The last few years have witnessed an explosion in the number of mathematical models for random graphs and networks, as well as models for dynamics on these network models. In this context I would like to exhibit the power of two well known philosophies in attacking problems in random graphs and networks: First, local weak convergence: The idea of local neighborhoods of probabilistic discrete structures (such as random graphs) converging to the local neighborhood of limiting infinite objects has been known for a long time in the probability community and has proved to be remarkably effective in proving convergence results in many different situations. Here we shall give a wide range of examples of the above methodology. In particular, we shall show how the above methodology can be used to tackle problems of flows through random networks, where we have a random network with nodes communicating via least cost paths to other nodes. We shall show in some models on the completely connected network how the above methodology allows us to prove the convergence of the empirical distribution of edge flows, exhibiting how macroscopic order emerges from microscopic rules. Also, we shall show how for a wide variety of random trees (uniform random trees, preferential attachment trees arising from a wide variety of attachment schemes, models of trees from Statistical Physics etc) the above methodology shows the convergence of the spectral distribution of the adjacency matrix of theses trees to a limiting non random distribution function. Second, scaling limits: For the analysis of critical random graphs, one often finds that properly associated walks corresponding to the exploration of the graph encode a wide array of information (including the size of the maximal components). In this context we shall extend work of Aldous on Erdos-Renyi critical random graphs to the context of inhomogeneous random graph models. If time permits we shall describe the connection between these models and the multiplicative coalescent, arising from models of coagulation in the physical sciences.

The cosmos and its laws have fascinated people since the ancient times. Many scientists and philosophers have tried to describe and explain what they saw in the sky. And almost all of them have used mathematics to formulate their ideas and compute predictions for the future. Today, we have made huge progress in understanding and predicting how planets, stars, and galaxies behave. But still, the mysteries of our universe are formulated and resolved in mathematical language and always with new mathematical methods and ideas. In this lecture, you will hear about two of the most famous physicists of all times, Isaac Newton (1643-1727) and Albert Einstein (1879-1955), and about their theories of the universe. You will learn about common features and central differences in their viewpoints and in the mathematics they used to formulate their theories. In passing, you will also encounter the famous mathematician Carl Friedrich Gauß (1777-1855) and his beautiful ideas about curvature.

In this talk, I will survey the recent understandings on the motion of water waves obtained via rigorous mathematical tools, this includes the evolution of smooth initial data and some typical singular behaviors. In particular, I will present our recently results on gravity water waves with angled crests.

Reliable information transmission between two sites of a network naturally leads to a percolation problem. When the information to be transmitted is quantum an exciting possibility arises: transform the network performing well chosen measurements to enhance the transmission probability. This idea, introduced recently by Acin, Cirac and Lewenstein is now systematically and successfully applied to a variety of two-dimensional networks, but open questions show that a complete theory is missing. The talk will involve some quanta, some network geometry, some percolation and, hopefully, some fun. No knowledge of quantum theory or percolation theory is assumed. Graduate students are encouraged to attend.

A useful tool to study tachyon condensation in string theory is the renormalization group flow of the associated relevant perturbation of a conformal field theory. One assumes that the worldsheet RG flow reflects the actual time evolution of the target space. The relationship is not completely direct: for example, the RG equations are first order differential equations, while the target space equations of motion are at least second order in time. To put the correspondence on firmer footing we discuss the coupling of non-conformal 2d field theories with $n\geq 25$ scalar fields to 2d gravity, following previous discussions of Polyakov and of Schmidhuber and Tseytlin. The conformal mode of the metric becomes a timelike target space direction with a Liouville action, so that the Liouville dressing of an operator is related to its scaling properties. We will give a conjecture for the equations of motion for the worldsheet couplings, which reduces to the standard 2d RG equations in the limit that the Liouville mode becomes semiclassical, and describe our current attempts to prove this conjecture.

The talk will be an introduction to the world of functional inequalities with a geometric content. I will in particular focus on the family of log-Sobolev and Sobolev inequalities and show how these inequalities are closely connected to the geometry of the ambient space. I will mainly follow the Bakry-Ledoux approach to these inequalities which is is based on the notion of intrinsic curvature of a diffusion operator and at the end of the presentation will explain how these ideas have recently been used in sub-Riemannian geometry.

The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and disease transmission only occurs within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds) then the critical model survives for time \(O(N^{1/2})\). Here we improve that result by proving the convergence of \(i_N(t)=I(tN^{1/2})/N^{1/2}\) to a limiting diffusion. We do this by showing that in the first O(1), this four dimensional process collapses to two dimensions: the number of SI and II partnerships are constant multiples of the the number of infected singles \(I_t\). The other variable \(Y_t\), the total number of singles, behaves like an Ornstein-Uhlenbeck process on a time scale O(1) and averages out of the limit theorem for \(i_N(t)\). This is joint work with Anirban Basak and Eric Foxall.

In this talk, I will discuss the problem of computing the cohomology of the general sheaf in a moduli space of sheaves on a surface. I will concentrate on the case of rational and K3 surfaces. The case of rational surfaces is joint work with Jack Huizenga and the case of K3 surfaces is joint work with Howard Nuer and Kota Yoshioka.

Out-of-equilibrium behavior is the characteristic feature of the long-time dynamics of nonlinear dispersive equations on compact domain. This means that solutions typically do not exhibit any form of long-time stability near equilibrium solutions or configurations. We shall survey several aspects of this behavior both from a dynamical systems and statistical mechanics point of view.

Mathematics is being used in many ways to improve the analysis and interpretation of DNA and other molecules that can affect our health. I will describe how math was used to identify genes associated with tumor progression, and to develop methods to identify and quantify genetic variations without expensive and time-consuming sequencing. resulting in a rapid, economical test for transplant compatibility, a cancer therapy, and numerous clinical diagnostic assays. I will also discuss some surprising mathematical connections discovered in the course of this work.

Rankin-Selberg convolution L-functions are important functions in number theory. Their properties play a central role in many of deepest works on the Ramanujan-Petersson Conjecture. In a recent paper, Hoffstein and Hulse defined generalizations of these L-functions, the so-called "shifted-convolution" L-functions. They obtained the meromorphic continuation of the functions in many cases. Here we consider symmetrizations of these L-functions, and we exactly evaluate their special values at diagonal weights for all shifts. This is joint work with Michael Mertens.

Given a stationary Yang-Mills connection A, we are interested in studying its singular structure. In this talk we introduce a quantitative way to stratify the singular sets. Our main results include a Minkowski Volume estimate and the rectifiability of this quantitative stratification, which leads to the rectifiability of the classical stratifications S^k(A) for all integer k. To be precise, we first recall certain background preliminaries needed for this talk. After giving the statements of the main results, I will briefly describe the machinary used in the proof, and explain the new points and the major difficulty that we have faced. The main results in the talk are based on a work by myself last year. If time allows I will further discuss some open problems regarding global gauge in this field, and recent progress in those directions made jointly with Aaron Naber.

A conference in honor of Harold Layton's 60th Birthday (Afternoon Session)

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.

We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions. We prove that the set of Lagrangian torus fibers (with weak bounding cochain data) with non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space. We can apply tropical geometry to locate such Lagrangian torus fibers in the moment map. We show that these Lagrangian submanifolds are nondisplaceable by equivariant Hamiltonian diffeomorphisms.

We present the construction of a general type of topological sigma-model on generalized Calabi-Yau (GCY) manifolds, which includes the familiar A-model and B-model as special examples. We discuss the deformation theory of generalized complex structures on GCY manifolds from the viewpoint of the topological sigma-model. This investigation leads to an extension of the famous Tian-Todorov theorem to the realm of generalized geometry.

I will describe a linear time algorithm for computing the Riemann map from the unit disk onto an n-gon. The method depends on results from computational geometry (fast computation of the medial axis) and hyperbolic geometry (a theorem of Dennis Sullivan about convex sets in hyperbolic 3-space), as well as classical conformal and quasiconformal theory. Conversely, the fast mapping algorithm implies new results in computational geometry, e.g., (1) quadrilateral meshing for polygons and PSLGs (planar straight line graphs) with optimal time and optimal angle bounds, (2) the first polynomial time algorithm for refining general planar triangulations into non-obtuse triangulations (no angles > 90 degrees; this is desirable for various applications and 90 is the best bound that can be achieved in polynomial time). The talk is intended to be a colloquium-style overview, but I would be happy to discuss more technical details, as requested.

A very classical shape optimization problem consists in optimizing the topology and geometry of an elastic structure subjected to fixed boundary loads. One typically aims to minimize a weighted sum of material volume, structure perimeter, and structure compliance (a measure of the inverse structure stiffness). This task is not only of interest for optimal designs in engineering, but e.g. also helps to better understand biological structures. The high nonconvexity of the problem makes it impossible to find the globally optimal design; if in addition the weight of the perimeter is chosen small, very fine material structures are optimal that cannot even be resolved numerically. However, one can prove an energy scaling law that describes how the minimum of the objective functional scales with the model parameters. Part of such a proof involves the construction of a near-optimal design, which typically exhibits fine-scale structure in the form of branching and which gives an idea of how optimal geometries look like. (Joint with Robert Kohn)

Multi-scale kinetic equations can be compressed: in certain regimes, the Boltzmann equation is asymptotically equivalent to the Euler equations, and the radiative transfer equation is asymptotically equivalent to the diffusion equation. A lot of detailed information is lost when the system passes to the limit. In linear algebra, it is equivalent to being of low rank. I will discuss such transition and how it affects the computation: mainly, in the forward regime, inserting low-rankness could greatly advances the computation, while in the inverse regime, the system being of low rank typically makes the problems significantly harder.