In this talk I will introduce two stochastic voting systems and results we proved. (i) Axelrod's model generalizes the voter model in which individuals have one of Q possible opinions about each of F issues and neighbors interact at a rate proportional to the fraction of opinions they share. We proved that on large two-dimensional torus if Q/F is small, then there is a giant component of individuals who share at least one opinion and consensus develops on this percolating cluster. (ii) The latent voter model allows a latent period after each site flips its opinion. We will present Shirshendu's result on a random r-regular graph with n vertices that as the rate of exponential latent period $\lambda \gg \log n$, dynamics converge to coexistence behavior with quasi-stationary density = 1/2 at $O(\lambda)$ times. Using different technologies one can generalize it to the varying degree case, a.k.a. the configuration models. Joint work with Rick Durrett and Shirshendu Chatterjee

Renal autoregulation stabilizes kidney functions and provides protection against blood pressure fluctuations. Autoregulation is mediated by two mechanisms: the *myogenic response,* where increased blood pressure elicits vascular constriction, and *tubuloglomerular feedback,* where salt excretion is balanced by adjustments of filtration rate. Coupling of the two mechanisms give rise to complex behaviours that are challenging to analyse. In the talk, I will describe a mathematical model of renal autoregulation, which represents both mechanisms and thus can be used to study the interactions developed among them. I will provide the necessary physiological background, and I will focus on the mathematical formulation of the involved processes. The talk will be accessible to everyone with basic understanding of differential equations.

Motivated by the dynamic feedback systems in the kidney, we consider time-delayed transport equations with stochastic left-hand boundary conditions. We first prove the existence and uniqueness of the steady-state solution for the deterministic case with sufficiently small delay (e.g., zero left-hand boundary). Likewise, we prove those for the stochastic case using the similar analytic techniques. In this talk we will show the process of model formulation with biological motivation and address the role of time-delay and stochastic boundary on the solution behaviors. The talk should be accessible to all graduate students who are familiar with basic ODE theory.

We will describe the configuration model and discuss what happens when the people's opinions and the connections in the network coevolve. Despite the combined efforts of James Gleeson, Peter Mucha, Bill Shi, David Sivakoff, Josh Socolar, Chris Varghese and myself, we cannot prove any rigorous results so the talk should be accessible to almost anyone.

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Motivated by several biological questions, we consider ODEs and PDEs with stochastically switching right-hand sides and PDEs with stochastically switching boundary conditions. In a variety of situations, we prove that the system exhibits surprising behavior. In this talk we will outline this class of problems and highlight some of the most interesting and unexpected results. The talk will be accessible to all graduate students.

Given a closed, convex curve C in the plane, a billiard path on C is a polygon P inscribed in C such that, at each vertex v of P, the two edges of P incident with v make equal angles with the tangent line to C at v. (Intuitively, this is the path a billiard ball would follow on a frictionless pool table bounded by C.) For most convex curves C, there are only a finite number of triangular billiard paths on C, a finite number of quadrilateral billiard paths, and so on. Obviously, when C is a circle, there are infinitely many closed billiard n-gons inscribed in C, but, surprisingly, the same is true when C is an ellipse. (This is a famous theorem due to Chasles.) The interesting question is whether there are other convex curves, besides ellipses, for which there are infinitely many closed billiard n-gons for some n. In this talk, I'll discuss the above-mentioned phenomenon and show how it is related to the geometry of non-holonomic plane fields (which will be defined and described). This leads to some surprisingly beautiful geometry, which will require nothing beyond multivariable calculus from the audience.

I will begin by recounting how I got involved in an industrial collaboration with a company that makes glass tempering furnaces and how younger grad students can seek such opportunities. I will then describe the mathematical model I developed for the company while highlighting challenges that arose due to differences in culture and priorities between academia and industry.

The advection-diffusion equation commonly arises in the tracking of substances that are carried by and diffuse in a fluid. My interest lies in models of biological flows, where the equation is defined on domains representing the lumen of blood vessels, tubules, or ducts, which owing to its complex geometry and deformation, has a significant impact on the underlying substance that cannot be ignored. In the talk, I will go through the basic steps involved in the development of a numerical method for the solution of the advection-diffusion equation on such domains with an emphasis on domains with moving boundaries. To handle the difficulties introduced by the boundary, the proposed method will follow the finite volume formulation.

The study of surfaces with constant mean curvature (CMC) goes back to 1841 when Delaunay classified all CMC surfaces of revolution. There has been consistent work on finding CMC hypersurfaces in various ambient manifolds. In this talk, we will discuss some nice properties of CMC surfaces, and then the existence of CMC surfaces in the Schwarzschild, and in general, asymptotically flat manifolds.

According to the NSF Survey of Earned Doctorates, of the Mathematics PhDs earned in 2011, 37% had an offer for postdoctoral research, 33% had definite employment in either academic, government, or industry positions, and 28% were seeking employment/postdoc. This means that although 100% of you are doing research right now, most likely only a third of you will continue to do research in a university setting. That means (a) getting a postdoc will be competitive and/or (b) you need to familiarize yourself with non-postdoc opportunities. To help you in this process, we've assembled a panel of local experts who have recently been through the job application process: Christine Berkesch (research faculty), Emily Braley (teaching faculty), Liz Munch (postdoc and non-academic), as well as senior faculty who can talk about qualities that are sought in both university (Mike Reed) and government research (Bill Pardon) settings. Come and learn what you can do now and in the future to make your job search more successful!

Tropical Geometry is a relatively recent field, that uses combinatorics to study algebraic geometry. In this talk, I'll introduce the tropical semiring, and the idea of tropical monomials and polynomials. We'll look at some examples of tropical hypersurfaces, and if there is time I will introduce the idea of a tropical variety, and it's relationship to the traditional algebraic variety of a curve.

Is it true that two polygons in the plane have the same area if and only if they can be decomposed into congruent polygons? What about in three and higher dimensions? And what about the analogous questions for polygons in the hyperbolic plane and polyhedra in higher dimensional hyperbolic spaces? Some aspects of the subject are elementary, while others involve Abel's dilogarithm and arcane subjects, such as algebraic K-theory. This talk will be largely elementary, and fun.

More than 95% of the present day curvature of the universe is not a result of regular baryonic matter represented by the periodic table of elements. About 73% is well described by a geometrically natural cosmological constant, also referred to as dark energy, which results in a very small amount of curvature uniformly spread throughout the universe. We will explore the possibility that the remaining 23%, commonly referred to as dark mater, could also be explained very naturally from a geometric point of view.

All right, brain. You don't like me and I don't like you, but let's just do this and I can get back to killing you with trace formulae. -Homer Simpson (misquoted) We will discuss trace formulae starting with the Poisson summation formula and working towards the case of compact locally symmetric spaces. No background is assumed. Oh, and I'll bring beverages (both the big kid and little kid kind).

Let $E$ be a non-CM elliptic curve defined over $\mathbb{Q}$. For each prime $p$ of good reduction, $E$ reduces to a curve $E_p$ over the finite field $\mathbb{F}_p$. In this talk, we are particularly interested in ssquare-free values of two sequences: $f_p(E) =p + 1 - a_p(E)$ and $f_p(E) = a_p(E)^2 - 4p$, where $a_p(E)=p+1-|E(\mathbb{F}_p)|$. More precisely for any fixed curve $E$, we first give an upper bound for the number of primes $p$ up to $X$ for which $f_p(E)$ is square-free. Second, we show that the average results on this prime counting function are compatible with the corresponding conjectures at the level of the constants, i.e., whether the average of the conjectured constants is equivalent to the constant obtained via the average conjecture. This is joint work with S. Akhtari, C. David and L. Thompson.

Abstract: In this talk I will give a quick overview of Einstein's theory of general relativity. I will then move on to discuss the mystery of dark matter: why astrophysicists think it's out there in the universe and what phenomena any successful theory of dark matter will have to explain. One such phenomenon is the Tully-Fisher relation, an intriguing correlation between the visible mass of galaxies and the rotational velocities of their stars. I will wrap up by describing a theory of wave dark matter and how it could possibly explain the Tully-Fisher relation.

The theory of theta functions, which are defined by certain Fourier series, was developed by great mathematicians like Jacobi and Riemann. Among the numerous applications of this theory are certain results in projective geometry pertaining to complex tori. In this talk, we will focus on the 1-dimensional case and briefly discuss the higher-dimensional story towards the end.

We will discuss techniques for approximating a point in high-dimensional Euclidean space which is close to a known low-dimensional compact submanifold when only a compressed linear sketch of the point is available. More specifically, given a point, x, close to a known submanifold of R^D, we will consider linear measurement operators, M: R^D -> R^m, which have associated nonlinear inverses, A: R^m -> R^D, so that || x - A(Mx) || is small even when m << D. Both the design of good linear operators, M, and the design of stable nonlinear inverses, A, will be discussed. An algorithmic implementation of a particular nonlinear inverse will be presented, along with related stability bounds for the approximation of manifold data.

Many algebraic objects are notorious for being easy to define, but hard to find explicitly. However, certain algebraic objects, when viewed with the "correct" combinatorial framework, become much easier to actually find. This allows us to compute much larger examples by hand, and often gives us insight into the object's underlying structure. In this talk, we will define irreducible decompositions of ideals, and explore their underlying combinatorial structure in the special case of monomial ideals in polynomial rings. As time permits, we will look at recent results in the case of binomial ideals. This talk will be accessible to anyone who has taken a course in Abstract Algebra.

A body of literature has developed concerning “cloaking by anomalous localized resonance”. Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. In this talk, we will discuss a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source plays a crucial role in determining whether or not resonance occurs. The talk assumes minimal background knowledge.

Given a (n x n)-matrix M over a commutative integral domain R, one can try to associate to it a diagonal matrix called the Smith Normal Form of M. This can be done when R is the ring of integers, or the polynomial ring F[x] over a field F, and various applications of the existence of the Smith Normal Form are discussed in matrix theory. Which commutative integral domains R have the property that every matrix with coefficients in R admits a Smith Normal form? This is a very old question, as for instance Wedderburn in 1915 already discussed the case where R is the ring of holomorphic functions. I will review all necessary concepts, and discuss several easily stated open problems in this circle of ideas.

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Group theory is one of the basic topics of abstract algebra and therefore probably well-known. In this talk, we are going to introduce left-orders on groups and expand the fundamental theorem of homomorphisms from groups to ordered groups. We will see some examples of left-ordered groups that show different levels of orderability and, time permitting, formulate a theorem of A. Rhemtulla which discusses the existence of torsion-free groups without any order.

We introduce a novel kind of percolation on graphs called jigsaw percolation intended as a simple model for collaborative problem solving and merging of ideas. Each node in a network (regarded as a person in a social network) has a unique piece of a jigsaw puzzle. At each stage, components of connected puzzle pieces merge if at least two people are adjacent in the social network and at least two puzzle pieces can join. I will discuss our recent results on this model, outline the proofs, and discuss some open problems. This is joint work with Charles Brummitt, Shirshendu Chatterjee and Partha Dey.

Biological physical networks, especially those involved in resource delivery and distribution, often exhibit a hierarchical structure. Quantifying this structure is crucial to obtaining a better understanding of the processes underlying the network formation, and such a quantification has long been obtained using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the ``root'' of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. In this talk I will present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate with flow capacity. The new method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. I will show that the sensitivity of the hierarchical levels to weight perturbations can be analyzed in a rigorous way. I will also discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.

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Math Slam

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