The Jacobian of any compact Riemann surface carries a natural theta divisor, which can be defined as the zero locus of an explicit function, the Riemann theta function. I will describe a generalization of this idea, which starts by replacing the Jacobian with the moduli space of higher rank bundles. These moduli spaces also carry theta divisors, described via "generalized" theta functions. In this talk, I will describe recent progress in the study of generalized theta functions.

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties of a contact structure are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact (r) surgery is symplectically/Stein fillable for r in (0,1]. This is joint work with James Conway and John Etnyre.

For a finite 2-generated group G, one can consider the moduli of elliptic curves equipped with G-structures, which is roughly a G-Galois cover of the elliptic curve, unramified away from the origin. The resulting moduli spaces are quotients of the upper half plane by possibly noncongruence subgroups of SL(2,Z). When G is abelian, it is easy to see that such level structures are equivalent to classical congruence level structures, but in general it is difficult to classify the groups G which yield congruence level structures. In this talk I will focus on a recent joint result with Pierre Deligne, where we show that for any metabelian G, G-structures are congruence in an arithmetic sense. We do this by studying the monodromy action of the fundamental group of the moduli stack of elliptic curves (over Q) on the pro-metabelian fundamental group of a punctured elliptic curve.

FDS (fds.duke.edu) is a content management system (CMS) widely used across Duke for schools and departments to effectively maintain their faculty research and teaching related web pages and reports. In this talk we'll cover some fundamentals of FDS and give a short tutorial on the FDS templates. We hope this talk will help everyone (either webmasters, web developers and designers, and FDS group managers, or interested faculty/staff members) to use FDS better.

We report the results of our integrated experimental and theoretical investigations of biological systems dominated by interfacial effects. Particular attention is given to elucidating natural strategies for water-repellency, walking on water, underwater breathing, and drinking.

Suppose ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data (i.e. Cech complexes) as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends on more than just the fibrewise homotopy type of the region covered by sensors. In the setting of planar sensors that also measure weak rotation information, we provide necessary and sufficient conditions for the existence of an evasion path, and we pose an open question concerning Cech and alpha complexes. Joint with Gunnar Carlsson.

Oh rhythm of my heart is beating like a drum...... with the words "I love you" rolling off my tongue....... No never will I roam for I know my place is home..... where the ocean meets the sky...... I'll be sailing.....

The talk presents an explicit classification of the ALE Ricci flat K\"ahler surfaces, generalizing previous classification results of Kronheimer. The manifolds are related to a special class of deformations of quotient singularities of type $\mathbb C^2/G$, with $G$ a finite subgroup of $U(2)$. I will also explain the relation with the Tian-Yau construction of complete Ricci flat Kahler manifolds.

Each number field (finite extension of the rational numbers) has an invariant associated to it called the class number (the cardinality of the class group of the field). Class numbers pop up throughout number theory, and over the last two hundred years people have been considering questions about the growth and divisibility properties of class numbers. WeÂ?ll focus on class numbers of quadratic extensions of the rationals, surveying some key results in the two centuries since the pioneering work of Gauss, and then turning to very recent joint work of the speaker with Roger Heath-Brown on averages and moments associated to class numbers of imaginary quadratic fields.

TBA

A body of literature has developed concerning Â?cloaking by anomalous localized resonanceÂ?. 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.

The digital nature of biology is crucial to its functioning as an information system, as well in building hierarchical components in a repeatable way. We explain how protein systems can function as discrete components, despite the importance of non-specific forces due to the hydrophobic effect. That is, we address the question of why proteins bind to ligands predictably and not in a continuous distribution of places, the way grease forms into blobs. We will give a detailed description of how data mining in the PDB can reveal how proteins interact. We highlight the role of the hydrophobic effect, but we see that it works inversely to the usual concept of hydrophobic interaction. Our work suggests the need for a more accurate model of the dielectric effect in the vicinity of a protein surface, and we discuss some advances in this direction. Our research also provides an understanding of how molecular recognition and signaling can evolve. We give an example of the use of our ideas in drug design.

I will describe how representation theory and the geometry of homogeneous spaces may be used to determine the degenerations of a given Hodge structure. This work is part of a larger program to understand the degenerations of a smooth variety that is being pursued, in various subset of collaboration, by Mark Green, Phillip Griffiths, Matt Kerr, Greg Pearlstein and me.

Given a smooth irreducible complex curve $C$, there are several isomorphism invariants one can attach to $C$. One invariant is the genus of $C$, that is, the number of handles in the corresponding Riemann surface. A subtler invariant is the gonality of $C$, that is, the minimal degree of a dominant map from $C$ of $\mathbb{P}^1$. A lower bound for either invariant has diophantine consequences when $C$ can be defined over a number field, but the ability to give non-trivial lower bounds depends on how $C$ is presented. In this talk we will consider a sequence $C_1,C_2,\ldots$ of finite unramified covers of $C$ and give spectral criteria for the gonality of $C_n$ to tend to infinity.

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.

We establish an analog of the sphere theorem in the setting of contact geometry. Specifically, if a given three dimensional contact manifold admits a compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure is tight. The proof is a blend of topological and geometric techniques. A necessary technical result is a lower bound for the radius of a tight ball in a contact 3-manifold. We will also discuss geometric conditions in dimension three for a contact structure to be universally tight in the nonpositive curvature setting. This is joint work with Rafal Komendarczyk and Patrick Massot.

Points fall into the unit interval according to a certain rule, splitting it up into fragments. An example rule is the following: at each step, two points are randomly drawn from the unit interval and the one that falls into the smaller (or larger) interval is discarded, while the other one is kept. This process is inspired by the so-called "power of choice" paradigm originating in the computer science literature on balanced load allocation models. The question of interest is how much the rule affects the geometry of the point cloud. With Elliot Paquette [1] we introduced a general version of this interval fragmentation model and showed that the empirical distribution of rescaled interval lengths converges almost surely to a deterministic probability measure. I will report on this work as well as on work in progress [2] where we show that the empirical measure of the points converges almost surely to the uniform distribution. The proofs involve techniques from stochastic approximation, non-linear integro-differential equations, ergodic theory for Markov processes and perturbations of semigroups on L^p spaces, amongst other things. [1] Maillard, P., & Paquette, E. (2016). Choices and intervals. Israel Journal of Mathematics, 212(1), 337Â?384. [2] Maillard, P., & Paquette, E. (in preparation). Interval fragmentations with choice: equidistribution and the evolution of tagged fragments

Real-world data often consists of the superposition of multiple informative signals. Examples include an image of the night sky containing both stars and galaxies; a communications message with impulsive noise; and a low rank matrix obscured by sparse corruptions. Demixing is the problem of determining the constituent signals from the observed superposition. Convex optimization offers a natural framework for solving demixing problems. This talk describes a geometric characterization of success in this framework that, when coupled with a natural incoherence model, leads into the realm of random geometry. A powerful result from spherical integral geometry then provides an exact formula for the probability that the convex demixing approach succeeds. Analysis of this formula reveals sharp phase transitions between success and failure for a large class of demixing methods. We apply our results to demixing the superposition of sparse vectors in random bases, a stylized robust communications protocol, and determining a low rank matrix corrupted by a matrix that is sparse in a random basis. Empirical results closely match our theoretical bounds. Joint work with Joel A. Tropp.

I will describe joint work with E. Giroux in which we show that every Weinstein domain admits a Lefschetz fibration over the disk (that is, a singular fibration with Weinstein fibers and Morse singularities). We also prove an analogous result for Stein domains in the complex analytic setting. The main tool used to prove these results is Donaldson's quantitative transversality.

Granular materials exist all around us, from avalanches in nature to the mixing of pharmaceuticals, yet the behavior of these ``fluids'' is poorly understood. Their flow can be characterized by the continuous forming and breaking of a strong force network resisting flow. This jamming/unjamming behavior is typical of a variety of systems, including granular flows, and is influenced by factors such as grain packing fraction, applied shear stress, and the random kinetic energy of the particles. I'll present experiments on quasi-static shear and free-surface granular flows under the influence of external vibrations. By using photoelastic grains, we are able to measure both particle trajectories and the local force network in these 2D flows. We find through particle tracking that dense granular flow is composed of comparable contributions from the mean flow, affine, and non-affine deformations. During shear, sufficient external vibration weakens the strong force network and reduces the amount of flow driven by sidewalls. In a rotating drum geometry, large vibrations induce failure as might be expected, while small vibration leads to strengthening of the pile. The avalanching behavior is also strongly history dependent, as evident when the rotating drum is driven in an oscillatory motion, and we find that sufficient vibration erases the memory of the pile. These results point to the central role of the mobilization of friction in quasi-static granular flow.

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.

Einstein's General Relativity is a geometric theory of space, time, and gravitation. In some sense, it is the successor of Newton's famous theory of gravitation -- the theory Newton is said to have come up with when an apple fell onto his head. But although Einstein's theory is much better at predicting gravitational effects in our universe, Newton's theory is not at all outdated or even obsolete. In fact, many astrophysical measurements and simulations still heavily rely on Newtonian intuitions, calculations, and concepts. In the talk, I will explain how and to what extent this usage of Newtonian theory in astrophysics and related fields is motivated and mathematically justified. This will lead us to the notion of Newtonian limit. We will also see some examples for the behavior of relativistic quantities like mass and center of mass under this Newtonian limit.

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same grading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We will introduce the construction, discuss its geometrical content, and give an overview over the various preserved properties. Toric fiber products have been applied most successfully to families of ideals parametrized by combinatorial objects like graphs. We will show how to use toric fiber product to prove structural theorems about classes of ideals from algebraic statistics.

The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.

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.

Suppose that (d_1, ..., d_n) is an n-tuple of integers satisfying sum_j d_j = 0. Eliashberg posed the problem of computing the class of the locus in the moduli space of n-pointed, genus g curves [C;x_1,...,x_n] where sum d_j x_j = 0 in the jacobian of C. In this talk I will give the solution and sketch the proof, which uses known facts about the structure of mapping class groups.

Embedded contact homology (ECH) is an invariant of a closed contact 3-manifold, but proving its invariance is not so straightforward: the only known proof (due to Taubes) is to show that it is isomorphic to monopole Floer homology, which only depends on the underlying manifold. Colin, Ghiggini, Honda, and Hutchings defined a version of ECH for contact 3-manifolds with boundary, which are naturally sutured manifolds, and conjectured that this is also an invariant of the underlying sutured manifold. In this talk I will show that sutured ECH is indeed an invariant and discuss exactly what kind of invariant it is. This is joint work with Cagatay Kutluhan.

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.

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe my recent proof of the Gross-Stark conjecture, a p-adic version of Stark's Conjecture that relates the leading term of the Deligne-Ribet p-adic L-function to a determinant of p-adic logarithms of p-units in abelian extensions. Next I will state my refinement of the Gross-Stark conjecture that gives an exact formula for Gross-Stark units. I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a p-adic solution to Hilbert's 12th problem.