I will survey some modern questions in geometry that were solved or that could be solved using tools of gauge theory. This talk should be accessible to first year grad students, and of interest to anyone who is curious about what happens in the field of geometry.

Wavelets are widely used in signal processing (e.g. analysis of sounds and music) and imaging, for tasks such as denoising and compression (ever wondered how jpeg works?). In harmonic analysis they have been used to understand and solve problems involving integral operators motivated by PDEs. In numerical PDEs they lead to fast algorithms for solving certain types of integral equations and PDEs. I will give a gentle introduction to wavelets and some of their motivating applications, accompanied by live demos. If time allows, I will discuss shortcomings and how they have been addressed in more recent developments and generalizations.

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.

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.

I will first explain with a few simple examples a beautiful picture in geometric invariant theory which relates quotient constructions in symplectic and algebraic geometry. Then we will naively apply this picture in a suitable infinite dimensional setting, leading us to the notion of extremal Kahler metrics and the Yau-Tian-Donaldson conjecture, which is an important problem in Kahler geometry today.

One of the exciting results of applied algebraic topology for data analysis has been a formulation of the field of "Stochastic topology." This is an intersection of topology and probability/statistics. I will present some research directions in this field: 1) Euler integration for stochastic models of surfaces and shapes: how topological summaries such as persistence homology or Euler characteristics curves can be used to model surfaces and measure distances between bones. 2) Random simplicial complex models: Given m points drawn from a distribution on manifold construct the union of balls of radius r around these points. As m goes to infinity and r goes to zero what can we say about the limiting distribution of Betti numbers or critical points of this random process ? 3) Spectral theory of simplicial complexes: There is a well developed spectral theory for graphs that provides insights on random walks, spectral clustering of graphs, and near linear time algorithms for solving a system of linear equations. How do these ideas extend to simplicial complexes, in particular: a) is there a notion of a Cheeger inequality for clustering to preserve holes ? b) how does one define a random walk on simplicial complexes that have limiting distributions related to the Harmonics of the (higher order) Hodge Laplacian ? c) we conjecture that the question of near linear time algorithms for linear systems is related to a notion of discrete Ricci curvature for graphs. I just expect knowledge of basic math and will focus on motivating concepts rather than details.

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.

Kick off this year's flu season with a better understanding of within-host influenza dynamics. Influenza A is a rapidly-evolving RNA virus that typically escapes herd immunity through the generation of new antigenic variants every 3 to 8 years. An important part of this antigenic evolution is believed to occur at the intrahost level. I will present two competing models of intrahost dynamics and compare their predictions to empirical observations.

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.

Cartan's theorem on maximal tori in compact Lie groups can be thought of as a generalization of the spectral theorem for unitary matrices. The goal of this talk will be to sketch the `topological' proof of this theorem, based on the Lefschetz fixed point theorem. Along the way, we'll encounter the flag variety, an interesting object whose geometry encodes the representation theory of the Lie group. Those who don't specialize in geometry or topology fear not, we will give examples showing that these concepts are very concrete objects familiar from linear algebra.

Periodic electromagnetic scattering problems are interesting and challenging for various reasons. I will outline these problems and discuss my research in how to deal with singularities that arise. My methods include some analysis, some asymptotics, some numerics, a bunch of pictures I ripped off the web, and - as long as there are no follow up questions - a little bit of geometry.

Morse theory relates the topology of a manifold M to the critical point set of a generic real-valued function on M. Most of this talk will be a non-rigorous review of the basic ideas of Morse theory and some simple applications. There will be lots of pictures of tori and other fun manifolds! In the last part of the talk, I’d like to discuss briefly Witten’s novel approach to Morse theory from the 1980’s, which involves studying the asymptotics of a perturbed Laplace operator. I hope this will be interesting and new to most of the audience, even those who have seen some classical Morse theory. It should also illustrate the important role that analysis can play in topological problems.

The notion of conformal mapping is of fundamental importance in complex analysis. Conformal maps are used by mathematicians, physicists and engineers to change regions with complicated shapes into much simpler ones, and to do so in a way that preserves shape on a small scale (that is, when viewed up close). This makes it possible to ``transpose’’ a problem that was formulated for the complicated-looking region into another, related problem for the simpler region(where it can be easily solved) -- then one uses conformal mapping to ``translate'' the solution of the problem over the simpler region, back to a solution of the original problem (over the complicated region). The beauty of conformal mapping is that its governing principle is based on a very simple idea that is easy to explain and to understand (much like the statement of Fermat's celebrated last theorem) . In the first part of this talk I will introduce the notion of conformal mapping and will briefly go over its basic properties and some of its history (including a historical mystery going back to Galileo Galilei). I will then describe some of the many real-life applications of conformal maps, including: cartography; airplane wing design (transonic flow); art (in particular, the so-called ``Droste effect’’ in the work of M. C. Escher). Time permitting, I will conclude by highlighting a 2013 paper by McArthur fellow L. Mahadevan that uses the related notion of quasi-conformal mapping to link D'Arcy Thompson's iconic work On Shape and Growth (published in 1917) with modern morphometric analysis (a discipline in biology that studies, among other things, how living organisms evolve over time). No previous knowledge of complex analysis is needed to enjoy this talk.

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 talk is about the problem of packing subspaces of real/complex space so that they are as far apart as possible, sometimes called Grassmann packing. For a particular distance metric, the problem is equivalent to finding a matrix with minimum group coherence. I will give a fundamental lower bound on group coherence, and then describe some matrix constructions based on Kronecker products which achieve the lower bound on group coherence, and which therefore give optimal subspace packings. Then I will talk about results which show that random subspace packings are close to optimal asymptotically. I will also say something about why good subspace packings are currently of interest in signal processing applications. This work is joint with Robert Calderbank and Yao Xie (now at Georgia Tech). The math involved is mainly linear algebra and random matrix theory, but I will take care to make the talk accessible to all.

TBA

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.

The weak law of large numbers states that for a sequence of independent identically distributed random variables of finite mean, their average converges to this mean in probability as the number of terms tends to infinity. Well, how fast? That is, how many draws must we make before we see this behavior? For many distributions, the convergence is exponentially fast in the number of terms. Such behavior is a hallmark of concentration of measure: under suitable conditions, well behaved functions of many random variables do not deviate much from a particular value. In this talk, we'll show that such properties are not mysterious, but can be derived from a simple recipe using a few choice inequalities. Examples in both the discrete & continuous settings will be given, making connections with convex geometry.

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.

In this talk we will explore a mathematical data analysis tool called persistent homology and look specifically into how we can turn topological information into useful data for statistical techniques. The problem is one of translation: persistent homology outputs a module, but statistics is formulated for objects in metric, vector, Banach, and Hilbert spaces. We'll see some of the ways this issue can be dealt with in a special case (single-parameter persistence) and discuss which of those techniques are viable for a more general case (multiparameter persistence).

Beyond Krull dimension, rings and modules have various "dimensions": depth, height, projective/injective dimensions, flat dimension, global dimension, weak dimension, among others. These notions are defined homologically in terms of lengths of resolutions, and Ext and Tor functors provide a way to measure them. I'll talk about how they are related with each other. I'll start from geometic interpretation of Krull dimension and height, followed by regular sequences and depth. Then I'll introduce Cohen-Macaulay modules and Gorenstein modules as modules having particularly nice homological properties. Just as in the case of completion where analysis is introduced to algebra to prove some highly nontrivial results, homological techniques have proved to be very powerful in modern commutative algebra, producing such surprising results as homological characterization of regular rings (Serre, Auslander, Buchsbaum). I'll briefly introduce the notion of canonical modules and the question of finiteness of injective resolution. Finally, I'll talk about how these notions can be globalized to scheme and sheaves, which makes geometry "kind of equivalent" to algebra.

Consider the following three questions: How many uniformly random permutations are needed to invariably generate the symmetric group? How many iterations of a Monte-Carlo algorithm are needed to decide whether a polynomial in Z[x] splits? How many sumsets formed from independent Poisson multistep are needed to have a chance of an empty intersection? And their answer: Four. What is so special about four? Not much. These all are special cases of the ubiquitous Ewens sampling formula. Come and find out why.

In representation theory, you learn about compact Lie groups. In algebraic geometry, you learn about complex algebraic varieties. In this talk, we will see how these two notions come together in the world of Abelian varieties. These are important objects in algebraic geometry, which carry some pretty remarkable and unexpected properties. We will try to cover as many of these as possible. The talk will focus on complex Abelian varieties, so some familiarity with complex analysis would be helpful. Those currently in the Riemann surfaces course may find some interesting parallels.

The world is in dire need of more formalisms with which to do differential geometry. Just kidding! More seriously, over the years I have encountered some complicated formulae in differential geometry and have, successfully I believe, used canonical objects sitting on frame bundles to simplify them and their proofs. I will give at least one example of such a formula, namely Simon's calculation of the Laplacian of the second fundamental form of a minimal submanifold and show how the formalism makes it and its proof simple.

We'll undertake a gentle introduction to class field theory by investigating cyclotomic fields, including a proof of quadratic reciprocity. The results we'll discuss complement Les Saper's Grad Faculty seminar talk, though by no means is the latter a prerequisite. As a special treat, I will reveal a completely new, elementary proof of Fermat's Last Theorem.

The goal of this talk is to show how the search for a proper definition of the second derivative of a function leads us to Yang Mills theory, the mathematics underlying modern particle physics and the source of powerful tools in topology and geometry. Along the way, we will introduce vector bundles, connections, and curvature. We assume a knowledge of multivariable calculus (and the first derivative).

Persistent Homology is an emerging field of Computational Topology that is developing tools to discover the underlying structure in high-dimensional data sets. I will discuss the origins and main concepts involved in Persistent Homology in an accessible way, with illustrations and comprehensive examples. If time allows, I will also describe some current, as well as, future applications of Persistent Homology.