Mathematics Colloquium Fall 2014
September 12: Moon Duchin (Tufts University)
Geometry and counting in the Heisenberg group
The growth function of a finitely-generated group enumerates how many words can be spelled with each possible number of letters-- this should be thought of as a sort of volume growth in any geometric model of the group. A major theorem of Gromov tells us exactly which groups have growth in the polynomial range: those that are (virtually) nilpotent. But we can still wonder how regular the growth of a nilpotent group is: is it actually a polynomial? Or could it exhibit some transcendentality together with pretty slow growth?
I'll talk about some themes and techniques in the study of group growth and outline a geometry of numbers for nilpotent groups, including a recent result with M. Shapiro settling a long-standing question: the Heisenberg group -- the simplest non-abelian nilpotent group -- has rational growth in any generating set.
September 19: Gregory G. Smith (Queen's University)
Nonnegative sections and sums of squares
A polynomial with real coefficients is nonnegative if it takes on only nonnegative values. For example, any sum of squares is obviously nonnegative. For a homogeneous polynomial with respect to the standard grading, Hilbert famously characterized when the converse holds, that is when every nonnegative homogeneous polynomial is a sum of squares. After reviewing some history of this problem, we will examine this converse in more general settings such as global sections of a line bundles. This line of inquiry has unexpected connections to classical algebraic geometry and leads to new examples in which every nonnegative homogeneous polynomial is a sum of squares. This talk is based on joint work with Grigoriy Blekherman and Mauricio Velasco.
September 26: Jack Xin (UC Irvine)
G-equations and Front Motion in Fluid Flows
G-equations are level set Hamilton-Jacobi equations (HJE) for modeling flame fronts in turbulent combustion where a fundamental problem is to characterize the turbulent flame speeds s_T. The existence of s_T is connected with the homogenization of HJE, however classical theory does not apply due to the non-coercive and non-convex nature of the level set Hamiltonian. We shall illustrate the asymptotic properties of s_T from both Eulerian and Lagrangian perspectives in the case of two dimensional periodic incompressible flows, in particular cellular flows.
Analytical and numerical results demonstrate that G-equations capture well the enhancement, slow down and quenching phenomena observed in fluid experiments. We also comment on s_T in chaotic flows. This is joint work with Yifeng Yu and Yu-Yu Liu.
October 3: Pham Huu Tiep (Arizona)
The notion of adequate subgroups was introduced by Thorne. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown recently by Guralnick, Herzig, Taylor, and Thorne that if the degree is small compared to the characteristic then all absolutely irreducible representations are adequate. We will discuss extensions of this result obtained recently in joint work with R. M. Guralnick and F. Herzig. In particular, we show that almost all absolutely irreducible representations in characteristic p of degree less than p are adequate. We will also address a question of Serre about indecomposable modules in characteristic p of dimension less than 2p-2.
October 10: Alejandro Adem (UBC)
Topology of Commuting Matrices
In this talk we will describe basic topological properties of the space of commuting unitary matrices. In particular we will show that they can be assembled to form a space which classifies commutativity for vector bundles and which has very interesting homotopy-theoretic properties.
October 17: Roseanna Zia (Cornell)
A micro-mechanical study of coarsening and rheology of colloidal gels: Cage building, cage hopping, and Smoluchowski’s ratchet
Reconfigurable soft solids such as viscoelastic gels have emerged in the past decade as a promising material in numerous applications ranging from engineered tissue to drug delivery to injectable sensors. These include colloidal gels, which microscopically comprise a scaffoldlike network of interconnected particles embedded in a solvent. Network bonds can be permanent or reversible, depending on the nature and strength of interparticle attractions. When attractions are on the order of just a few kT, bonds easily rupture and reform. On a macroscopic scale, bond reversibility allows a gel to transition from solidlike behavior during storage, to liquidlike behavior during flow (e.g., injection or shear), and back to solidlike behavior in situ. On a microscopic scale, thermal fluctuations of the solvent are occasionally strong enough to break colloidal bonds, temporarily allowing particles to migrate and exchange neighbors before rebonding to the network, leading to structural evolution over time. Prior studies of colloidal gels have examined evolution of length scales and dynamics such as decorrelation times. Left open were additional questions such as how the particle-rich regions are structured (liquidlike, glassy, crystalline), how restructuring takes place (via bulk diffusion, surface migration, coalescence of large structures), and the impact of the evolution on rheology. In this talk I discuss these themes as explored in our recent dynamic simulations. We find that the network strands comprise a glassy, immobile interior near random-close packing, enclosed by a liquidlike surface along which the diffusive migration of particles drives structural coarsening. We show that coarsening is a three-step process of cage forming, cage hopping, and cage arrest, where particles migrate to ever-deeper energy wells via “Smoluchowski’s ratchet.” Both elastic and viscous high-frequency moduli are found to scale with the square-root of the frequency, similar to the perfectly viscoelastic behavior of non-hydrodynamically interacting, purely repulsive dispersions. But here, the behavior is elastic over all frequencies, with a quantitative offset between elastic and viscous moduli, which owes its origin to the hindrance of diffusion by particle attractions. Propagation of this elasticity via the network gives rise to age-stiffening as the gel coarsens. This simple phenomenological model suggests a rescaling of the moduli on dominant network length scale that collapses moduli for all ages onto a single curve. We propose a Rouse-like theoretical model and, from it, derive an analytical expression that predicts the effects of structural aging on rheology whereby linear response can be determined at any age by measurement of dominant network length scale—or vice versa.
October 24: Almut Burchard (Toronto)
Symmetrization, sharp inequalities, and geometric stability for integral functionals
Many integral functionals are maximized (under appropriate constraints) by radially symmetric functions. For example, the Coulomb energy of a positive charge density --- the double integral of the Newton potential against the density --- increases under symmetrization. The physical reason is that the interaction energy between the charges grows as the typical distance between the charges shrinks. The energy increases strictly, unless the charge density is already radially decreasing about some point. Is this characterization of equality cases "stable"? In other words, must near-maximizers be close to maximizers?
Such stability questions have been well-studied for the isoperimetric inequality and other functionals that involve gradients since the 1990's; the first results in that direction are due to Bonnesen in the 1920's. For example, the excess perimeter of a set (as compared to a ball of the same volume) controls its difference from a suitable translate of that ball. Much less is known about convolutions and other multiple integrals that describe "non-local" interactions. In some cases, not even a complete list of maximizers is known. I will discuss very recent developments (due to M. Christ, Figalli, Jerison, and others), mention open problems, and present joint work with Greg Chambers on the Coulomb energy.
October 31: Bao Chau Ngo (Chicago)
L-function, trace formula, and moduli space
In his PhD thesis, J. Tate recast the construction of Riemann's zeta function in term of harmonic analysis on the group of ideles. This construction was generalized by Godement and Jacquet to principal L-function of automorphic forms. In a minimalistic view, Langlands program consists in understanding analytic properties of all automorphic L-functions. Braverman and Kazhdan proposed a generalization of Godement-Jacquet's construction. I will talk about these construction in connection with the trace formula and the geometry of certain moduli spaces.
(Monday) November 17: Kenneth Ho (Stanford)
Fast direct methods for structured matrices
Many linear systems arising in practice are governed by rank-structured matrices. Examples include PDEs, integral equations, Gaussian process regression, etc. In this talk, we describe our recent work on fast direct algorithms that exploit such structure. These methods are of particular interest due to their exceptional robustness and high capacity for information reuse. Our main technical achievement is a linear-complexity matrix factorization as a generalized LU decomposition. This factorization permits fast multiplication/inversion and furthermore supports rapid updating. We anticipate that such techniques will be game-changing in environments requiring the analysis of many right-hand sides or the solution of many closely related systems, such as in protein design or other inverse problems. Similar applications abound in computational statistics and data analysis.
(Wednesday) November 19: Craig Schroeder (UCLA)
Tackling the robustness problem in physically-based simulation
Robustness is an important part of any practical numerical method. I will discuss two different aspects of robustness that are relevant to modern physically-based simulation and the solutions that we have devised for each. The first is the development of a numerical integrator for high-deformation solids that exhibits exceptional stability and reliability while being faster than comparable existing methods. The second example is a provably robust intersection algorithm for cutting meshes using floating point arithmetic. I will also talk about the method we developed with Disney that created the most dramatic snow scenes in Disney's "Frozen" and some of the things that we are doing to make it robust..
November 21: Hung Tran (Univ. of Chicago)
Selection problems for a discounted degenerate viscous Hamilton--Jacobi equation
I will give first a brief overview on the selection problem for solutions of Hamilton--Jacobi equations, which leads to the theory of viscosity solutions. Then I will describe the cell/ergodic problem of interest and its interesting phenomena. Finally, I will state the corresponding selection problem, the main result, and explain some key ideas. This is a joint work with Hiroyoshi Mitake.
(Monday) November 24: Qingtao Chen (ICTP, Italy)
In knot theory, Jones, HOMFLY and Kauffman polynomials share the common feature that they can be defined via a purely combinatorial method called skein relation. By using a skein relation, a knot polynomial is defined recursively by reducing its crossings. From the discovery of quantum invariants, it is widely believed that such simple skein relations do not exist anymore due to the complexity of computation of quantum invariants. Recently, we proposed several very interesting congruent skein relations for colored HOMFLY invariants, as well as for colored Jones polynomials and su(n) invariants. We have proved series of infinite examples for these new conjectures, especially the knot case for congruent skein relation of colored Jones, as well as tested lots of highly nontrivial examples by using programming techniques. The motivation behind this phenomenon involves several areas of mathematics and string theory. We hope this will shed some light on the Volume conjecture and related topics. This is a joint work with Kefeng Liu, Pan Peng and Shengmao Zhu.
(Tuesday) November 25: Qin Li (Caltech)
Intrinsic Sparse Mode Decomposition of High Dimensional Random Fields with Applications to Stochastic Elliptic PDEs
Inspired by the recent developments in data sciences, we introduce an intrinsic sparse mode decomposition method for high dimensional random fields. This sparse representation of the random field allows us to break a high dimensional stochastic field into many spatially localized modes with low stochastic dimension locally. Such decomposition enables us to break the curse of dimensionality in our local solvers. To obtain such representation, we first decompose the covariance function into low-rank part plus sparse parts. We then extract the spatially localized modes from the sparse part by solving an $L^0$ minimization. We further relax this $L^0$ minimization problem into an $L^1$ minimization and prove rigorously the equivalence of the two formulations. Moreover, we provide an efficient algorithm to solve it. As an application, we apply our method to solve elliptic PDEs with random media having high stochastic dimension. Using this localized representation, we propose various combinations of local and global solvers that achieve different level of accuracy and efficiency. At the end of the talk, I will also discuss other applications of the intrinsic sparse mode extraction. This work is in collaboration with Thomas Y. Hou and Pengchuan Zhang.
(Monday) December 1: Joseph Neeman (UT Austin)
Some phase transitions in the stochastic block model
The stochastic block model is a random graph model that was originally 30 years ago to study community detection in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.
(Wednesday) December 3: Artem Chernikov (GTM, Paris)
Applications of model theory to geometric Ramsey theory
In a series of papers by Alon, Conlon, Fox, Gromov, Naor, Pach, Pinchasi, Radoicic, Sharir, Sudakov, Lafforgue, Suk and others, it was demonstrated that families of graphs with the edge relation given by a semialgebraic relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and can be decomposed into very homogeneous semialgebraic pieces modulo a small mistake (for example, the incidence relation between points and lines on the real plane, or higher dimensional analogues). We show that in fact the whole theory can be developed for families of graphs whose edge relation is uniformly definable in a structure satisfying a certain model theoretic property called distality, with respect to a large class of measures. Moreover, distality characterizes these strong regularity properties.
The result is similar to Tao's recent algebraic regularity lemma, but covers an orthogonal class of examples (and applies in particular to definable graphs in o-minimal theories and in p-adics).
This is joint work with Sergei Starchenko.
(Monday) December 8: Alden Walker (Univ. of Chicago)
Gromov's surface subgroup question
Gromov asked whether every one-ended hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed surface. This question is open in general, but the answer is known to be "yes" for several notable classes of hyperbolic groups. I'll give some background on the question and describe the construction of surface subgroups of random groups, including why one might care about the case of random groups. I'll also explain some (interestingly superficial) similarities with the construction of surface subgroups of closed hyperbolic 3-manifold groups due to Kahn and Markovic. This is joint work with Danny Calegari.
(Tuesday) December 9: Yao Yao (UW Madison)
Singularity and mixing in 2D incompressible fluid equations
The question of global regularity v.s. finite time blow-up remains open for many fluid equations, and even in the cases that the global regularity is known, solutions can develop small scales as time progresses. In this talk, I will first discuss a fluid equation in 2D with a parameter alpha, where for alpha=0 it becomes the 2D Euler equation, and for alpha=1/2 it becomes the surface geostrophic equation. We study patch dynamics for this equation in the half-plane, and prove that when alpha is small but positive, the solution can have a finite-time singularity. I will also discuss a transport equation where the density is passively transported by some incompressible flow, and we study how well a given initial density can be mixed if the flow satisfies some physically relevant quantitative constraints. This talk is based on joint works with A. Kiselev, L. Ryzhik and A. Zlatos.
(Wednesday) December 10: Jennifer Hom (Columbia)
The knot concordance group
Under the operation of connected sum, the set of knots in the 3-sphere forms a monoid. Modulo an equivalence relation called concordance, this monoid becomes a group called the knot concordance group. We will consider various algebraic methods -- both classical and modern -- for better understanding the structure of this group.