Many classes of mathematical objects turn out to be classified in the same way --- two infinite series and 3 "exceptional" objects. These include symmetries of 3-dimensional solids, rigid singularities, certain types of Lie algebras, positive definite even intersection forms, etc. Discovering why such classes should have the same classification has led to many beautiful ideas and observations. I will give a review of some of the basic ideas (assuming very little in the way of prerequisites) and I may have time to say why string theory has been important in this context.

The Positive Mass Theorem in general relativity states that a spacelike hypersurface of a spacetime satisfying the dominant energy condition must have nonnegative total mass. In the special case in which the hypersurface is totally geodesic, local energy density coincides with scalar curvature, and the above theorem becomes a purely geometric statement about complete, asymptotically flat Riemannian manifolds. I will try to present the necessary background for one to understand the statement of the theorem. I will also discuss attempts to better understand the relationship between scalar curvature and total mass. Note that this talk is especially geared towards early graduate students and people specializing in other fields, and thus I will assume no previous knowledge of smooth manifolds, Riemannian geometry or general relativity.

I'll be talking about how to construct the moduli space for genus g Riemann surfaces with r boundary components. I'll draw lots of pictures and focus a lot of attention on genus 1 Riemann surfaces with 1 boundary component. As an application, I'll probably talk about H^1(SL2(Z)) with coefficients in various representations--and the correspondence to modular forms (briefly, and without a whole lot of proofs).

Many systems employ sensors to interpret the environment. The target-tracking task is to gather sensor data from the environment and then to partition these data into tracks that are produced by the same target. The goal of sensor fusion is to gather data from a heterogeneous collection of sensors (e.g, audio and video) and fuse them together in a way that enriches the performance of the sensor network at some task of interest. This talk summarizes two recent efforts that incorporate mildly sophisticated mathematics into the general sensor arena, and also comments on the joys and pitfalls of trying to apply math for customers who care much more about the results than the math. First, a key problem in tracking is to 'connect the dots:' more precisely, to take a piece of sensor data at a given time and associate it with a previously-existing track (or to declare that this is a new object). We use topological data analysis (TDA) to form data-association likelihood scores, and integrate these scores into a well-respected algorithm called Multiple Hypothesis Tracking. Tests on simulated data show that the TDA adds significant value over baseline, especially in the context of noisy sensor data. Second, we propose a very general and entirely unsupervised sensor fusion pipeline that uses recent techniques from diffusion geometry and wavelet theory to compress and then fuse time series of arbitrary dimension arising from disparate sensor modalities. The goal of the pipeline is to differentiate classes of time-ordered behavior sequences, and we demonstrate its performance on a well-studied digit sequence database. This talk represents joint work with many people. including Chris Tralie, Nathan Borggren, Sang Chin, Jesse Clarke, Jonathan deSena, John Harer, Jay Hineman, Elizabeth Munch, Andrew Newman, Alex Pieloch, David Porter, David Rouse, Nate Strawn, Adam Watkins, Michael Williams, Lihan Yao, and Peter Zulch.

I will review some basics of crystal bases for highest-weight representations for a semisimple Lie algebra. I will also point to some connections with number theory through Fourier coefficients of Eisenstein series, mostly in type A.

The Patlak-Keller-Segel equations (PKS) are widely applied to model the chemotaxis phenomena in biology. It is well-known that if the total mass of the initial cell density is large enough, the PKS equations exhibit finite time blow-up. In this talk, I will present some recent results on applying additional fluid flows to suppress chemotactic blow-up in the PKS equations.

This talk will be concerned with two problems. The first is the zeros of the derivatives of. Kac random polynomials K_n, which is a random polynomial with i.i.d. coefficients. It has been shown that the empirical measure of zeros of K_n will converge to the uniform measure on the unit circle of complex plane. Same convergence holds true for nay fixed order of derivative of K_n. In a joint work with Renjie Feng, we show if we consider the N_n-th order of derivative of K_n, then asymptotic behavior of empirical measure of this derivative will depend on the limit of \frac{N_n}{n}. In particular, as long as this ratio is greater than 0, the phenomenon of ‘zeros clustering around unit circle’ breaks down. The second talk is about Average Nearest Neighbor Degree (ANND), which is a measure for the degree-degree correlation for complex network. We shall be concerned with the probabilistic properties of ANND in the configuration model. We prove if the variable X generating the network has order of moment larger than 2, then the ANND(k) will converge uniformly to μ2/μ1, where μ2 is the second moment of X, and μ1 is the first moment. For the case that X has infinite variance, we show the pointwise (i.e., for fixed k) scaled convergence of ANND(k) to a stable random variable. This is joint work with Nelly Litvak and Pim van der Hoorn. More recently, Clara Stegehuis showed that when X is sample from the Pareto distribution, then one can obtain a complete spectrum of ANND(k) for the erased configuration model.

Mapping class groups are topological objects which can be used to describe the continuous symmetries of a surface. On the other hand, every compact orientable surface has a moduli space, a complex variety whose points parametrize all of its inequivalent complex structures. These concepts turn out to be closely related. In this talk, we'll cover the basics of both mapping class groups and moduli of Riemann surfaces, as well as explore their relationship.

Locally Linear Embedding(LLE), is a well known manifold learning algorithm published in Science by S. T. Roweis and L. K. Saul in 2000. In this talk, we provide an asymptotic analysis of the LLE algorithm under the manifold setup. We establish the kernel function associated with the LLE and show that the asymptotic behavior of the LLE depends on the regularization parameter in the algorithm. We show that on a closed manifold, asymptotically we may not obtain the Laplace-Beltrami operator, and the result may depend on the non-uniform sampling, unless a correct regularization is chosen. Moreover, we study the behavior of the algorithm on a compact manifold with boundary. This talk is based on the joint work with Hau-tieng Wu.

THIS JUST IN - An Abstract: I plan to speak about the good old Calabi Conjecture, and its beautiful solution by Yau, that gave gim the Fields Medal. I will start with some background material, and see how far we can get into the proof!

An introductory lecture to the Conley Index theory. We consider the flow case and introduce the key object of study: an index pair of an isolated invariant set. Index pairs are robust under perturbations and their homotopy type is invariant, making them an ideal tool for problems with error terms or even data-generated systems. The relevant tools are algebraic topology and some knowledge of continuous flows.

The ear from the outside in. Eardrum, middle ear, cochlea, 8th nerve, brainstem, cortex. What happens anyway when you listen to Mozart or Van Halen? How do pressure waves become electrical signals? What happens next? Is there deep mathematics in the auditory system? And what are those carteliginous things doing flapping in the breeze on the side of your head? Who says an abstract has to have declarative sentences? Will some of these questions be answered? Come and see!

No abstract yet.

The prevailing numerical methods for solving the reaction-diffusion systems in models of cardiac electrical activity currently use second-order adaptive mesh refinement, refining the spatial and temporal meshes only near the traveling wavefront(s). However, in two and three spatial dimensions under biologically relevant initial conditions and forcing, these wavefronts can constitute a relatively high percentage of the computational domain, limiting the effectiveness of the scheme. In this talk, I will present a numerical scheme based on higher order finite elements and spectral deferred correction designed to improve the efficiency in computing for domains of cardiac cells.

I will motivate the need to analyze data sets in high dimensions, their geometrical properties and the properties of functions on them with several examples. I will focus on techniques based on random walks on data sets, and introduce a new nonlinear system of coordinates based on heat kernels, similar in spirit to the GPS system, for parametrizing data sets. If time allows, I will also discuss simple but surprisingly successful applications of the heat kernel to fit functions on data, that performs at the state-of-art or better as a classifier on a variety of benchmark data sets.

The nephron is the functional unit of the kidney. The flow rate in each nephron is regulated, in part, by tubuloglomerular feedback, a negative feedback loop. In some parameter regimes, this feedback system can exhibit oscillations that approximate limit-cycle oscillations. However, nephron flow in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the associated complex power spectra in SHR. A bifurcation analysis of the TGF model equation was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. Four potential sources of spectral complexity in SHR were identified: (1) bifurcations that produce qualitative changes in solution type, leading to multiple spectrum peaks and their respective harmonic peaks; (2) continuous lability in delay parameters, leading to broadening of peaks and their harmonics; (3) episodic lability in delay parameters, leading to multiple peaks and their harmonics; and (4) coupling of small numbers of nephrons, leading to broadening of peaks, multiple peaks, and their harmonics. We conclude that the complex power spectra in SHR may be explained by the inherent complexity of TGF dynamics, which may include solution bifurcations, variation in TGF parameters, and coupling between small numbers of neighboring nephrons.

I will give an introduction to the analysis, geometry, and topology of vector bundles for a general (i.e. nongeometric) audience. I will range from how vector bundles arise in Math 103 to how we can use partial differential equation techniques to extract interesting physical, algebraic, and topological information from them.

I'm sure it will be great!

Coarsening is the phenomenon where many objects (water drops, molecular islands, particles in a freezing liquid) becoming a smaller number of objects in an orderly way. This talk will examine modeling one such system, tiny liquid drops, through three models: a PDE for the fluid, a coarsening dynamical system for the drops, and an LSW-type ensemble model for the group of drops. We will study self-similarity in the dynamics and extensions of the model to examine very long times when drops grow large enough that gravity distorts their shape.

After a brief introduction of modular forms on the upper half plane and vector-space valued differential forms, we will explore a very classical result (independently due to Eichler and Shimura) which relates certain cohomology groups to cusp forms on the upper half plane of corresponding weight. We will then put our algebraic hat on, and recast this result in modern light, using the theory of Automorphic forms developed by (among others) Harish-Chandra and Langlands. I hope to make the talk accessible to most graduate students. Though the topics we will talk about are related to my research, it is not a research talk, more of an exposition. The first part of the talk should be a breeze for anyone with understanding of some fundamental concepts in Complex analysis and Algebraic Topology (holomorphic functions, differential forms, deRham cohomology.) A course in Representation Theory would be helpful in relating to the concepts in the second part of the talk.

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.

What do vector arrangements, discrete graphs, and perfect matchings have in common? These seemingly unrelated objects (and many others) have a very similar underlying structure, known as a matroid. As a result, studying matroids allows you to simultaneously study many different objects from all over mathematics. In addition, many properties and constructions from these various objects, such as loops, duals, bases, cycles, rank, polynomial invariants, and minors (subgraphs), generalize naturally to matroids. In this talk, we will give a general definition of a matroid, and motivate their study by examining some of these constructions in detail. The only prerequisite for this talk is basic linear algebra.

In 1873, a man named Francis Galton posed a question in Educational Times, calling for the mathematical study of the extinction of family surnames over time. Within a year, mathematician Henry Watson replied with a solution. But instead of ending there, this question opened up a new direction of mathematics: the study of branching processes. A branching process is a particle system in which the particles undergo splitting or branching events dictated by particular rules. This talk will introduce some examples of these systems (from the basic Galton-Watson model to more general branching-selection models), interesting questions people ask about branching processes, and some recent research done in this area.

Optimization problems in material science often require quickly varying composites of materials. In this talk, basic homogenized properties of composites and laminates are discussed. The basic theory is applied to construct and optimize interesting structures, such as field rotators and conducting wheels. The talk will be accessible to everyone.

Abstract: Modular forms, a kind of SL2(Z)-invariant holomorphic functions defined on the upper half plane, are one of the most important objects studied in modern number theory. This talk will start from the basic definitions of modular forms and give some examples and important theorems associated with Eisenstein series. Finally we will use the power of modular forms to solve some classical problems on partitions of integers in additive number theory, including the Ramanujan congruence and sums of squares.