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).

Data lying in a high-dimensional ambient space are commonly thought to have a much lower intrinsic dimension. In particular, the data may be concentrated near a lower-dimensional subspace or manifold. There is an immense literature focused on approximating the unknown subspace, and in exploiting such approximations in clustering, data compression, and building of predictive models. Most of the literature relies on approximating subspaces using a locally linear, and potentially multiscale, dictionary. In this talk, we propose a simple and general alternative, which instead uses pieces of spheres, or spherelets, to locally approximate the unknown subspace. Theory is developed showing that spherelets can produce dramatically lower covering numbers and MSEs for many manifolds. We develop spherical principal components analysis (SPCA) and spherical multiscale methods. Results relative to state-of-the-art competitors show dramatic gains in ability to accurately approximate the subspace with orders of magnitude fewer components. This leads to substantial gains in data compressibility, few clusters and hence better interpretability, and much lower MSE based on small to moderate sample sizes. A Bayesian nonparametric model based on spherelets will be introduced as an application.

In this talk we look at a specific problem in probability related to the stochastic versions of the Burgers' and Navier-Stokes equations, and the path taken to construct sufficient assumptions to prove the desired properties, specifically the existence of an invariant distribution. This talk covers material in Stochastic Differential Equations and Stochastic Partial Differential Equations, but also in Real Algebraic Geometry and Perturbation Theory.

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.

Periods are numbers that can be expressed as integrals of algebraic differential forms over domains defined by polynomial inequalities with rational coefficients. They form a subring of complex numbers, which contains multiple zeta values and multiple modular values. Although some periods are transcendental, one can work out a Galois theory for them using their defining algebraic data, which is how the classical Galois theory for algebraic numbers were developed. I will discuss Francis Brown's results on multiple zeta values and more recent work on multiple modular values.

We show how algebraic relations can be encoded in suggestive topological diagrams and use this to prove various algebraic equations in a purely pictorial way. We will first go over a few canonical examples: monoids, self-dual objects, Frobenius algebras, and monads. Then we will briefly discuss the underlying theory that makes this miracle rigorous.

We study the dynamics of waves in media with a local periodic structure which varies adiabatically (over many periods of the periodic lattice) across the medium. We focus in particular on the case where symmetries of the periodic structure lead to degeneracies in the Bloch band dispersion surface. An example of such symmetry-induced degeneracies are the `Dirac points’ of media with `honeycomb lattice’ symmetry, such as graphene. Our results are as follows: (1) A systematic and rigorous derivation of the `anomalous velocity’ of wave-packets due to the Bloch band’s Berry curvature. The Berry curvature is large near to degeneracies, where it takes the form of a monopole. We also derive terms which do not appear in the works of Niu et al. which describe a `field-particle’ coupling effect between the evolution of observables associated with the wave-packet and the evolution of the wave packet envelope. These terms are of the same order as the anomalous velocity. (2) Restricting to one spatial dimension, the derivation of the precise dynamics when a wave-packet is incident on a Bloch band degeneracy. In particular we derive the probability of an inter-band transition and show that our result is consistent with an appropriately interpreted Landau-Zener formula. I will present these results for solutions of a model Schr\”{o}dinger equation; extending our results to systems described by Maxwell's equations is the subject of ongoing work. This is joint work with Michael Weinstein and Jianfeng Lu.

Graphics processing units (GPUs) are powerful accelerators that can launch many processes in parallel. Over the past decade, they have been utilized for scientific computation, including molecular dynamics, fluid mechanics, machine learning, and stochastic differential equations. Although dependent on the algorithm, GPUs may execute code faster than CPUs by several orders of magnitude. The mathematics department at Duke hosts 4 older generation GPUs on two nodes that are available for department use. In this seminar I will briefly introduce how GPUs are different than CPUs; the bulk of my time will be a tutorial on how to code CUDA so that attendees may begin to take advantage of these departmental resources for their research. Depending on the attendance, it may be a hands-on tutorial so bring your laptop.

In this talk, I will use my recent project (a fast algorithm for calculating the energy of a many-body quantum system in the random phase approximation) as an outline to present two cool techniques in applied math and show their actual applications in the project. First, we'll see that the trapezoid rule you teach in Calculus, when applied to periodic functions, is far more impressive than you thought. We'll also get a taste of the surprisingly nice properties that come from combining matrix decompositions with randomized algorithms. Finally, as an added bonus, we'll see how Cauchy's integral formula can be used (in this project) to sum N^2 things in O(N) time.

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.

Problems in computational geometry, such as parameterizing a surface or computing shape deformation under geometric constraints, pose various challenges. I will give brief overview of related problems, highlighting the link between discrete differential geometry, optimization and computer graphics. Then, we will see how convex optimization can be used to approximate a specific class of geometric problems, that include shape mapping, bending a cube (https://youtu.be/iOwPGG5-54Q) and perhaps matching lemur teeth.

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We find exact solutions of Maxwell's equations which yield discrete Bragg-peak-type diffraction patterns for helical structures, in the same way in which plane waves yield discrete diffraction patterns of crystals. We call these waves 'twisted X-rays', on account of its 'twisted' waveform. As in the crystal case, the atomic structure can be determined from the diffraction pattern. We demonstrate this by recovering the structure of the Pf1 virus (Protein Data Bank entry 1pfi) from its simulated diffraction data under twisted X-rays.

The twisted waves are found in a systematic way, by first answering a simpler question: could we derive plane waves from the goal that the diffraction pattern crystals is discrete? The answer is yes. Constructive interference at the intensity maxima trivially comes from the fact that the waves share the discrete translation symmetry of crystals. Destructive interference off the maxima is much more subtle, and - as I will explain in the talk - can be traced to the fact that the waves have a larger, continuous translation symmetry. Replacing the continuous translation group by the continuous helical group which extends the discrete symmetry of helical structures leads to twisted waves.

Once the waveforms are found, discreteness (or mathematically, extreme sparsity) of the diffraction pattern of helices under these waves can be proven by appealing to the generalisation of the Poisson summation formula to abelian groups which goes back to A. Weil, whose motivation came from number theory rather than structural biology.

Joint work with Dominik Juestel (TUM) and Richard James (University of Minnesota), SIAM J. Appl. Math. 76 (3), 2016, and Acta Cryst. A72, 190, 2016.

Contact geometry is the study of geometric structures on odd dimensional smooth manifolds given by a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability; these hyperplane fields are called contact structures. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. Contact and symplectic geometry are closely intertwined and we explain how one can make use of J-holomorphic curves to obtain contact invariants. This talk will have lots of examples, cool pictures, and animations illustrating these fascinating concepts in contact geometry.

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We introduce a new quasi-local mass functional for a surface in spacetime and use it to prove the null Penrose conjecture under fairly generic conditions.

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.

Khovanov homology is a special case of a process known as categorification. The idea of categorification is to lift a known polynomial invariant of links to a homology theory whose isomorphism type is an invariant of links and whose ¡°Euler characteristic¡± is the original polynomial. In the case of Khovanov homology, this Euler characteristic is the famous Jones polynomial. After reviewing some basic knot theory and the construction of the Jones polynomial, we discuss the construction of Khovanov homology. Finally, we will discuss some topological applications of Khovanov homology.

What do eigenvalues have to do with geometry and topology? The first part of the talk will provide a few answers to that very broad question, including a discussion of the Euler characteristic from a spectral theory perspective. The second part of the talk will be a brief introduction to my research in analytic torsion, a topological invariant defined in terms of eigenvalues. In particular I'll explain some similarities and differences between analytic torsion and Euler characteristic.

Surface hopping is a very popular approach in theoretical chemistry for mixed quantum-classical dynamics. Yes, the above sentence looks scary. Let us start over again ... We will examine from a mathematical point of view how stochastic trajectories can be used to approximate solutions to a Schrodinger equation (which is different from what Feynman did). Besides some applications in chemistry, this is a nice topic since it combines ideas from asymptotic analysis, applied probability, and applied harmonic analysis. The only background assumed in this talk is "separation of variables" (and of course some PDEs where separation of variables is applied to).

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.

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.

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two ends. We study the functor and establish a long exact sequence relating the corresponding Legendrian cohomology categories of the two ends. As applications, we prove that the functor between augmentation categories is injective on objects, and find new obstructions to the existence of exact Lagrangian cobordisms. The main technique is a recent work of Chantraine, Dimitroglou Rizell, Ghiggini and Golovko on Cthulhu homology.

The kidney not only filters metabolic wastes and toxins from the body, but it also regulates the body's water balance, electrolyte balance, and acid-base balance, blood pressure, and blood flow. Despite intense research, aspects of kidney functions remain incompletely understood. I will discuss how our group use mathematical modeling techniques to address a host of previously unanswered questions in renal physiology and pathophysiology: Why is the mammalian kidney so susceptible to hypoxia, despite receiving ~25% of the cardiac output? What are the mechanisms underlying the development of acute kidney injury in a patient who has undergone cardiac surgery performed on cardiopulmonary bypass? What is the effect of inhibiting sodium-glucose transport, a novel treatment for reducing renal glucose update in diabetes, on renal NaCl transport and oxygen consumption?

Early in the history of complex analysis, it was realized that there are no continuous versions of the square root or the logarithm on the entire complex plane; instead, analysts invented multi-valued functions to deal with these strange behaviors. The "graphs" of these multi-valued functions can get very interesting, and can be interpreted topologically. In general, the space of solutions to a "nice" system of holomorphic ordinary differential equations on the non-zero complex numbers will not be made up of functions, but of multi-functions. Studying these spaces of solutions have led to several ideas in algebraic topology, especially monodromy, and the relationship between systems of ODE and possible monodromies is called the Riemann-Hilbert Correspondence.

Suppose you need to compute an integral over a general surface numerically. How would you do it? You could triangulate the surface, or you might use coordinate charts. Either way is a lot of work, maybe more than you want to do if you have a large number of surfaces. I will describe a fairly simple method, appropriate for smooth, closed surfaces, developed by a former grad student here, Jason Wilson, in his Ph.D. thesis, including proofs that his algorithm works. I will then discuss the extension to integrals for potentials defined by densities on surfaces, such as harmonic functions. In that case the integrand has a singularity; special treatment is needed, and some interesting math comes in. Another of our former Ph.D.'s, Wenjun Ying, has contributed to that work (among many projects of his). Such integrals occur in several scientific contexts; I will especially mention Stokes flow (fluid flow dominated by viscosity), appropriate for modeling some aspects of biology on small scales. For more information, see J. t. Beale, W. Ying, and J. R. Wilson, A simple method for computing singular or nearly singular integrals on closed surfaces at my web site or at http://arxiv.org/abs/1508.00265

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Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic representation of $\GL_n(\A_F)$ where $F$ is a global field.