# 2017 Colloquia

Colloquium

## An Introduction to Research in the Department of Mathematical Sciences

### Fred Peck, Emily Stone

In this colloquium members of the faculty give short talks on their research.  If you are a graduate student who has not yet picked an MA project/thesis advisor, this is an excellent opportunity to learn a bit about what research questions interest us.

Colloquium

## Peter Coffee, VP for Strategic Research, Salesforce

### Acquiring Intelligence: Assembling Tomorrow's Data-Driven, AI-Everywhere World

It's been said that AI is the discipline that kicks its successes out of the nest: that once it works, it's called "language processing" or "image recognition" or "predictive recommendation" or some other specific description of the useful result. (Whatever's left behind as AI, it's been said, is what's "Almost Implemented.") Peter Coffee, global VP for Strategic Research at cloud computing leader Salesforce, will be with us once again, this time to discuss the component skills and disciplines that actually are implemented in the company's Einstein machine-intelligence portfolio. He will also share the process of identifying and acquiring the companies and teams that have been assembled into this real-world layer of "Assistive Intelligence" (perhaps a more useful definition of "AI"), across the Salesforce Platform, and to discuss the academic and career preparation that leads toward tomorrow's opportunities.

Colloquium

## James Tipton Visiting Assistant Professor

### Infinite Product Representations of Kernel Functions on Fractals

A fractal is, loosely speaking, an image which exhibits a degree of self-similarity. In the early eighties Benoit Mandelbrot published the classic, Fractal Geometry of Nature, in which he argued that many natural phenomena were better modeled by fractals.  The fractals we will consider are obtained through the iteration of quadratics, although the results presented here hold in much greater generality.

Shortly after Mandelbrot's publication, mathematicians began looking for ways to extend various branches of analysis to fractals.  Among these branches is functional analysis, and in particular we will look at a recently developed method for constructing a kernel function on a given fractal.

Uniquely associated to any kernel function is a Hilbert space of functions in which every linear evaluation functional is bounded.  These spaces are called reproducing kernel Hilbert spaces, and as a Hilbert space, notions of length and angle can be defined.  The construction we follow represents the kernel function as an infinite product involving iterations of the quadratic defining the fractal of interest.  We will determine precisely which quadratics this construction holds for.

Colloquium

## Katharine Shultis Gonzaga University

### Systems of parameters and the Cohen-Macaulay property

Let $$R$$ be a commutative, Noetherian, local ring and $$M$$ a finitely generated $$R$$-module. Consider the module of homomorphisms $$\operatorname{Hom}_R(R/\mathfrak{a},M/\mathfrak{b} M)$$ where $$\mathfrak{b}\subseteq\mathfrak{a}$$ are parameter ideals of $$M$$. When $$M=R$$ and $$R$$ is Cohen-Macaulay, Rees showed that this module of homomorphisms is always isomorphic to $$R/\mathfrak{a}$$. Recently, K. Bahmanpour and R. Naghipour showed that if $$\operatorname{Hom}_R(R/\mathfrak{a},R/\mathfrak{b})$$ is isomorphic to $$R/\mathfrak{a}$$ for every pair of parameter ideals $$\mathfrak{b}\subseteq\mathfrak{a}$$ then $$R$$ is Cohen-Macaulay. In this talk, we will define the terms above and discuss the structure of $$\operatorname{Hom}_R(R/\mathfrak{a},M/\mathfrak{b}M)$$ for general $$R$$.

Colloquium

## Eric Chesebro University of Montana

### Combinatorial polynomials and the geometry of 2-bridge links

We will discuss a combinatorial family of 1 variable integer polynomials indexed by the rational numbers.  Along the way, we will talk about Fibonacci numbers, Chebyshev polynomials, the Farey graph, and the hyperbolic geometry of 2-bridge links.

Colloquium

## Ryan Grady Montana State University

### An intro to quantum BV theory

Starting from elementary differential topology, I will introduce quantum field theory in the Batalin-Vilkovisky (BV) formalism.  I will discuss the relationship between quantum BV theories and projective volume forms.  In particular, I will illustrate (via example) how to build volume forms on interesting moduli spaces using BV theory.

Colloquium

## Diana Schepens Montana State University – PhD Candidate

### The effects of metabolite production cost on cooperation in microbial communities

Metabolic cross-feeding between microbes is observed in many microbial communities. It has been experimentally observed that cross-feeding synthetic communities have increased level of fitness and cell growth as compared to wild type cells. There are also numerous examples of cross-feeding communities in nature.

Our goal is to develop a model to analyze the effects that resource investment into metabolite production have on the evolution of syntrophy in a microbial community. We first analyze the investment into the substrates and enzymes that are used to produce the metabolite in a metabolic pathway in order to formulate a representation of the cost of producing the metabolite. We then combine this cost function together with a model of a microbial community containing a variety of phenotypes to observe conditions under which cooperation occurs.

Colloquium

## Linh Nguyen University of Idaho

### Mathematics of Photoacoustic Tomography

Photoacoustic tomography (PAT) is a hybrid method of imaging. It combines the high contrast of optical imaging and high resolution of ultrasound imaging. A short pulse of laser light is scanned through the biological object of interest. The photoelastic effect produces an ultrasound pressure propagating throughout the space, which is measured by transducers located on an observation surface. The goal of PAT is to find the initial pressure inside the object, since it contains helpful information of the object.

The mathematical model for PAT is an inverse source problem for the wave equation. In this talk, we will discuss several methods for solving this inverse problem. They include inversion formulas, time reversal techniques, and iterative methods.

Colloquium

## Andrew Hoegh Montana State University

### Predictive Spatiotemporal Modeling

Statistical modeling is often used for one of two distinct paradigms: explanatory or predictive inference. Data science or predictive analytics based applications are often concerned with prediction. With an emphasis on methodology for predictive inference in spatiotemporal settings, this talk will provide an overview of a multiscale spatiotemporal framework developed to predict outbreaks of social unrest in Central and South America. Civil unrest is a complicated, multifaceted social phenomenon that is difficult to forecast. Relevant data for predicting future protests consist of a massive set of heterogenous data sources, primarily from social media. A modular approach to extract pertinent information from disparate data sources is implemented to develop a Bayesian multiscale framework to fuse prediction from algorithms mining social media.

Special Event

Colloquium

## Charlie Katerba University of Montana – PhD Candidate

### From Hyperbolic to Algebraic Geometry through Character Varieties

In this expository lecture, we will review some of the basics of hyperbolic geometry from the perspective of representation theory and use this point of view to motivate the study of a 3-manifold's character variety.  Along the way we will touch on some classic results in low-dimensional topology, explore concrete examples, and discuss various ways to continue applying character varieties to 3-manifold topology.

This lecture will review the background material for the speaker's dissertation defense.

## Glen Satten, Centers for Disease Control Joint work with Yijuan Hu, Department of Biostatistics and Bioinformatics, Emory University

### Analyzing Data on the Microbiome using Linear Models based on Approximate Singular Value Decompositions

Data from a microbiome study is often analyzed using approaches developed by Ecologists: a matrix of pairwise distances is calculated between each observation, followed by ordination (graphical representation of each observation as a point in a low-dimensional space, using either principal components or multidimensional scaling based on the distance matrix). Ordination is frequently successful in separating meaningful groups of observations (e.g., cases and controls). Although distance-based analyses such as Permanova can be used to test whether explanatory variables (such as case-control status, batch or sample pH) have a significant effect on the distance matrix, the connection between data on individual species (or operational taxonomic units, OTUs) and the information in the distance matrix is lost, and there is no way to know which species contribute to the patterns seen in ordination for the high-dimensional data we gather in a microbiome study. To provide a single analysis path that includes distance-based ordination, global hypothesis tests of any effect of the microbiome, and hypotheses tests of the effects of individual OTUs, we present a novel approach we call the linear decomposition model (LDM). Using simulations we show that the LDM can have higher power than solely distance-based methods, while avoiding some technical difficulties that plague existing methods when a non-Euclidean distance is used. Finally, we show how the effects of confounding covariates can be accounted for by a novel 'peeling' approach.

## Oliver Serang, Department of Computer Science

### Moment-based approximations of algorithms on semirings

Many problems can be computed more efficiently through divide-and-conquer approaches by exploiting inverse operations on a ring (e.g., fast matrix multiplication and fast convolution via Karatsuba or FFT); however, these same well-studied problems often have poor solutions on semirings, because the lack of an inverse operation renders many divide-and-conquer techniques impossible. For this reason, the fastest known algorithms for max-matrix multiplication (which can be used with an adjacency matrix to find the shortest/longest paths in a graph) and max-convolution (used to compute max-product marginals / MAP of random variables with sum relationships Y=X_1+X_2+\cdots) are substantially slower than the ring-based algorithms. A generic method is presented, which uses the moments of distributions containing the correct solution. This method allows ring-based, divide-and-conquer algorithms to approximate the solutions on the semiring (\times, max).

## Ian Parker Renga, Western State Colorado University

### “We want to bring them into what we love”: Exploring Desire in Teacher Education

What is it we love about teaching and learning? Are teacher candidates expected to share that love? Such questions were on my mind as I observed a master Montessori teacher instructing teacher candidates on how to teach pre-algebra. The candidates were transfixed as the instructor systematically arranged beads on a board and then transformed the beautiful design into a tool for discovering variables and polynomial expression. The candidates remarked how typically dry math became more appealing, more enticing. The instructor explained that teaching math or any discipline required bringing children into what the teacher loved about it and piquing their desire to learn more. The preparation exercise, it appeared, was designed with the same intention. As I discuss in this talk, whether intended or not, all teacher preparation activity can be seen as reflecting particular educational desires and orienting candidates to those desires. To explore this claim, I first establish the compatibility of a liturgical framing of desire with Vygotskian sociocultural theory; I then present findings from an in-depth qualitative examination of two teacher preparation programs, one for Montessori teachers and the other for teachers planning to serve in urban, Title I schools. In an educational climate preoccupied with talk of meeting needs, building skills, and producing outcomes, questions of eros, or passionate desire, may seem superfluous; nice to consider, perhaps, but unlikely to address the difficult realities faced by K-12 teachers. Yet, as I argue, failing to directly engage eros may insidiously conscript teachers and those who prepare them in perpetuating those realities and task them with fulfilling educational desires they don’t always share.

## Peter Liljedahl, Simon Fraser University Burnaby, British Columbia

### Illumination in Mathematics: An Affective Experience?

What is the nature of illumination in mathematics? That is, what is it that sets illumination apart from other mathematical experiences? In this talk the answer to this question is pursued through a qualitative study that seeks to compare and contrast the AHA! experiences of preservice teachers with those of prominent research mathematicians. Using a methodology of analytic induction in conjunction with historical and contemporary theories of discovery, creativity, and invention along with theories of affect the anecdotal reflections of participants from these two populations are analysed. Results indicate that, although manifested differently in the two populations, what sets illumination apart from other mathematical experiences are the affective aspects of the experience.

## Laramie Paxton, Washington State University

### Measuring Level Sets of $$C^{1,1}$$ Functions

Singular integrals comprise a rich area of analysis, the most well known example being the Hilbert Transform. In this talk, we will discuss a singular integral that also intersects geometric measure theory. For functions $$f:\mathbb{R}^n\rightarrow\mathbb{R}$$ that are $$C^{1,1}$$ (i.e. the first derivative is Lipschitz continuous), for which 0 is a regular value (i.e. the gradient $$\nabla f$$ does not vanish on the 0-level set), and whose 0-level set is bounded, there is a not too hard proof that our singular integral computes $$\mathcal{H}^{n-1}(\{f^{-1}(0)\})$$, the $$(n-1)$$-dimensional Hausdorff measure of the 0-level set of $$f$$. We will also briefly mention the simple analysis problem that inspired this research.

## Arvind Saibaba, North Carolina State University

### D-optimal Experimental Design for Bayesian Inverse Problems

Optimal Experimental Design seeks to control experimental conditions in order to maximize the amount of information gained about parameters of interest, subject to physical or budgetary constraints. The parameters we wish to infer are represented on fine-scale grids; consequently, the experimental design problem is extremely computationally challenging and efficient algorithms are needed. We develop a computational framework for the D-optimality criterion in PDE based inverse problems.  Our approach exploits a certain low-rank structure in the covariance matrices using novel randomized estimators. This approach allows us to reduce the computational costs by several orders of magnitude compared to naive approaches. We demonstrate our algorithms on an optimal sensor placement problem from contaminant source identification.

Joint work with Alen Alexanderian, Ilse CF Ipsen (both at Department of Mathematics, NCSU)

## Carla Farsi, University of Colorado – Boulder

### Representations, wavelets, and spectral triples for k-graphs: an infinite path space approach.

I will introduce k-graphs and talk about some of their properties. In particular I will talk about the infinite path space of a k-graph starting from the example of one-graphs, and explain how this object is central to many properties of k-graphs. No expertise in the area needed, I will introduce all the relevant concepts from scratch. This talk touches upon and highlights joint work with Gillaspy, Kang, Jorgensen, Julien, and Packer.

## Kyle Austin, Ben Gurion University

### Inverse Approximation of Groupoids

In this talk, I will first discuss inverse approximation: what it means and how does one use it in practice. Then I will discuss groupoids and give an overview of their convolution and C*-algebras. My focus will be on understanding what the convolution product means and how it translates "dynamical" information of the groupoid into algebra.  I will focus on giving lots of examples so that the unfamiliar audience member can get a proper feel of these concepts. After giving lots of examples, I will explain a current project of Magdalena Georgescu and I in which we have devised a method of taking inverse limits of groupoids that preserve their harmonic analysis (i.e. the pullback morphism induces a C*-morphism of groupoid C*-algebras). I will briefly discuss what we have shown and its applications.

## Abhishek Methuku, Central European University

### On subgraphs of 2k-cycle-free graphs

Kühn and Osthus showed that every bipartite 2$$k$$-cycle-free graph $$G$$ contains a four-cycle-free subgraph with at least $$1/(k-1)$$ fraction of the edges of $$G$$. We give a new and simple proof of this result.

In the same paper Kühn and Osthus also showed that a $$2k$$-cycle-free graph which is obtained by pasting together four cycles has average degree at most $$16k$$ and asked whether there exists a number $$d=d(k)$$ such that every $$2k$$-cycle-free graph which is obtained by pasting together $$2l$$-cycles has average degree at most $$d$$ if $$k > l \ge 3$$ are given integers. We answer this question negatively.

We show that for any $$\varepsilon>0$$, and any integer $$k \ge 2$$, there is a $$2k$$-cycle-free graph $$G$$ which does not contain a bipartite subgraph of girth greater than $$2k$$ with more than $$\left(1-\frac{1}{2^{2k-2}}\right)\frac{2}{2k-1}(1+\varepsilon)$$ fraction of the edges of $$G$$. Győri et al. showed that if $$c$$ denotes the largest constant such that every 6-cycle-free graph $$G$$ contains a bipartite subgraph which is 4-cycle-free having $$c$$ fraction of edges of G, then $$\frac{3}{8}\le c\le\frac{2}{5}$$. Putting $$k=3$$, our result implies that $$c=\frac{3}{8}$$.

Our proof uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős: For any $$\varepsilon>0$$, and any integers $$a,b, k\ge2$$, there exists an $$a$$-uniform hypergraph $$H$$ of girth greater than $$k$$ which does not contain any $$b$$-colorable subhypergraph with more than $$\left(1-\frac{1}{b^{a-1}}\right)\left(1+\varepsilon\right)$$ fraction of the hyperedges of $$H$$.

Joint work with Grósz and Tompkins.