reff (College at Brockport)

**Title:** Gain Graphs, Oriented Hypergraphs and Matrices

**Time:** Thursday September 11, 11am-12pm

**Location:** B6 Holmes

**Abstract:**

A gain graph is a graph where each orientation of an edge is given a group element,
which is the inverse of the group element assigned to the opposite orientation. Complex
unit gain graphs have particularly nice matrix properties which I will discuss.

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or -1. Recently, there has been an interest in finding a suitable way to study matrices associated to hypergraphs. I will show how oriented hypergraphs provide a very natural setting to study matrices associated to hypergraphs, and introduce some eigenvalue properties.

If time permits I will mention some potential projects related to these structures and their applications.

**Speaker:** Dr. Aleksey S. Polunchenko (Binghamton University)

**Title:** Suspect Something Fishy? How Statistics Can Help Detect It, Quickly

**Time:** Tuesday, April 22, 2014 12:30pm-1:30pm

**Location:** 105 Edwards

**Abstract:**

Statistics is a branch of mathematics concerned with rational decision-making among
uncertainty. This talk aims to provide an introduction to the nook of statistics that
deals with cases when a solution has to be worked out "on-the-go", i.e. when time
is a factor. The talk will focus on the quickest change-point detection problem, aka
sequential change-point detection.

Dr. Polunchenko will conduct a one-hour student discussion after the presentation!

**Speaker:** Dr. Ernest Fokoue (Rochester Institute of Technology)

**Title:** Discovering the Fascinating World of Big Data Predictive Analytics and Some Mathematical
and Statistical Tools for Conquering It

**Time:** Thursday, March 27, 2014 11am-12pm

**Location:** Holmes B6

**Abstract:**

Dr. Fokoue will present attractive and appealing big data analytics problems and
will identify some of the mathematical and statistical concepts that tend to appear
almost ubiquitously in most statistical machine learning methods. Dr. Fokoue hopes
to provide guidelines to mathematics, statistics and computer science professors as
to some of the things they should emphasize in order to prepare students adequately
for a potential career in modern data science.

**Speaker:** Jonathan Lottes (College at Brockport)

**Title:** Chaos and Dynamical Systems

**Time:** Thursday, March 6, 2014 11am-11:30am

**Location:** Holmes B6

**Abstract:**

This talk will give an introduction to some basic terms in dynamical systems and
will focus on the sawtooth and reverse sawtooth functions. In particular, the fixed
points, eventually fixed points, periodic points, and eventually periodic points will
be discussed, as well as how the points of the two functions are related. Other orbits
will be looked at to demonstrate the chaotic behavior of the functions.

**Speaker:** Michelle Anderson (College at Brockport)

**Title:** Modeling Cell Arrangement in Epithelial Tissue

**Time:** Thursday, March 6, 2014, 11:30am-12pm

**Location:** Holmes B6

**Abstract:**

We developed an off-lattice, 3D, particle-based model to simulate cell rearrangement
in epithelial sheets. Our model assigns cells orientation and polarization in addition
to position, volume, and shape. Including orientation and polarization in our model
allowed us to add another facet of realism to individual cells and cell-to-cell interaction
allowing us to more realistically simulate important developmental processes in collections
of cells.

**Speaker:** Dr. Mihai Bailesteanu (University of Rochester)

**Title:** Spin(7) Manifolds - Old and New

**Time:** Thursday, November 21, 2013, 11am-12pm

**Location:** Holmes B6

**Abstract:**

Spin(7) Manifolds are 8 dimensional Riemannian manifolds that have a cross product
on their tangent bundles which generate a 4-form. We can define some canonical vector
fields on these manifolds, which in turn allow us to define some type of moment map.
The goal of having a moment map is to study the topology of the underlying manifold.
We will discuss some recent developments.

**Speaker:** Zhuang Hou (University of Rochester)

**Title:**Delay Differential Equations (DDEs) and Stochastic DDEs: Explosion property and applications

**Time:** Thursday, November 14, 2013, 11am-12pm

**Location:** Holmes B6

**Abstract:**

A DDE or SDDE is a differential equation where the increment of the solution not
only depends on the value of the solution in the current time but also in the past.
This type of equation is getting more and more important in several models. In this
talk, I will talk about recent results about the explosion property of SDDE. And I
have also joined the work to construct high dimensional DDE model of Genome-wide Dynamics
Regulatory Networks. I will also talk about the model in Bio-statistics.

**Speaker:** Dr. Julius Esunge (University of Mary Washington)

**Title:** Generating Functions and the Moment Problem

**Time:** Tuesday, October 29, 2013, 11am-12pm

**Location:** Holmes 205

**Abstract:**

How much information is sufficient for a random variable? In particular, if we know
the mean and variance can we uniquely specify the probability distribution of a random
variable? We will consider such questions, together with more general items, and their
applications. We will see the impact of these ideas in gambling, reproduction, actuarial
science, and many other fields.

**Speaker:** Dr. Howard Skogman

**Title:** Towards a more realistic graph theory model

**Time:** Thursday, September 26, 2013, 11am-12pm

**Location:** Holmes B6

**Abstract:**

We will consider solutions to the "Heat Equation" on finite graphs. In particular,
we will first consider graphs with constant edge weights, then deterministic but non-constant
edge weights, and finally we will add a stochastic (random noise) component.

**Speaker:** Dr. Gabriel Prajitura

**Title:** The Leibniz Test

**Time:** Tuesday, September 10, 2013, 11am-12pm

**Location:** Holmes 205

**Abstract:**

We will present a general version of the Leibniz Test for series in which the usual
ingredients (alternation of signs, and decreasing terms) are no longer present.