**2017-2018**

**Tuesdays 4:10 p.m. in Carver 205 - Tea
and cookies starting at 3:45 p.m. in Carver 404
**

The ISU Department of Mathematics Colloquium is organized by

Pablo Raúl Stinga (stinga@iastate.edu)

__ __

**Fall 2017
**

_________________________________________________________________________________________________________

**September 5**

**Speaker: Tin-Yau Tam**

Auburn University

**Title: Orbital geometry - from matrices to
Lie groups **** **

**Abstract:**Given an $n\times n$ matrix $A$, the celebrated Toeplitz-Hausdorff theorem asserts that the classical numerical range $\{x^*Ax: x\in {\mathbb C}^n: x^*x=1\}$ is a convex set, where ${\mathbb C}^n$ is the vector space of complex $n$-tuples and $x^*$ is the complex conjugate transpose of $x\in {\mathbb C}^n$. Schur-Horn Theorem asserts that the set of the diagonals of Hermitian matrices of a prescribed eigenvalues is the convex hull of the orbit of the eigenvalues under the action of the symmetric groups. These results are about unitary orbit of a matrix. Among interesting generalizations, we will focus our discussion on those in the context of Lie structure, more precisely, compact connected Lie groups and semisimple Lie algebras. Some results on convexity and star-shapedness will be presented.

**September 12
**

**Speaker: Dennis Kriventsov**

Courant Institute (NYU)

**Title: Spectral optimization and free
boundary problems**

**Abstract:**A classic subject in analysis is the relationship between the spectrum of the Laplacian on a domain and that domain's geometry. One approach to understanding this relationship is to study domains which extremize some function of their spectrum under geometric constraints. I will give a brief overview of some of these optimization problems and describe the (very few) explicit solutions known. Then I will explain how to approach these problems more abstractly, using tools from the calculus of variations to find solutions. A key difficulty with this approach is showing that the solutions (which are a priori very weak) are actually smooth domains, which I address in some recent work with Fanghua Lin. Our method revolves around relating spectral optimization problems to certain vector-valued free boundary problems of Bernoulli type.

**September 19
**

**Speaker: Deanna Haunsperger**

Carletton College

**Title: Stories from Math Horizons **

**Abstract:**In this talk, Deanna will talk about becoming involved in the Mathematical Association of America as an editor of Math Horizons, some of the cool mathematics she learned in this process, and opportunities for participating in the national mathematical community.

**October 3
**

**Speaker: Hien Nguyen**

Iowa State University

**Title: Mean curvature flow, its long-time
existence, self-similar surfaces, and some related
problems**

**Abstract:**For curves in the plane, the curvature at a point measures how fast the direction is changing. If each point moves perpendicularly to the loop at a speed equal to the curvature, the resulting flow is called the curve shortening flow. To visualize this flow, one can think of the loop as the edge of a thin layer of ice floating on water. Corners will round out instantly; skinny offshoots disappear fast; inlets fill in; and flat edges move slowly. If the ice layer is circular, it will shrink while remaining circular and disappear eventually. This last example is called a self-similar solution because its shape does not change under the flow. In higher dimensions, because so many paths go through a point, one considers the mean curvature, which is an average of all the curvatures of all the curves through a point, and defines the mean curvature flow.

In this talk, I will explore the properties of the mean curvature flow and some classical results. I will then present more recent development about its long-time existence and self-similar surfaces. In particular, I will focus on the gluing techniques and talk about the main steps and difficulties for gluing pieces of surfaces in order to construct new self-similar solutions.

**October 5 ****(Room: Carver 018)**

**Speaker: Michael Young**

Iowa State University

**Title: Polychromatic colorings of hypercubes
and integers **

**Abstract:**Given a graph

*G*which is a subgraph of the

*n*-dimensional hypercube

*Q_n*, an edge coloring of

*Q_n*with $

*r\ge2*$ colors such that every copy of

*G*contains every color is called

*G-polychromatic*. Originally introduced by Alon, Krech and Szabó in 2007 as a way to prove bounds for Turán type problems on the hypercube, polychromatic colorings have proven to be worthy of study in their own right. This talk will survey what is currently known about polychromatic colorings and introduce some open questions. In addition, there are some natural generalizations and variations of the problem that will also be discussed. One generalization will be polychromatic colorings of the integers, which we use to prove a conjecture of Newman.

**October 10
**

**Speaker: Jack H. Lutz**

Iowa State University

**Title: Who asked us? How the theory
of computing answers questions that weren't about
computing **

**Abstract:**It is rare for the theory of computing to be used to answer open mathematical questions whose statements do not involve computation or related aspects of logic. This talk discusses recent developments that do exactly this. After a brief review of algorithmic information and dimension, we describe the

*point-to-set principle*(with N. Lutz) and its application to two new results in geometric measure theory. These are (1) N. Lutz and D. Stull's strengthened lower bound on the Hausdorff dimensions of generalized Furstenberg sets, and (2) N. Lutz's extension of the fractal intersection formulas for Hausdorff and packing dimensions in Euclidean spaces from Borel sets to arbitrary sets.

**October 12** **(Room: Carver 202)**

**Speaker: Bernard Lidicky
**

Iowa State University

**Title: Flag algebras and applications**

**Abstract:**We give a short introduction to flag algebras and we discuss several of its applications. Flag algebras is a method, developed by Razborov to attack problems in extremal combinatorics. Razborov formulated the method in a general way which made it applicable to various settings. In this talk we give a brief introduction of the basic notions used in flag algebras and demonstrate the method on Mantel's theorem. Then we discuss applications of the flag algebras in different settings. In particular, we mention applications to crossing numbers, iterated extremal structure and Ramsey numbers.

**October 13 (Room: Carver 202)
**

**Speaker: Ralph McKenzie**

Vanderbilt University

**Title: P or NP-complete: a very successful
application of general algebra to a fundamental
graph homomorphism problem **

**Abstract:**With any finite relational structure A we have a computational problem: input a finite structure B of the same signature as A; accept B if and only if there is a homomorphism from B to A. The CSP-dichotomy conjecture of Feder and Vardi states that given any template A (a finite relational structure), the described computational problem either admits a polynomial-time algorithm, or is NP-complete. Feder and Vardi proved that this general conjecture is equivalent to the restricted conjecture where the template is simply a di-graph.

My talk will sketch developments in both directions, new algorithms for large families of CSP problems, and surprising algebraic results offering unexpected insight into the diversity of deep structures in finite algebras.

**October 17
**

**Speaker: Emille Lawrence**

University of San Francisco

**Title: Topological symmetry
groups of graphs in S^3**

**Abstract:**The study of graphs embedded in S^3 was originally motivated by chemists’ need to predict molecular behavior. The symmetries of a molecule can explain many of its chemical properties, however we draw a distinction between rigid and flexible molecules. Flexible molecules may have symmetries that are not merely a combination of rotations and reflections. Such symmetries prompted the concept of the topological symmetry group of a graph embedded in S^3. We will discuss recent work on what groups are realizable as the topological symmetry group for several families of graphs, including the Petersen family and Möbius ladders.

**October 24
**

**Speaker: Tathagata Basak**

Iowa State University

**Title: From sphere packing in R^24 to a
monster manifold via hyperbolic geometry**

**Abstract:**Last year it was proved that the Leech lattice provides the densest way to pack spheres in 24 dimensional Euclidean space. We shall talk about the infinite complex reflection group G whose special properties are borrowed from the Leech lattice. The group G acts naturally on the unit ball in complex thirteen dimensional vector space preserving a negative curvature metric. Let Y be the part of the ball on which G acts freely. The monstrous proposal conjecture states that the fundamental group of the quotient Y/G maps onto a close cousin of the monster simple group and this leads to a construction of a 12 dimensional complex manifold with a natural monster action. We shall descibe generators and relations for the fundamental group of Y/G providing strong evidence for this conjecture. Part of this is joint work with Daniel Allcock. There are two main ingredients in the proofs: (1) making use of a still poorly understood analogy between complex reflection groups and Weyl groups of complex simple Lie algebras and (2) making use of special properties of the Leech lattice. Along the way we shall explain how this project leads us to some other work of independent interest: like constructing new arithmetic lattices in U(n,1) or explicit uniformization of hermitian symmetric spaces by automorphic forms.

**October 26 (Room: Carver 001)
**

**Speaker: Songting Luo**

Iowa State University

**Title: Mathematical modeling and simulation
of wave-matter interactions **

**Abstract:**Simulating wave-matter interactions is important for various practical applications. Depending on the wavelengths and the matter sizes, different models (PDE models) are required and/or applied. For each model, effective numerical methods are highly desirable for applications. I will talk about some problems rising from nano optics, kinetic theory, and high frequency wave propagation that cover a wide range of scales, along with discussion of the numerical methods.

**October 31**

**Krishna Athreya
**

Iowa State University

**Title: On the sums of powers of the
likelihood function of random walks on the integer
lattice in d dimension**

**Abstract:**In their classic book on problems in analysis written around 1925 (republished in the US in 1945 and 1970) Polya and Szego give the asymptotics of sums of powers of binomial coefficients as n goes to infinity. In this talk we prove an analog of their result for sums of powers of probabilities involving random walks on the integer lattice in

*d*dimension. We deduce some results for the binomial and multinomial probabilities. We also outline some open problems.

** November 7
**

**Speaker: Alicia Prieto Langarica
**

Youngstown State University

**Title: A mathematical
model of the effects of temperature on human sleep
patterns**

**Abstract:**Sleep is on of the most fundamental, across species, and less understood processes. Several studies have been done on human patients that suggest that different temperatures, such as room temperature, core body temperature, and distal skin temperature, have an important effect on sleep patterns, such as length and frequency of REM bouts. A mathematical model is created to investigate the effects of temperature on the REM/NonREM dynamics. Our model was based on previous well-established and accepted models of sleep dynamics and thermoregulation models.

[

**Authors**

**:**Selenne Bañuelos (California State University Channel Islands) - Janet Best (The Ohio State University) -

**Alicia Prieto Langarica**(Youngstown State University) - Pamela B. Pyzza (Ohio Wesleyan University) - Markus Schmidt (Ohio Sleep Medicine Institute) - Shebly Wilson (Morehouse College)]

**November 14
**

**Speaker: Yong Zeng
**

National Science Foundation

**Title: Bayesian inference via filtering
equations for financial ultra-high frequency data**

**Abstract:**We propose a general partially-observed framework of Markov processes with marked point process observations for ultra-high frequency (UHF) transaction price data, allowing other observable economic or market factors. We develop the corresponding Bayesian inference via filtering equations to quantify parameter and model uncertainty. Specifically, we derive filtering equations to characterize the evolution of the statistical foundation such as likelihoods, posteriors, Bayes factors and posterior model probabilities. Given the computational challenge, we provide a convergence theorem, enabling us to employ the Markov chain approximation method to construct consistent, easily-parallelizable, recursive algorithms. The algorithms calculate the fundamental statistical characteristics and are capable of implementing the Bayesian inference in real-time for streaming UHF data, via parallel computing for sophisticated models. The general theory is illustrated by specific models built for U.S. Treasury Notes transactions data from GovPX and by Heston stochastic volatility model for stock transactions data. This talk consists joint works with B. Bundick, X. Hu, D. Kuipers and J. Yin.

[

**Bio.**Yong Zeng serves as a program director in Division of Mathematical Sciences DMS at National Science Foundation. He is also a professor in the Department of Mathematics and Statistics at University of Missouri - Kansas City.]

**December 5**

**Speaker: Oyita Udiani
**

National Institute for Mathematical and Biological Synthesis (NIMBioS)

**Title: Mathematical models of social advocacy
on networks**

**Abstract:**

**Members of the public hold opposing positions on a multitude of social issues. For example, although there is scientific consensus that climate change is occurring and driven by human activity, a sampling of popular media makes it clear that opposing discourses are alive and well in the United States. As with many social issues, there are several organizations working to affect policy on climate change through grassroots advocacy. Achieving success depends on a number of factors, but none is perhaps more important than understanding how to design campaigns for persuading and mobilizing stakeholders within communities of interest. In this talk, I will introduce a theoretical framework to study this question using models of adaptive influence on networks. Examples discussed include campaigns to convert skeptics (maximizing prevalence of beliefs) vs. mobilize converts (maximizing extremism of beliefs).**

[

**Bio.**Oyita Udiani is an applied mathematician and NSF postdoctoral fellow at NIMBioS. His research develops models to study questions related to the organization of social and biological systems.]

**Spring 2018**

**January 8 (4:10pm -- Carver 268)
**

**Leili Shahriyari
**

Mathematical Biosciences Institute - The Ohio State
University

**Title: Discovering
effective cancer treatments through
Computational models**

**Abstract:**

Carcinogenesis is a complex stochastic evolutionary process. One of the key components of this process is evolving tumors, which interact with and manipulate their surrounding microenvironment in a dynamic spatio-temporal manner. Recently, several computational models have been developed to investigate such a complex phenomenon and to find potential therapeutic targets. In this talk, we present novel computational models to gain some insight about the evolutionary dynamics of cancer. Furthermore, we propose an innovative framework to systematically employ a combination of mathematical methods and bioinformatics techniques to arrive at unique personalized targeted therapies for cancer patients.

**January 10 (4:10pm -- Carver 268)
**

**Claus Kadelka
**

Institute
of Medical Virology, University of Zurich,
Switzerland -- Division of Infectious Diseases and
Hospital Epidemiology, University Hospital Zurich,
Zurich, Switzerland

**Title: Computational HIV Vaccinology and the Robustness of Gene Regulatory Networks**

** **

**Abstract:**

My talk will be split
into two parts: first, my current work in
computational HIV vaccinology; and second,
robustness analyses of gene regulatory networks
(GRNs).

HIV broadly
neutralizing antibodies (bnAbs) are the major hope
for an effective HIV vaccine and therapy
development, but are only elicited at low frequency
in natural HIV infection. We recently conducted a
systematic survey of bnAb activity in 4,484 HIV-1
infected individuals, and identified several viral
and disease parameters associated with bnAb
development, as well as antibody binding patterns
predictive of bnAb existence. Through phylogenetic
HIV sequence analysis, we further identified more
than 300 likely transmission pairs, and exhibited,
for the first time, that parts of the HIV antibody
response are heritable.

**January 12 (4:10pm -- Carver 268)**

**Erica Rutter
**

Center for Research in Scientific Computation -- North Carolina State University

**Title: ****Modeling and
Estimating Biological Heterogeneity in
Spatiotemporal Data**

**Abstract:**

Heterogeneity in biological populations, from cancer to ecological systems, is ubiquitous. Despite this knowledge, current mathematical models in population biology often do not account for inter-individual heterogeneity. In systems such as cancer, this means assuming cellular homogeneity and deterministic phenotypes, despite the fact that heterogeneity is thought to play a role in therapy resistance. Glioblastoma Multiforme (GBM) is an aggressive and fatal form of brain cancer notoriously difficult to predict and treat due to its heterogeneous nature. In this talk, I will discuss several approaches I have developed towards incorporating and estimating cellular heterogeneity into partial differential equation (PDE) models of GBM growth. In particular, I will discuss the use of random differential equations for modeling purposes and the Prohorov metric framework for estimating parameter distributions from data.

**January 16**

**Philip Ernst
**

Rice University

**Title: Yule's "Nonsense
Correlation" Solved!**

**Abstract:**

In this talk, I will discuss how I recently resolved a longstanding open statistical problem. The problem, formulated by the British statistician Udny Yule in 1926, is to mathematically prove Yule's 1926 empirical finding of "nonsense correlation". We solve the problem by analytically determining the second moment of the empirical correlation coefficient of two independent Wiener processes. Using tools from Fredholm integral equation theory, we calculate the second moment of the empirical correlation to obtain a value for the standard deviation of the empirical correlation of nearly .5. The "nonsense" correlation, which we call "volatile" correlation, is volatile in the sense that its distribution is heavily dispersed and is frequently large in absolute value. It is induced because each Wiener process is "self-correlated" in time. This is because a Wiener process is an integral of pure noise and thus its values at different time points are correlated. In addition to providing an explicit formula for the second moment of the empirical correlation, we offer implicit formulas for higher moments of the empirical correlation. The full paper is currently in press at The Annals of Statistics and can be found at https://projecteuclid.org/euclid.aos/1498636874

**January 17 (4:10pm -- Carver 268)
**

**Michael Catanzaro
**

University of Florida

** **

**Title: Topological data analysis and a geometric
approach to multiparameter persistent homology**

**Abstract:**

The prevalence of ever-increasing sources of data demands the development of new tools for analysis. Topological data analysis (TDA) provides one such toolbox, relying on geometric and topological methods to highlight features of data that are not apparent using other approaches. A central idea of TDA is to determine the features that persist across multiple scales. These persistent features can be completely described and conveniently visualized due to a structure theorem closely related to the structure theorem for finitely generated abelian groups. In this talk, I will discuss a generalized version of persistence inspired by a parameterized form of Morse theory, and discuss how it can be used in practice.

**January 19 (4:10pm -- Carver 268)
**

**Rana Parshad
**

Clarkson University

**Title:** **Finite Time
Blow-Up and "Ecological" Damping: Applications to
Invasive Species Control**

**Abstract:**

In this talk I will present some recent finite time blow-up results in heat and wave equations, as a lead into studying certain explosive invasive populations, such as the Burmese python in South Florida. I will next introduce the idea of "ecological" damping as a means of controlling such invasive populations. These novel controls have the advantage of avoiding non-target effects due to classical chemical and biological control. I will conclude with future directions in the biological control of invasive species.

**January 22 (4:10pm -- Carver 268)
**

**Zahra Aminzare
**

Princeton University

**Title****: ****Synchronization
patterns in networks of nonlinear dynamical
systems**

**Abstract:**

The analysis of synchronization in networks of nonlinear systems is important in a variety of research fields in science and engineering as well as in mathematics. In the human nervous system, synchronization can be beneficial, allowing for production of a vast range of behaviors such as generation of circadian rhythms and emergence of organized bursting in pancreatic beta-cells; or detrimental, causing disorders such as Parkinson’s disease and epilepsy.

In realistic networks that feature heterogeneous nodes and nonuniform coupling structure, complex patterns of synchronization emerge. Finding the conditions that foster synchronization in networked systems is critical to understanding a wide range of biological and mechanical systems. In this talk I introduce several synchronization patterns and identify when synchronization occurs and explain its dependance on parameters such as network structure, coupling weights, and intrinsic nodal dynamics.

As a real application of synchronization in biological settings, I show synchronous phenomena in central pattern generators (CPGs). CPGs are sophisticated circuits that can generate complex locomotor behaviors and even switch between different gaits. I discuss the mechanism of gait transition in an oscillator model of CPGs in insects.

**January 24 (4:10pm -- Carver 268)
**

**Yoonsang Lee
**

Lawrence Berkeley National Laboratory

**Title: Uncertainty
Quantification of Physics-constrained Problems**

**Abstract:**

Observation data along with mathematical models play a crucial role in improving prediction skills in science and engineering. In this talk we focus on the recent development of uncertainty quantification methods, data assimilation and parameter estimation, for Physics-constrained problems that are often described by partial differential equations. We discuss the similarities shared by the two methods and their differences in mathematical and computational points of view and future research topics. As applications, numerical weather prediction for geophysical flows and parameter estimation of kinetic reaction rates in the hydrogen-oxygen combustion are provided.

**January 26 (4:10pm -- Carver 268)
**

**Amy Veprauskas
**

University of Louisiana at Lafayette

**Title**: **Changes
in population outcomes resulting from phenotypic
evolution and environmental disturbances**

**Abstract:**

We develop an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection. For this evolutionary model, we use bifurcation analysis to establish the existence and stability of a branch of positive equilibria that bifurcates from the extinction equilibrium when the inherent growth rate passes through one. We then present an application to a daphnia model to demonstrate how the evolution of resistance to a toxicant may change persistence scenarios. We show that if the effects of a disturbance are not too large, then it is possible for a daphnia population to evolve toxicant resistance whereby it is able to persist at higher levels of the toxicant than it would otherwise. These results highlight the complexities involved in using surrogate species to examine toxicity. Time permitting, we will also consider a nonautonomous matrix model to examine the possible long-term effects of environmental disturbances, such as oils spills, floods, and fires, on population recovery. We focus on population recovery following a single disturbance, where recovery is defined to be the return to the pre-disturbance population size. Using methods from matrix calculus, we derive explicit formulas for the sensitivity of the recovery time with respect to properties of the population or the disturbance.

**January 29 (4:10pm -- Carver 268)**

**Mark Kempton
**

Harvard University

**Title: Quantum walks on
graphs**

**Abstract:**

Algebraic and spectral techniques in graph theory have recently found important application in quantum information theory via the study information transfer through networks of interacting qubits. Of particular interest is the problem of determining when a quantum state can be transferred perfectly through such a network, and this has been shown to be modeled by a so-called "quantum walk" on a graph. I will discuss results on perfect and approximately perfect state transfer in this context in perturbations of various classes of graphs.

**January 31 (4:10pm -- Carver 268)
**

**Thomas Fai
**

Harvard University

**Title: ****The Lubricated Immersed Boundary
Method**

**Abstract:**

Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.

**February 2 (4:10pm -- Carver 268)**

**Alexey Miroshnikov
**

University of California Los Angeles

**Title: ****Inference of demographic histories
for populations of variable size**

**Abstract:**

Studying the demographic histories of humans or other species and understanding their effects on contemporary genetic variability is one of the central tasks of population genetics. The genealogical relationship of a sample of several individuals is commonly modeled by the ancestral recombination graph (ARG). The ARG has a very complex structure and using it for inference is computationally prohibitive. In this talk I will present novel Hidden Markov Models used to approximate the ARG and an effective computational methodology for inference of demographic histories based on these models.

**February 7 (4:10pm -- Carver 268)**

**Joey Iverson
**

University of Maryland

**Title: ****Optimal line packings from finite
group actions**

**Abstract:**Frames are spanning sets of vectors that allow for stable analysis and reconstruction of data in Hilbert space. In applications such as compressed sensing and quantum information theory, it is critically important to find examples of frames whose vectors are spread wide apart. In particular, we would like the interior angles between the lines they span to be as wide as possible, as measured by the coherence. This is an old problem, going back at least to the work of van Lint and Seidel in the 1960s, and it remains an active and challenging area of research today. In this talk we will present a new recipe for converting transitive actions of finite groups into tight frames, many of which have optimal coherence. The main idea is to use an association scheme as a kind of converter to pass from the discrete world of permutation groups into the continuous setting of frames. This process is easy to implement in a computer program like GAP. We will present several examples of optimally incoherent frames produced in this way, including the first infinite family of equiangular tight frames with Heisenberg symmetry. (These are not SIC-POVMs, but they appear to be related.)

**February 8 (4:10pm -- Pearson 2105)**

**Achilles Beros
**

University of Hawaii, Manoa

**Title: The complexity of
algorithmic teaching and learning**

**Abstract:**

I will discuss some recent contributions to the research program I initiated with my thesis in 2013: classifying models in algorithmic learning theory according to their arithmetic complexity. Specifically, I will discuss the complexity of models of teaching. I will discuss the connections between algorithmic learning theory, grammatical inference and machine learning as well as the way research in the applied and theoretical research influence one another in the study of machine intelligence.

** February 13**

**Daphne Der-Fen Liu
**

California State University, Los Angeles

**Title: From
Integral
Sequences with Forbidden Differences to Graph
Coloring Problems **

**Abstract:**

Sequences with special patterns possess phenomenal properties. For instance, the Fibonacci sequence is directly related to the Golden Ratio and appears frequently in nature. In the early 70’s Cantor and Gordon introduced the parameter called density of integral sequences with forbidden differences. For a given set of positive integers D, a D-sequence is a sequence of integers such that the difference between any two terms does not fall in D. The maximum density of such a D-sequence is called the density of D, denoted by m(D). The parameter m(D) is closely related to the parameter k(D) involved in the so called "lonely runner conjecture" [Wills 1967, and Bienia et al. 1998], and coloring parameters of distance graphs introduced by Eggleton, Erdös, Skilton in mid-80’s. (For a set of positive integers D, the distance graph generated by D has all integers as the vertex set, and two vertices are adjacent if their absolute difference falls in D.)

We introduce close connections among the above parameters, and show how these connections are used to solve some open problems in these areas. In addition, we discuss recent results and open problems.

**February 19 (4:10pm -- Carver 268)**

**David Sivakoff
**

Ohio State University

**Title: Stochastic Dynamics
on Graphs**

**Abstract:**

Many physical and intangible structures, such as the internet and collaboration networks, can be abstracted as graphs. With the goal of understanding the various processes that take place on these structures, such as the spread of a virus or the cultivation of an idea, we study stochastic processes on graphs. I will discuss a few of these models, and outline some directions for future discovery.

**February 20 (4:10pm -- Pearson 2105)
**

**Megan Bernstein
**

Georgia Institute of Technology

**Title: ****Progress in showing cutoff for
random
walks on the symmetric group**

**Abstract:**

Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed distribution. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff with window for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk, generated by permutations with a binomially distributed number of two-cycles. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group. Results include joint work with Nayantara Bhatnagar, Evita Nestoridi, and Igor Pak.

**March 20**

**Susan Kelly
**

University of Wisconsin - La Crosse

** **

**Title: Two Women in
Mathematics who helped create a Path for Others**

**Abstract:**

**April 3**

**Anthony Bonato
**

Ryerson University

**Title: Graph Searching
Games and Probabilistic Methods**

**Abstract:**

The intersection of graph searching and probabilistic methods is a new topic within graph theory, with applications to graph searching problems such as the game of Cops and Robbers and its many variants, Firefighting, graph burning, and acquaintance time. Graph searching games may be played on random structures such as binomial random graphs, random regular graphs or random geometric graphs. Probabilistic methods may also be used to understand the properties of games played on deterministic structures. A third and new approach is where randomness figures into the rules of the game, such as in the game of Zombies and Survivors. We give a broad survey of graph searching and probabilistic methods, highlighting the themes and trends in this emerging area. The talk is based on my book (with the same title) co-authored with Pawel Pralat published by CRC Press.

**April 10**

**Shelby Nicole Wilson
**

Morehouse College

**Title: An ODE mixed-effect model of
vascular tumor growth with anti-angiogenic treatment
**

**Abstract:**

**April 17**

**Vladimir Sverak
**

University of Minnesota

** **

**Title: Calculations and theory for the
Navier-Stokes equations and their simpler models -
some recent examples**

**Abstract:**

**Numerical calculations can give important hints for theoretical PDE analysis. In this talk, we will discuss a few examples of calculations related to new theorems for PDEs of incompressible fluid mechanics.**

**April 24
**

**Daniel Erman
**

University of Wisconsin-Madison

**Title: Big polynomials rings**

**Abstract:**

There is a meta-principle in algebra that limits of free objects tend to be free themselves. We will consider this principle in the context of polynomial rings, taking the limit as the number of variables goes to infinity. I will then discuss how the principle sheds light on some famous conjectures in algebraic geometry and commutative algebra, such as Stillman’s Conjecture on projective dimension and Hartshorne’s Conjecture on complete intersections. This is joint work with Steven Sam and Andrew Snowden.