Abstract: Newton wrote in Lemma 28 of his Principia that there is no oval figure in R^2, whose cutoff areas by lines can be found by means of polynomial equations involving parameters of the line. Arnold formulated a general conjecture that the only infinitely smooth convex domains in R^n, whose cutoff volume by hyperplanes is an algebraic function of the parameters of the hyperplane, are ellipsoids in odd dimensions n. Recently, in a paper by Agranovsky and in our paper with A.Merkurjev and V.Yaskin, Arnold's conjecture was confirmed in a special case where the cutoff volume is a polynomial of the parameters of the hyperplane. In this talk I will present the history of the problem and the main ideas of the proof.

• Xiaoyuan Chang (Harbin University of Science and Technology)

Title: Analysis of a stoichiometric bacteria-grazer reaction-diffusion model for organic matter biodegradation Abstract: The dynamical behaviors of a bacteria-carbon-nitrogen reaction-diffusion system with grazers and without grazers are investigated in this paper. It is shown that the bacteria death rate and the grazer death rate play a decisive role in the dynamics of the system. The existence, stability and persistence of the solutions are proved by applying the theorem of existence of eigenvalue, the local and global bifurcation theory, and the abstract persistence theory. Through numerical simulations, the transient dynamic and the oscillation phenomenon are obtained.

• Junping Shi (William & Mary)

Title: Modeling animal movement with memory with partial differential equations with time-delay Abstract: Animal populations often self-organize into territorial structure from movements and interactions of individual animals. Memory is one of cognitive processes that may affect the movement and navigation of the animals. We will review several mathematical approaches of animal spatial movements, and also introduce our recent work using partial differential equations with time-delay to model and simulate the memory-based movement. We will show the bifurcation and pattern formation for such models. It is based on joint work with Chuncheng Wang, Hao Wang, Xiangping Yan, Qingyan Shi and Yongli Song.

• Wei Meng (W&M): Linear connectivity for tournaments to be highly linked

Abstract: A digraph is k-linked if for any two disjoint sets of vertices x_1,..., x_k and y_1,...,y_k there are vertex disjoint paths P_1,...,P_k such that P_i is directed from x_i to y_i for i=1, ..., k. Pokrovskiy in 2015 proved that every strongly 452k-connected tournament is k-linked. In this paper, we significantly reduce this connectivity bound and show that any (24k-19)-connected tournament is k-linked. This is joint work with Martin Rolek, Yue Wang, and Gexin Yu.

• Yue Wang (W&M): Enhancing Erdos-Lovasz Tihany conjecture for line graphs of multigraphs

Abstract: let s,t be integers. A graph G is (s,t)-splittable if the vertices of G can be partitioned into two sets S and T such that the chromatic number of subgraph induced by S (denoted by G[S]) is at least s and the chromatic number of subgraph induced by T (denoted by G[T]) is at least t. The well-known Erdos-Lovasz Tihany Conjecture from 1968 states that every graph G whose chromatic index equals s+t-1 and larger than its clique number is (s,t)-splittable. This conjecture is hard, and only known to be true for line graphs, quasi-lines, and graphs with independent number 2. In this paper, we prove an enhanced version of the conjecture for line graphs of multigraphs. More precisely, we show that for s,t greater than 3.5r+2, every graph G whose chromatic index equals s+t-1 and larger than its clique number is (s,t+r)-splittable. Moreover, when r=1, we obtain the same result for s,t at least 4. This is joint work with Gexin Yu.

Abstract: We shall present recent development in sparse machine learning. In particular, we shall elaborate the needs of developing sparse machine learning methods with a illustration by simulation results. Machine learning methods in reproducing kernel Banach spaces will be discussed. Numerical results will be presented to demonstrate the effectiveness of sparse learning methods.

Abstract: Rigidity, arising from mechanics, is the property of a framework that does not flex. A combinatorial characterization of rigidity in the Euclidean plane has been obtained by Laman in 1970. In this talk, we will present sufficient conditions of rigidity as well as some applications in graph theory.

Abstract: One of the deepest problems in graph theory is Hadwiger's Conjecture, which asserts that every k-chromatic graph contains a clique minor of order k for each positive integer k. The conjecture is confirmed to be true when k is at most 6, and in fact, it is equivalent to the Four Color Theorem when k equals 5 or 6, and remains wide open for k at least 7. In this talk, we will explore a weaker version of the conjecture: what's the function f(k) so that an f(k)-chromatic graph contains a k-connected minor? No prior knowledge in graph theory will be assumed in this talk. This is based on joint work with Runrun Liu and Martin Rolek.

Abstract: Reaction-advection-diffusion models that capture the interactions between plants, surface water and soil moisture can qualitatively reproduce community-scale vegetation patterns that are observed in dryland ecosystems. On gently sloped terrain, these patterns often appear as bands of vegetation growth alternating with bare soil. The vegetation bands can be tens of meters thick with spacing on the order of a hundred meters, and form a regular striped pattern that often occupy tens of square kilometers on the landscape. I will focus on aspects of the surface/subsurface water dynamics within these models. Capturing these hydrological processes on appropriate timescales may allow us to better utilize observational data as we work to identify the dominant mechanisms underlying the formation of dryland vegetation patterns and understand how environmental factors influence pattern characteristics.

Abstract: It is a fundamental principle in Analysis that taking into account the symmetries of an object simplifies the study of functions on that object. Starting from elementary examples and the classical theory of Fourier series, we will explore various instances of this principle and its natural connections to the theory of groups and their representations. Following Alain Connes' philosophy of noncommutative geometry, certain spaces of representations will be studied by means of algebras of linear operators, leading to the characterization of a new class of representations, of particular topological relevance. Unexpected symmetries will be shown to appear in that context.

Description: In this special mathematics colloquium, several recent mathematics alumni will discuss their career path and how mathematics has helped them to achieve their goals.

Abstract: Let G be a finite group. The Frobenius--Schur indicator of an irreducible character $\chi$, denoted $\varepsilon(\chi)$, is defined as $\varepsilon(\chi) = \frac{1}{|G|} \sum_{g \in G} \chi(g^2)$. It is known that $\varepsilon(\chi)=1, -1,$ or $0$, where $\varepsilon(\chi) = 0$ precisely when $\chi$ is not real-valued. When $\chi$ is real-valued, $\varepsilon(\chi) = 1$ if $\chi$ is afforded by a representation that may be defined over the real numbers, otherwise $\varepsilon(\chi) = -1$. In this talk we outline a computational method used to prove that the exceptional groups $F_4(q)$, $E_7(q)_{\textrm{ad}}$, and $E_8(q)$ have no irreducible characters with Frobenius--Schur indicator -1.

Abstract: When F is a field and R an F-algebra, it is often important to classify the R-module structures carried by some fixed F-vector space V. We will describe, in down-to-earth terms, a classical approach to that problem, which involves linear algebra, group theory, algebraic geometry, and homological algebra. Using tools from algebraic topology, we will then sketch new developments of these techniques and their application to the proof of a result in commutative algebra, which completely answers a conjecture posed by Vasconcelos in 1974.

Abstract: The study of moments of the Riemann zeta-function is important and has been very active. For example, fine estimates of these moments would yield the Lindelof Hypothesis, which is one of the central unsolved problems in number theory. In this talk, I will discuss the history and some recent progress of moments.

Abstract Abstract: We study efficient quantum error correction schemes for the fully correlated channel on an n-qubit system with error operators that assume the form $\sigma_x^{\otimes n}$, $\sigma_y^{\otimes n}$, $\sigma_z^{\otimes n}$. In particular, when 2k+1 is odd, we have a quantum error correction scheme using one arbitrary qubit $\sigma$ to protect the data state $\rho$ in the 2k-qubit system. When n=2k+2 is even, we have a hybrid quantum error correction scheme that protects a 2k-qubit state $\rho$ and 2-classical bits. The scheme was implemented using Matlab, Mathematica and the IBM's quantum computing framework qiskit. Note: Problems and results will be described in terms of elementary matrix theory. No quantum mechanics background is needed. Co-authors: Seth Lyles, and Yiu-Tung Poon

Abstract: For a graph G, a graph labeling is an assignment of natural numbers to the vertices V(G) and/or edges E(G) of G in such a way that certain conditions hold. Many classic and well studied problems in graph theory and computer science can be phrased in terms of graph labelings. We will discuss some specific graph labelings, in particular the complexity of transforming one labeling into another subject to certain conditions. Many special cases of this have interesting connections to group theory.

Abstract: In reaction-diffusion models describing biological and chemical interactions, some dispersal and interaction can be of nonlocal nature. First we show that in some models from cellular biology or ecology depending on the spatial average of density functions instead of local density functions, such nonlocal spatial average can induce instability of constant steady state, which is different from classical Turing instability. In particular, for systems of two equations containing spatial averages, spatially non-homogeneous time-periodic orbits could occur through bifurcations from the constant steady state. Examples from a nonlocal predator-prey model and a pollen tube tip model will be used to demonstrate such bifurcations. In another direction, we show that when a averaging nonlocal dispersal occurs instead of classical diffusion, how the mechanism of Turing diffusion-induced instability and pattern formation changes.

Abstract: Advancements in geographical information systems have enabled scientists to collect data of unprecedented size over space. Nowadays, these spatial data commonly arise in diverse fields as biology, engineering, health sciences, environment and information technology. In many of these studies, data are collected on a count or binary response with spatial covariate information. In this talk, we will introduce a new class of generalized geo-additive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is used to analyze the crash counts data in the Tampa-St. Petersburg urbanized area in Florida.

Abstract: Quantum mechanics is a physical theory that offers a description for how physical systems (particularly very small ones) behave. It is at once curious, counter-intuitive, and remarkably accurate. In this talk, I will discuss the theory of quantum information, which provides an abstraction of the information-theoretic aspects of quantum mechanics. This abstraction forms the mathematical foundation of quantum information science, including quantum computation and quantum cryptography, which hold the potential for transformative changes in the way we process information. The theory of quantum information is also a beautiful and highly interdisciplinary subject that incorporates techniques from many areas of mathematics, such as functional analysis, matrix theory, abstract algebra, algebraic geometry, information theory, and computational complexity theory. In addition to describing some of its basic aspects, I will also discuss a few active areas of research and tantalizing open problems in quantum information theory.

Abstract: In this talk, I will present two nonlinear models for microbial dynamics vis-a-vis environmental health. Firstly, I will present a stoichiometric organic matter decomposition model in a chemostat culture that incorporates the dynamics of grazers. This mechanistic biodegradation model leads to reliable and suggestive ecological insights in the preservation and restoration of our fragile ecosystems. Using the model, I answer the following research questions: (i) What mechanisms allow microbes and resources to persist uniformly or go extinct? (ii) How do grazing and dead microbial residues affect decomposition? Secondly, I will talk about a stoichiometric mathematical model to predict methane emissions from oil sands tailings. Microbial metabolism of fugitive hydrocarbons produces greenhouse gas (GHG) emissions from oil sands tailings ponds and end pit lakes that retain semisolid wastes from surface mining of oil sands ores. Predicting GHG production, particularly methane, would help oil sands operators mitigate tailings emissions and would assist regulators evaluating the trajectory of reclamation scenarios.

3-4pm: Mikhail Chebotar (Kent State University): Around the Koethe Conjecture. Host: Chi-Kwong Li

Abstract: The Koethe Conjecture (whether a sum of two left nil ideals is nil) is one of the most famous open problems in Ring Theory and it inspired many interesting questions. We will discuss some recent progress and new directions for research in this area.

Abstract: The strange attractors of chaotic systems are among the most beautiful and intricate objects in all of mathematics (picture the famous Lorenz ``Butterfly'' attractor). Yet, rarely are these attractors completely understood, primarily because of the contrasting behaviors that these systems exhibit at local vs. global levels. How can something be both locally unstable and globally stable? In this talk, we will explore the local geometry of chaotic attractors in an effort to better understand their unpredictable nature. To do this, we will develop a method for determining the chaoticity of local regions around an attractor, and we will tailor the method according to the underlying nonlinear geometry of these attractors.

Abstract: The talk will discuss recent development on the existence of two- and three-dimensional solitary or multi-solitary surface waves on the water of finite depth with or without suface tension using the exact governing equations (called Euler equations). It will be shown that when the nondimensional wave-speed and surface tension are in various regions, the Euler equations possess several different kinds of two- or three-dimensional solitary or multi-solitary wave solutions. Moreover, some stability results for these waves will be addressed, such as transverse instability, spectral stability, asymptotic linear stability or conditional stability. The talk is accessible to non-experts or graduate students.

Abstract: The Riemann Hypothesis is an important unsolved problem in mathematics. It asserts that all the nontrivial zeros of the Riemann zeta-function have real part 1/2. The critical points of the Riemann zeta-function are also interesting, because they are intimately connected to the zeta zeros. In this talk, we will give an introduction to the Riemann zeta-function, discuss the distribution of the zeros and critical points, and show how they are related to some central problems in analytic number theory.

Abstract: In order to understand a complicated object in mathematics or in the real world, a straightforward but very useful approach is to separate the noise from the essence. This naturally leads to the following question: given a function f, is it possible to decompose it as f=f1+f2, where f1 is a function having an easy-to-explain structure, and f2 behaves like a random function?

The establishment of the Structure Theorems provided theoretically supports for this simple philosophy. Nowadays, Structure Theorems have been established in various areas in mathematics, including combinatorics, number theory and dynamical systems, each of which proves to have many important applications.

In this talk, I will introduce the development of the Structure Theorems in the areas mentioned above, with an emphasize on their applications in dynamical systems. This talk is based on joint works with Sebastian Donoso.

Abstract: Motivated by recent advances in technology for medical imaging and high-throughput genotyping, we consider an imaging genetics approach to discover relationships between the interplay of genetic variation and environmental factors and measurements from imaging phenotypes. We propose an image-on-scalar regression method, in which the spatial heterogeneity of gene-environment interactions on imaging responses is investigated via an ultra-high-dimensional spatially varying coefficient model (SVCM). Bivariate splines on triangulations are used to represent the coefficient functions over an irregular two-dimensional (2D) domain of interest. For the proposed SVCMs, we further develop a unified approach for simultaneous sparse learning (i.e., G×E interaction identification) and model structure identification (i.e., determination of spatially varying vs. constant coefficients). Our method can identify zero, nonzero constant and spatially varying components correctly and efficiently. The estimators of constant coefficients and varying coefficient functions are consistent and asymptotically normal. The performance of the method is evaluated by Monte Carlo simulation studies and a brain mapping study based on the Alzheimer's Disease Neuroimaging Initiative (ADNI) data.

Abstract: This talk will discuss two closely related problems, one in graph theory and one involving matrix rings. Given vertices $u$ and $v$ in a directed graph (digraph) $\Gamma$, we say that the ordered pair $(u, v)$ is a reachable pair if there exists a path of directed edges from $u$ to $v$. One may ask: if the digraph $\Gamma$ has $n$ vertices, then how many reachable pairs could $\Gamma$ contain? To answer this question, we translate it into an algebraic form and consider the problem of counting the number of nonzero entries in certain rings of $n \times n$ matrices. No prior knowledge will be assumed, and the talk should be accessible to undergraduates. This is joint work with Eric Swartz.

Abstract: Amy Xia will share her research project that designs a sourcing plan with supply chain sustainability performance and operational factors (e.g. cost structure, capacity, supply risk, and delivery time of the supplier) as decision criteria. The sourcing plan selects suppliers from multiple potential candidates to form a supply chain and determines the investment and order allocation among these selected suppliers in order to achieve both high sustainability performance and cost efficiency. Environmental, Social, and Governance (ESG) index is adopted to quantify a supply chain's sustainability performance; a frontier approach is used to provide a set of effective solutions, of which none is superior or inferior. This is cast as a nonlinear integer-programming problem. Special features of the problem are discovered; an effective algorithm to solve the problem is then proposed. Numerical results verify that the proposed algorithm outperforms an existing algorithm, and provide managerial insights into how sustainability considerations alter the supply chain sourcing planning. A simulation of Apple's sourcing decisions with iPhone 6 provides additional confirmation of the effectiveness of the proposed approach.

Abstract: An L-function is a type of generating function with multiplicative structure which arises from either an arithmetic-geometric object (like a number field, elliptic curve, abelian variety) or an automorphic form. The Riemann zeta function is the prototypical example of an L-function. While L-functions might appear to be an esoteric and special topic in number theory, time and again it has turned out that the crux of a problem lies in the theory of these functions. Many equidistribution problems in number theory rely on one's ability to accurately bound the size of L-functions; optimal bounds arise from the (unproven!) Riemann Hypothesis for the Riemann zeta function and its extensions to other L-functions. I will discuss some motivating equidistribution problems along with recent work (joint with K. Soundararajan) which produces new bounds for L-functions by proving a suitable "statistical approximation" to the (extended) Riemann Hypothesis.

Abstract: Start with four circles, all tangent to one another; then fill in the gaps between them with additional tangent circles. If you keep filling the gaps with smaller and smaller circles, you will generate an Apollonian circle packing. This picture has a rich history, from ancient Greece to Rene Descartes, to Japanese temple geometry. Amazingly, if you start with four circles whose curvatures are integers, then all the circles in the packing have this property. In my talk I'll describe some of the number theory that's been inspired by Apollonian packings in the last 20 years: theorems and conjectures about the growth of curvatures in a packing, and which integers can appear as curvatures. I will finish with my own work on the domain of multivariable power series defined by Apollonian packings.

Abstract: How many edges may a graph with no triangle have? Given a graph F, the Turan problem asks to maximize the number of edges in a graph on n vertices subject to the constraint that it does not contain F as a subgraph. In this talk, we will discuss constructions for this problem coming from finite geometry (eg using projective planes), combinatorial number theory, and "random polynomials".

Abstract: We study an efficient numerical method for solving difficult `saddle point' linear systems that arise at every time step in the discretization of incompressible flow problems, including those modeled by the Navier-Stokes equations (e.g. water, oil, air under 220 mph) and magnetohydrodynamics (flows on conducting fluids). By combining an algebraic splitting of the block saddle point matrix, a particular approximation of the Schur complement system, and an incremental version of the associated time stepping algorithm, we are able to decompose the linear systems into smaller pieces that are easier to solve. We prove that the approximations made in the solve process are third (or fourth) order, and so are appropriate for use with second order time stepping methods. Numerical tests are performed which verify excellent performance of the methods on a variety of test problems.

Abstract: Ramsey theory dates back to the 1930's and computing Ramsey numbers is a notoriously difficult problem in combinatorics. We study Ramsey numbers of graphs under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph such that no triangle has all its edges colored differently. Given a graph H and apositive integer k, the Gallai-Ramsey number of H is the least positive integer N such that every Gallai coloring of the complete graph K_N using k colors contains a monochromatic copy of H. Gallai-Ramsey numbers of graphs are more well-behaved, though computing them is far from trivial. In this talk, I will present our recent results on Gallai-Ramsey numbers of cycles.

Abstract: Quantum information theory has emerged at the junction of multiple disciplines, blending concepts and techniques from physics, mathematics, and computer science. At the heart of this new field is the effort to understand and answer the following questions: How is information stored and manipulated in a quantum system, and how well is this information preserved under physical processes? In this talk, I will introduce some key notions in quantum information using mathematical formalism, including quantum states, quantum channels, and quantum entanglement. Some of the mathematical tools needed to understand the problems of detecting and manipulating entanglement will be presented.

Abstract: In finite dimensional quantum information, transformations between systems are represented by quantum channels: completely positive and trace preserving linear maps between matrix spaces. Single-shot quantum channel discrimination is the task of determining which of two known channels is acting on a system, given only a single use. We will review how entanglement can be used in this task to gain an advantage, and how this phenomenon is directly connected to properties of norms measuring the distance between the channels. In particular, the advantage provided by entanglement is quantified by the gap between the completely bounded trace norm and the induced trace norm. We will discuss recent results related to these norms and single-shot quantum channel discrimination, as well as open problems.

Abstract: A beautiful example of spontaneous pattern formation appears in the distribution of vegetation in some dry-land environments. Examples from Africa, Australia and the Americas reveal that vegetation, at a community scale, may spontaneously form into stripe-like bands, alternating with striking regularity with bands of bare soil, in response to aridity stress. A typical length scale for such patterns is 100 m; they are readily surveyed by modern satellites (and explored from your armchair in Google maps). These ecosystems represent some of Earth’s most vulnerable under threats of desertification, and some ecologists have suggested that the patterns, so easily monitored by satellites, may have potential as early warning signs of ecosystem collapse. I will describe efforts based in simple mathematical models, inspired by decades of physics research on pattern formation, to understand the morphology of the patterns. I will also describe efforts at analyzing the patterns via the satellite images, which, in some cases, we can accurately align with the aerial survey photographs from the 1950s to investigate details of the pattern evolution.

Abstract: Combining statistical parametric maps (SPM) from individual subjects is the goal in some types of group-level analyses of functional magnetic resonance imaging (fMRI) data. Brain maps are usually combined using a simple average across subjects, making them susceptible to subjects with outlying values. Furthermore, t tests are prone to false positives and false negatives when outlying values are observed. We propose a regularized unsupervised aggregation method for SPMs to find an optimal weight for aggregation, which aids in detecting and mitigating the effect of outlying subjects. We also present a bootstrap-based weighted t test using the optimal weights to construct an activation map robust to outlying subjects. We validate the performance of the proposed aggregation method and test using simulated and real data examples. Results show that the regularized aggregation approach can effectively detect outlying subjects, lower their weights, and produce robust SPMs.

Abstract: In the Hilbert space formulation, quantum states are density matrices, i.e., positive semidefinite matrices with trace one, and quantum channels are trace preserving completely positive linear maps on matrices. In this talk, we will present some results on the existence of quantum channels that send certain quantum states to other quantum states. Additional conditioons on the quantum channels may be imposed to satisfy certain specific requirements.

Abstract: We consider bootstrap percolation in tilings of the plane by regular polygons. First, we determine the percolation threshold for each of the infinite Archimedean lattices. More generally, let T denote the set of plane tilings t by regular polygons such that if t contains one instance of a vertex type, then t contains infinitely many instances of that type. We show that no tiling in T has threshold 4 or more. This material is self-contained, and requires no particular background. We'll share many open problems, as well as the intuition behind these results.

Title: Non-revisiting Paths in Polyhedral Map on Surfaces

Abstract: This will be an introduction to how the graph (of a real symmetric matrix, or a general matrix) constrains the multiplicities of its eigenvalues. The case of trees is most interesting and this will be described in some detail, including the maximum multiplicity, the minimum number of distinct eigenvalues, the possible lists of multiplicities and how they come about. This talk will be an overview of the subject and a second talk in the GAG seminar will continue the description. The subject has just been covered in a new book from Cambridge University Press (same title), and REU students have helped to make some very important contributions over the years. There is still plenty of work to be done in the area, which has been of interest to algebraic graph theorists and numerical analysts, as well as matrix theorists.