Dan Goldston : Small Gaps between Zeros of the Riemann Zeta-Function
- Number Theory ( 135 Views )We consider the complex zeros of the Riemann zeta-function &rho = &beta + i &gamma, &gamma > 0. The Riemann Hypothesis (RH) is that &beta = 1/2. If we consider the vertical distribution of these zeros, then the average vertical spacing between zeros at height T is 2&pi / log T. We expect theoretically and find numerically that the distribution of the lengths of these gaps follows a certain continuous GUE distribution where both very small and very large multiples of the average spacing occur. In contrast to this, the existence of a Landau Siegel-zero would force all the gaps in a certain large range to never be closer than half the average spacing, and also have even more bizarre and unlikely properties. There are three methods that have been developed to prove something about small gaps. First, Selberg in the 1940's using moments for the number of zeros in short intervals, was able to prove unconditionally that there are some gaps larger than the average spacing and others smaller than the average spacing. Next assuming RH Montgomery in 1972 introduced a pair correlation method for zeros and produced small gaps less than 0.67 times the average spacing. Finally, in 1981 Montgomery-Odlyzko assuming RH introduced a Dirichlet polynomial weighted method which found small gaps less then 0.5179 times the average spacing. (This method was further developed by Conrey, Ghosh, and Gonek.) These methods all exhibit the presumed barrier at 1/2 times the average spacing for small gaps. I will talk about two projects that are work in progress. The first is joint with Hugh Montgomery and is motivated by the observations that all the results mentioned above do not exclude the possibility that the small gaps found are all coming from multiple zeros and thus gaps of length zero, and at present we do not know if there are any non-zero gaps that are shorter then the average spacing. While we have not yet be able to prove there are any smaller than average non-zero gaps, we can quantify the relationship between non-zero gaps and multiple zeros and show there is a positive proportion of one or the other. The second project is joint work with Caroline Turnage-Butterbaugh where we have developed a Dirichlet Polynomial Weighted Pair Correlation Method which potentially can be applied to a number of questions on zeros.
Max Xu : Random multiplicative functions and applications
- Probability ( 207 Views )Random multiplicative functions are probabilistic models for multiplicative arithmetic functions, such as Dirichlet characters or the Liouville function. In this talk, I will first quickly give an overview of the area, and then focus on some of the recent works on proving central limit theorems, connections to additive combinatorics, as well as some other deterministic applications. Part of the talk is based on joint work with Soundararajan, with Harper and Soundararajan (in progress) and with Angelo and Soundararajan (in progress).
Julio Rebelo : On closed currents invariant by holomorphic foliations
- Geometry and Topology ( 151 Views )Let M be an algebraic complex surface equipped with a singular foliation F. We assume that F leaves invariant a closed current on M or, equivalently, that F possesses a transversely invariant measure. The purpose of this talk is two-fold. First we want to classify the pairs (M, F) as above, a problem that is usually regarded as a step towards developing a suitable Ergodic Theory for these foliations. On the other hand we want to explain the connection of this problem with the Kobayashi hyperbolicity of general type surfaces. In particular we shall sketch a new proof of McQuillan's theorem proving the Green-Griffiths conjecture for general type surfaces having positive Segre class.
Mahesh Kakde : Congruences between derivatives of geometric L-series
- Number Theory ( 191 Views )I will present a formulation of equivariant Tamagawa number conjecture for flat smooth sheaves on separated schemes of finite type over a finite field. After sketching a proof of this I will give application to Chinburg’s conjectures in Galois module theory and tower of fields conjecture. If time permits I will also give an application towards equivariant BSD for abelian varieties defined over global function fields. This is a joint work with David Burns.
Richard Bass : Uniqueness in law for parabolic SPDEs and infinite dimensional SDEs
- Probability ( 111 Views )We consider the heat equation on an interval with heat being introduced according to a random mechanism. When the random mechanism is space-time white noise, this equation has been much studied. We look at the case where the white noise is modified by a function A(u)(x) of the current temperatures u and where A is H\"older continuous as a function of u. Unlike other work along these lines, A(u)(x) can depend on the temperatures throughout the interval. Our method involves looking at the Fourier coefficients, which leads to an infinite dimensional system of stochastic differential equations. This is joint work with Ed Perkins.
Bulent Tosun : Fillability of contact surgeries and Lagrangian discs
- Geometry and Topology ( 149 Views )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.
Sayan Mukherjee : Random walks on simplicial complexes
- Probability ( 164 Views )The graph Laplacian has been of interest in statistics, machine learning, and theoretical computer science in areas from manifold learning to analysis of Markov chains. A common uses of the graph Laplacian has been in spectral clustering and dimension reduction. A theoretical motivation for why spectral clustering works is the Cheeger inequality which relates the eigenvalues of the graph Laplacian to how disconnected the graph is, Betti zero for topology. We ask how the Cheeger inequality extends to higher-order Laplacians, operators on simplicial complexes, and what clustering means for these higher-order operators. This is joint work with John Steenbergen Related to the graph Laplacian is the idea of random walks on graphs. We will define a random walk on simplicial complexes with a stationary distribution that is related to the k-dimensional Laplacian. The stationary distribution reveals (co)homology of the geometry of the random walk. We apply this random walk to the problem of semi-supervised learning, given some labeled observations and many unlabeled observations how does one propagate the labels.
Dave Schaeffer : Finite-length effects in Taylor-Couette flow
- Applied Math and Analysis ( 133 Views )Taylor-Couette flow provides one of the pre-eminent examples of bifurcation in fluid dynamics. This phrase refers to the flow between concentric rotating cylinders. If the rotation speed is sufficiently rapid, the primary rotary flow around the axis becomes unstable, leading to a steady secondary flow in approximately periodic cells. Assuming infinite cylinders and exact periodicity in his theory, Taylor obtained remarkable agreement with experiment for the onset of instability, agreement that remains unsurpassed in fluid mechanics to this day. This talk is concerned with incorporating the effect of finite-length cylinders into the theory, an issue whose importance was emphasized by Benjamin. Numerous experiments and simulations of the Navier Stokes equations all support to the following, seemingly paradoxical, observations: No matter how long the apparatus, finite-length effects greatly perturb many of the bifurcating flows but, provided the cylinders are long, hardly perturb others. We understand this paradox as a result of symmetry breaking. The relevant symmetry, which is only approximate, is a symmetry between two normal-mode flows with large, and nearly equal, numbers of cells.
Mark Jackson : Superstring Cosmology: New Physics in the Sky
- String Theory ( 206 Views )Striking advances in observational cosmology over the past two decades have ushered in a golden era in cosmology, where our focus has turned from what the universe is made of to why it has the form we observe. The leading theory capable of answering such a question, Superstring Theory, does not appear capable of being tested using conventional accelerator-based experiments, forcing us to be more creative in our goal to verify or dismiss it. Focusing on brane inflation as a string theory-inspired model of inflationary cosmology, I will review how the cosmic microwave background (CMB) will provide a deluge of high- precision data into otherwise inaccessible energy scales. These data include possible "Transplanckian" signatures in the power spectrum, indications of variable sound speed or extra dimensions in non- Gaussianity, or constraining the inflation model parameter space using polarization. I then describe how the production of cosmic (super)strings in brane inflation would provide an additional means to verify superstring theory, and which would yield much detailed information about the underlying theory parameters.
Tye Lidman : Homology cobordisms with no 3-handles
- Geometry and Topology ( 229 Views )Homology cobordisms are a special type of manifold which are relevant to a variety of areas in geometric topology, including knot theory and triangulability. We study the behavior of a variety of invariants under a particular family of four-dimensional homology cobordisms which naturally arise from Stein manifolds. This is joint work with Ali Daemi, Jen Hom, Shea Vela-Vick, and Mike Wong.
Hongkai Zhao : Approximate Separability of Greens Function for Helmholtz Equation in the High Frequency Limit
- Applied Math and Analysis ( 160 Views )Approximate separable representations of Greens functions for differential operators is a basic and important question in the analysis of differential equations, the development of efficient numerical algorithms and imaging. Being able to approximate a Greens function as a sum with few separable terms is equivalent to low rank properties of corresponding numerical solution operators. This will allow for matrix compression and fast solution techniques. Green's functions for coercive elliptic differential operators have been shown to be highly separable and the resulting low rank property for discretized system was explored to develop efficient numerical algorithms. However, the case of Helmholtz equation in the high frequency limit is more challenging both mathematically and numerically. We introduce new tools based on the study of relation between two Greens functions with different source points and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors to prove new lower bounds for the number of terms in the representation for the Green's function for Helmholtz operator in the high frequency limit. Upper bounds are also derived. We give explicit sharp estimates for cases that are common in practice and present numerical examples. This is a joint work with Bjorn Engquist.
Michael Mackey : Understanding, treating and avoiding hematological disease
- Mathematical Biology ( 147 Views )This talk will trace many years of work mathematical modeling hematological diseases. The understand part talks about the use of mathematical to figure out what causes cyclical neutropenia, and the treat part refers to work on treating cyclical neutropenia using recombinant cytokines. The avoid part deals with current ongoing work trying to obviate the deleterious effects of chemotherapy on blood cell productionone of the major negative side effects of chemotherapy.
Ingrid Daubechies : Surface Comparison With Mass Transportation
- Applied Math and Analysis ( 145 Views )In many applications, ranging from computer animation to biology, one wants to quantify how similar two surfaces are to each other. In the last few years, the Gromov-Haussdorff distance has been applied to this problem; this gives good results, but turns out to be very heavy computationally. This talk proposes a different approach, in which (disk-like) 2-dimensional surfaces (typically embedded in 3-dimensional Euclidean space) are first mapped conformally to the unit disk, and the corresponding conformal densities are then compared via optimal mass transportation,. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global Moebius transformations. The metric we construct also defines meaningful intrinsic distances between pairs of "patches" in the two surfaces, which allows automatic alignment of the surfaces. Numerical experiments on "real-life" surfaces to demonstrate possible applications in natural sciences will be shown as well. This is joint work with Yaron Lipman.
Yuanqing Cai : Fourier coefficients of theta functions on metaplectic groups
- Number Theory ( 152 Views )Kazhdan and Patterson constructed generalized theta representations on covers of general linear groups as multi-residues of the Borel Eisenstein series. These representations and their unique models were used by Bump and Ginzburg in the Rankin-Selberg constructions of the symmetric square L-functions for GL(r). In this talk, we will discuss the two other types of models that the theta representations may support. We first talk about semi-Whittaker models, which generalize the models used in the work of Bump and Ginzburg. Secondly, we determine the unipotent orbits attached to theta functions, in the sense of Ginzburg. We also determine the covers when these models are unique. Time permitting, we will discuss some applications in Rankin-Selberg constructions.
Yian Ma : Bridging MCMC and Optimization
- Applied Math and Analysis ( 205 Views )In this talk, I will discuss three ingredients of optimization theory in the context of MCMC: Non-convexity, Acceleration, and Stochasticity.
I will focus on a class of non-convex objective functions arising from mixture models. For that class of objective functions, I will demonstrate that the computational complexity of a simple MCMC algorithm scales linearly with the model dimension, while optimization problems are NP-hard.
I will then study MCMC algorithms as optimization over the KL-divergence in the space of measures. By incorporating a momentum variable, I will discuss an algorithm which performs "accelerated gradient descent" over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained.
Finally, I will present a general recipe for constructing stochastic gradient MCMC algorithms that translates the task of finding a valid sampler into one of choosing two matrices. I will then describe how stochastic gradient MCMC algorithms can be applied to applications involving temporally dependent data, where the challenge arises from the need to break the dependencies when considering minibatches of observations.
Ákos Nagy : From instantons to vortices on spherically symmetric ALF manifolds
- Geometry and Topology ( 114 Views )Yang-Mills theory on Asymptotically Locally Flat (ALF) 4-manifolds has been intensely studied by geometers and physicists since the late 70's. The most important examples are R^3 x S^1, the (multi-)Taub-NUT spaces, and the Euclidean Schwarzschild manifold. In this talk, I will outline the correspondence between spherically symmetric Yang-Mills instantons and planar Abelian vortices (following the ideas of Witten, Taubes, and Garcia-Prada), and then apply this instanton-vortex duality to spherically symmetric ALF 4-manifolds. Finally, I will show how this construction can be used to describe the low energy instanton moduli spaces of the Euclidean Schwarzschild manifold, and its generalizations. This is a joint work with Gonçalo Oliveira (IMPA).
Mateusz Michalek : Algebraic Phylogenetics
- Applied Math and Analysis ( 118 Views )Phylogenetics is a science that aims at reconstructing the history of evolution. Phylogenetic tree models are generalizations of well-known Markov chains. In my talk I will present so-called group-based models and their relations to algebra and combinatorics. To a model of evolution one associates an algebraic variety that is the Zariski closure of points corresponding to probability distributions allowed by the model. Many important varieties arise by this construction, e.g. secant varieties of Segre products of projective spaces. It turns out that group-based models provide toric varieties. In particular, they may be studied using tools from toric geometry relating to combinatorics of lattice polytopes.
David Andeerson : Stochastic models of biochemical reaction systems
- Probability ( 170 Views )I will present a tutorial on the mathematical models utilized in molecular biology. I will begin with an introduction to the usual stochastic and deterministic models, and then introduce terminology and results from chemical reaction network theory. I will end by presenting the deficiency zero theorem in both the deterministic and stochastic settings.
Jer-Chin Chuang : Subdivisions and Transgressive Chains
- Geometry and Topology ( 176 Views )Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those in Chern-Weil theory. In this talk, I characterize transgressions that are path-independent of subdivision sequence. The result is obtained by using a cohomology on posets that is shown to be equivalent to higher derived functors of the inverse (or projective) limit over the opposite poset.
John Pardon : Existence of Lefschetz fibrations on Stein/Weinstein domains
- Geometry and Topology ( 120 Views )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.