Colloquia/Spring2017
Spring 2017
Abstracts
September 16: Po-Shen Loh (CMU)
Title: Directed paths: from Ramsey to Pseudorandomness
Abstract: Starting from an innocent Ramsey-theoretic question regarding directed paths in graphs, we discover a series of rich and surprising connections that lead into the theory around a fundamental result in Combinatorics: Szemeredi's Regularity Lemma, which roughly states that every graph (no matter how large) can be well-approximated by a bounded-complexity pseudorandom object. Using these relationships, we prove that every coloring of the edges of the transitive N-vertex tournament using three colors contains a directed path of length at least sqrt(N) e^{log^* N} which entirely avoids some color. The unusual function log^* is the inverse function of the tower function (iterated exponentiation).
September 23: Gheorghe Craciun (UW-Madison)
Title: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture
Abstract: The Global Attractor Conjecture says that a large class of polynomial dynamical systems, called toric dynamical systems, have a globally attracting point within each linear invariant space. In particular, these polynomial dynamical systems never exhibit multistability, oscillations or chaotic dynamics.
The conjecture was formulated by Fritz Horn in the early 1970s, and is strongly related to Boltzmann's H-theorem.
We discuss the history of this problem, including the connection between this conjecture and the Boltzmann equation. Then, we introduce toric differential inclusions, and describe how they can be used to prove this conjecture in full generality.
September 30: Akos Magyar (University of Georgia)
Title: Geometric Ramsey theory
Abstract: Initiated by Erdos, Graham, Montgomery and others in the 1970's, geometric Ramsey theory studies geometric configurations, determined up to translations, rotations and possibly dilations, which cannot be destroyed by finite partitions of Euclidean spaces. Later it was shown by ergodic and Fourier analytic methods that such results are also possible in the context of sets of positive upper density in Euclidean spaces or the integer lattice. We present a new approach, motivated by developments in arithmetic combinatorics, which provide new results as well new proofs of some classical results in this area.
October 14: Ling Long (LSU)
Title: Hypergeometric functions over finite fields
Abstract: Hypergeometric functions are special functions with lot of symmetries. In this talk, we will introduce hypergeometric functions over finite fields, originally due to Greene, Katz and McCarthy, in a way that is parallel to the classical hypergeometric functions, and discuss their properties and applications to character sums and the arithmetic of hypergeometric abelian varieties. This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu.
Tuesday, October 25, 9th floor: Stefan Steinerberger (Yale)
Title: Three Miracles in Analysis
Abstract: I plan to tell three stories: all deal with new points of view on very classical objects and have in common that there is a miracle somewhere. Miracles are nice but difficult to reproduce, so in all three cases the full extent of the underlying theory is not clear and many interesting open problems await. (1) An improvement of the Poincare inequality on the Torus that encodes a lot of classical Number Theory. (2) If the Hardy-Littlewood maximal function is easy to compute, then the function is sin(x). (Here, the miracle is both in the statement and in the proof). (3) Bounding classical integral operators (Hilbert/Laplace/Fourier-transforms) in L^2 -- but this time from below (this problem originally arose in medical imaging). Here, the miracle is also known as 'Slepian's miracle' (this part is joint work with Rima Alaifari, Lillian Pierce and Roy Lederman).
October 28: Linda Reichl (UT Austin)
Title: Microscopic hydrodynamic modes in a binary mixture
Abstract: Expressions for propagation speeds and decay rates of hydrodynamic modes in a binary mixture can be obtained directly from spectral properties of the Boltzmann equations describing the mixture. The derivation of hydrodynamic behavior from the spectral properties of the kinetic equation provides an alternative to Chapman-Enskog theory, and removes the need for lengthy calculations of transport coefficients in the mixture. It also provides a sensitive test of the completeness of kinetic equations describing the mixture. We apply the method to a hard-sphere binary mixture and show that it gives excellent agreement with light scattering experiments on noble gas mixtures.
Monday, October 31: Kathryn Mann (Berkeley)
Title: Groups acting on the circle
Abstract: Given a group G and a manifold M, can one describe all the actions of G on M? This is a basic and natural question from geometric topology, but also a very difficult one -- even in the case where M is the circle, and G is a familiar, finitely generated group.
In this talk, I’ll introduce you to the theory of groups acting on the circle, building on the perspectives of Ghys, Calegari, Goldman and others. We'll see some tools, old and new, some open problems, and some connections between this theory and themes in topology (like foliated bundles) and dynamics.
November 7: Gaven Martin (New Zealand Institute for Advanced Study)
Title: Siegel's problem on small volume lattices
Abstract: We outline in very general terms the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram. This gives us the smallest regular tessellation of hyperbolic 3-space. This solves (in three dimensions) a problem posed by Siegel in 1945. Siegel solved this problem in two dimensions by deriving the signature formula identifying the (2,3,7)-triangle group as having minimal co-area.
There are strong connections with arithmetic hyperbolic geometry in the proof, and the result has applications in the maximal symmetry groups of hyperbolic 3-manifolds in much the same way that Hurwitz's 84g-84 theorem and Siegel's result do.
Wednesday, November 16 (9th floor): Kathryn Lindsey (U Chicago)
Title: Shapes of Julia Sets
Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. William Thurston asked "What are the possible shapes of polynomial Julia sets?" For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name? It turns out the answer to all of these is "yes!" I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes.
November 18: Andrew Snowden (University of Michigan)
Title: Recent progress in representation stability
Abstract: Representation stability is a relatively new field that studies somewhat exotic algebraic structures and exploits their properties to prove results (often asymptotic in nature) about objects of interest. I will describe some of the algebraic structures that appear (and state some important results about them), give a sampling of some notable applications (in group theory, topology, and algebraic geometry), and mention some open problems in the area.
Monday, November 21: Mariya Soskova (University of Wisconsin-Madison)
Title: Definability in degree structures
Abstract: Some incomputable sets are more incomputable than others. We use Turing reducibility and enumeration reducibility to measure the relative complexity of incomputable sets. By identifying sets of the same complexity, we can associate to each reducibility a degree structure: the partial order of the Turing degrees and the partial order of the enumeration degrees. The two structures are related in nontrivial ways. The first has an isomorphic copy in the second and this isomorphic copy is an automorphism base. In 1969, Rogers asked a series of questions about the two degree structures with a common theme: definability. In this talk I will introduce the main concepts and describe the work that was motivated by these questions.
Friday, December 2: Hao Shen (Columbia)
Title: Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?
Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.
Monday, December 5: Botong Wang (UW-Madison)
Title: Enumeration of points, lines, planes, etc.
Abstract: It is a theorem of de Bruijn and Erdos that n points in the plane determine at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization of this theorem, which confirms a “top-heavy” conjecture of Dowling and Wilson in 1975. I will give a sketch of the key ideas of the proof, which are the hard Lefschetz theorem and the decomposition theorem in algebraic geometry. I will also talk about a log-concave conjecture on the number of independent sets. These are joint works with June Huh.
Friday, December 9: Aaron Brown (U Chicago)
Lattice actions and recent progress in the Zimmer program
Abstract: The Zimmer Program is a collection of conjectures and questions regarding actions of lattices in higher-rank simple Lie groups on compact manifolds. For instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions. In particular—on manifolds whose dimension is below the dimension of all algebraic examples—Zimmer’s conjecture asserts that every action is finite.
I will present some background, motivation, and selected previous results in the Zimmer program. I will then explain two of my results within the Zimmer program: (1) a solution to Zimmer’s conjecture for actions of cocompact lattices in SL(n,R) (joint with D. Fisher and S. Hurtado); (2) a classification (up to topological semiconjugacy) of all actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang).
Monday, December 19: Andrew Zimmer (U Chicago)
Metric spaces of non-positive curvature and applications in several complex variables
Abstract: In this talk I will discuss how to use ideas from the theory of metric spaces of non-positive curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance non-increasing with respect to holomorphic maps. Moreover, this metric often satisfies well-known non-positive curvature type conditions (for instance, Gromov hyperbolicity or visibility) and one can then use these conditions to understand the behavior of holomorphic maps. Some of what I will talk about is joint work with Gautam Bharali.
Monday, January 9: Miklos Racz (Microsoft)
Statistical inference in networks and genomics
Abstract: From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.
I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data.
Friday, January 13: Mihaela Ifrim (Berkeley)
Two dimensional water waves
The classical water-wave problem consists of solving the Euler equations in the presence of a free fluid surface (e.g the water-air interface). This talk will provide an overview of recent developments concerning the motion of a two dimensional incompressible fluid with a free surface. There is a wide range of problems that fall under the heading of water waves, depending on a number of assumptions that can be applied: surface tension, gravity, finite bottom, infinite bottom, rough bottom, etc., and combinations thereof. We will present the physical motivation for studying such problems, followed by the discussion of several interesting mathematical questions related to them. The first step in the analysis is the choice of coordinates, where multiple choices are available. Once the equations are derived we will discuss the main issues arising when analysing local well-posedness, as well as the long time behaviour of solutions with small, or small and localized data. In the last part of the talk we will introduce a new, very robust method which allows one to obtain enhanced lifespan bounds for the solutions. If time permits we will also introduce an alternative method to the scattering theory, which in some cases yields a straightforward route to proving global existence results and obtaining an asymptotic description of solutions. This is joint work with Daniel Tataru, and in part with John Hunter.
Tuesday, January 17: Fabio Pusateri (Princeton)
The Water Waves problem
We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.
Friday, January 20: Sam Raskin (MIT)
Tempered local geometric Langlands
The (arithmetic) Langlands program is a cornerstone of modern representation theory and number theory. It has two incarnations: local and global. The former conjectures the existence of certain "local terms," and the latter predicts remarkable interactions between these local terms. By necessity, the global story is predicated on the local.
Geometric Langlands attempts to find similar patterns in the geometry of curves. However, the scope of the subject has been limited by a meager local theory, which has not been adequately explored.
The subject of this talk is a part of a larger investigation into local geometric Langlands. We will give an elementary overview of the expectations of this theory, discuss a certain concrete conjecture in the area (on "temperedness"), and provide evidence for this conjecture.
Monday, January 23: Tamas Darvas (Maryland)
Geometry on the space of Kahler metrics and applications to canonical metrics
A basic problem in Kahler geometry, going back to Calabi in the 50's, is to find Kahler metrics with the best curvature properties, e.g., Einstein metrics. Such special metrics are minimizers of well known functionals on the space of all Kahler metrics H. However these functionals become convex only if an adequate geometry is chosen on H. One such choice of Riemannian geometry was proposed by Mabuchi in the 80's, and was used to address a number of uniqueness questions in the theory. In this talk I will present more general Finsler geometries on H, that still enjoy many of the properties that Mabuchi's geometry has, and I will give applications related to existence of special Kahler metrics, including the recent resolution of Tian's related properness conjectures.
Friday, February 3: Melanie Matchett Wood (UW-Madison)
Random groups from generators and relations
We consider a model of random groups that starts with a free group on n generators and takes the quotient by n random relations. We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the Cohen-Lenstra heuristics, which predict the distribution of class groups of number fields. We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen. Finally, we discuss joint work with Yuan Liu on the non-abelian random groups built in this way, including the existence of a limit of the random groups as n goes to infinity.
Monday, February 6: Benoit Perthame (University of Paris VI)
Models for neural networks; analysis, simulations and behaviour
Neurons exchange informations via discharges, propagated by membrane potential, which trigger firing of the many connected neurons. How to describe large networks of such neurons? What are the properties of these mean-field equations? How can such a network generate a spontaneous activity? Such questions can be tackled using nonlinear integro-differential equations. These are now classically used in the neuroscience community to describe neuronal networks or neural assemblies. Among them, the best known is certainly Wilson-Cowan's equation which describe spiking rates arising in different brain locations.
Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. Several mathematical results will be presented concerning existence, blow-up, convergence to steady state, for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed.
One can also describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time $t$ the probability to find a neuron with time $s$ elapsed since its last discharge. Here, we can show that small or large connectivity leads to desynchronization. For intermediate regimes, sustained periodic activity occurs. A common mathematical tool is the use of the relative entropy method.
This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo and D. Smets.
February 10: Alina Chertock (NC State Univ.)
Numerical Method for Chemotaxis and Related Models
Chemotaxis is a movement of micro-organisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction- diffusion equation for the chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxis systems extremely delicate and challenging task.
In this talk, I will present a family of high-order numerical methods for the Keller-Segel chemotaxis system and several related models. Applications of the proposed methods to to multi-scale and coupled chemotaxis–fluid system and will also be discussed.
Friday, February 17: Gustavo Ponce(UCSB)
The Korteweg-de Vries equation vs. the Benjamin-Ono equation
In this talk we shall study the [math]\displaystyle{ k }[/math]-generalized Korteweg-de Vries [math]\displaystyle{ (k }[/math]-KdV) equation
[math]\displaystyle{ \partial_t u + \partial_x^3u +u^k\,\partial_xu=0,\;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+, }[/math]
and the [math]\displaystyle{ k }[/math]-generalized Benjamin-Ono ([math]\displaystyle{ k }[/math]-BO) equation
[math]\displaystyle{ \partial_t u-\partial_x^2\mathcal {H} u+u^k\,\partial_x u=0, \;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+, }[/math]
where [math]\displaystyle{ \mathcal {H} }[/math] denotes the Hilbert transform,
[math]\displaystyle{ \mathcal {H} f(x)=\frac{1}{\pi}\, {p.v.}\big(\frac{1}{x}\ast f\big)(x)=(-i\,sgn(\xi) \widehat{f}(\xi))^{\vee}(x). }[/math]
The goal is to review and analyze results concerning solutions of the initial value properties associated to these equations.
These include a comparison of the local and global well-posedness and unique continuation properties as well as special features of the special solutions of these models.
Monday, February 20, Amy Cochran (Michigan)
Mathematical Classification of Bipolar Disorder
Bipolar disorder is a chronic disease of mood instability. Longitudinal patterns of mood are central to any patient description, but are condensed into simple attributes and categories. Although these provide a common language for clinicians, they are not supported by empirical evidence. In this talk, I present patient-specific models of mood in bipolar disorder that incorporate existing longitudinal data. In the first part, I will describe mood as a Bayesian nonparametric hierarchical model that includes latent classes and patient-specific mood dynamics given by discrete-time Markov chains. These models are fit to weekly mood data, revealing three patient classes that differ significantly in attempted suicide rates, disability, and symptom chronicity. In the second part of the talk, I discuss how combined statistical inferences from a population do not support widely held assumptions (e.g. mood is one-dimensional, rhythmic, and/or multistable). I then present a stochastic differential equation model that does not make any of these assumptions. I show that this model accurately describes the data and that it can be personalized to an individual. Taken together, this work moves forward data-driven modeling approaches that can guide future research into precise clinical care and disease causes.
Friday, March 3, Ken Bromberg (Utah)
"Renormalized volume for hyperbolic 3-manifolds"
Motivated by ideas in physics Krasnov and Schlenker defined the renormalized volume of a hyperbolic 3-manifold. This is a way of assigning a finite volume to a hyperbolic 3-manifold that has infinite volume in the usual sense. We will begin with some basic background on hyperbolic geometry and hyperbolic 3-manifolds before defining renormalized volume with the aim of explaining why this is a natural quantity to study from a mathematician’s perspective. At the end will discuss some joint results with M. Bridgeman and J. Brock.
Tuesday, March 7: Roger Temam (Indiana University)
On the mathematical modeling of the humid atmosphere
The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations. We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities.
Wednesday, March 8: Roger Temam (Indiana University)
Weak solutions of the Shigesada-Kawasaki-Teramoto system
We will present a result of existence of weak solutions to the Shigesada-Kawasaki-Teramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely self-contained and does not rely on any earlier result. Based on an article with Du Pham, to appear in Nonlinear Analysis.
Wednesday, March 15: Enrique Zuazua (Universidad Autónoma de Madrid)
Control and numerics: Recent progress and challenges
In most real life applications Mathematics not only face the challenge of modelling (typically by means of ODE and/or PDE), analysis and computer simulations but also the need control and design.
And the successful development of the needed computational tools for control and design cannot be achieved by simply superposing the state of the art on Mathematical and Numerical Analysis. Rather, it requires specific tools, adapted to the very features of the problems under consideration, since stable numerical methods for the forward resolution of a given model, do not necessarily lead to stable solvers of control and design problems.
In this lecture we will summarize some of the recent work developed in our group, motivated by different applications, that have led to different analytical and numerical methodologies to circumvent these difficulties.
The examples we shall consider are motivated by problems of different nature and lead to various new mathematical developments. We shall mainly focus on the following three topics:
- Inverse design for hyperbolic conservation laws,
- The turnpike property: control in long time intervals,
- Collective behavior: guidance by repulsion.
We shall also briefly discuss the convenience of using greedy algorithms when facing parameter-dependence problems.
This lecture has been conceived for a broad audience. Accordingly, unnecessary technicalities will be avoided.
Friday, March 17: Lillian Pierce (Duke University)
P-torsion in class groups of number fields of arbitrary degree
Abstract: Fix a number field K of degree n over the rationals, and a prime p, and consider the p-torsion subgroup of the class group of K. How big is it? It is conjectured that this p-torsion subgroup should be very small (in an appropriate sense), relative to the absolute discriminant of the field; this relates to the Cohen-Lenstra heuristics and various other arithmetic problems. So far it has proved extremely difficult even to beat the trivial bound, that is, to show that the p-torsion subgroup is noticeably smaller than the full class group. In 2007, Ellenberg and Venkatesh shaved a power off the trivial bound by assuming GRH. This talk will discuss several new, contrasting, methods that recover or improve on this bound for almost all members of certain infinite families of fields, without assuming GRH.
Wednesday, March 29: Sylvia Serfaty (NYU)
Microscopic description of Coulomb-type systems
We are interested in systems of points with Coulomb, logarithmic or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. After reviewing the motivations, we will take a point of view based on the detailed expansion of the interaction energy to describe the microscopic behavior of the systems. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes are given. This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions. The main results are joint with Thomas Leblé and also based on previous works with Etienne Sandier, Nicolas Rougerie and Mircea Petrache.
Friday, April 7: Hal Schenck (UIUC)
Hyperplane Arrangements: Algebra, Combinatorics, Topology
A hyperplane arrangement is a collection of hyperplanes in affine space, usually the real or complex numbers. The complement X of the hypersurfaces has very interesting topology. In 1980 Orlik and Solomon determined that the cohomology ring is a quotient of an exterior algebra, with a generator for each hyperplane. Surprisingly, all relations are determined by the combinatorics of the arrangement. Nevertheless, there remain many interesting open questions, which involve a beautiful interplay of algebra, combinatorics, geometry, and topology. I'll spend much of the talk discussing this interplay, and close by discussing several conjectures in the field, along with recent progress on those conjectures, where the Bernstein-Gelfand-Gelfand correspondence plays a key role. Joint work with Dan Cohen (LSU) and Alex Suciu (Northeastern).
Friday, April 14: Wilfrid Gangbo (UCLA)
On intrinsic differentiability in the Wasserstein space [math]\displaystyle{ P_2(R^d) }[/math]
We elucidate the connection between different notions of differentiability in [math]\displaystyle{ P_2(R^d) }[/math]: some have been introduced intrinsically by Ambrosio-Gigli-Savare, the other notion due to Lions, is extrinsic and arises from the identification of [math]\displaystyle{ P_2(R^d) }[/math] with the Hilbert space of square-integrable random variables. We mention potential applications such as uniqueness of viscosity solutions for Hamilton-Jacobi equations in [math]\displaystyle{ P_2(R^d) }[/math], the latter not known to satisfy the Radon–Nikodym property. (This talk is based on a work in progress with A Tudorascu).
Monday, April 17: Ravi Vakil (Stanford)
The Mathematics of Doodling
Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I'll begin by doodling, and see where it takes us. It looks like play, but it reflects what mathematics is really about: finding patterns in nature, explaining them, and extending them. By the end, we'll have seen some important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century; and ongoing work continuing today.
Tuesday, April 18: Ravi Vakil (Stanford)
Cutting and pasting in (algebraic) geometry
Given some class of "geometric space", we can make a ring as follows.
Additive Structure: When U is an open subset of X we set [X] = [U] + [U \ X].
Multiplicative Structure: [X x Y] = [X][Y]
In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising "stabilization" structure, connecting geometry to arithmetic and topology. I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. This talk is intended for a broad audience. This is joint work with Melanie Matchett Wood.
Friday, April 28: Thomas Yizhao Hou (Caltech)
The interplay between theory and computation in the study of 3D Euler equations
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities. To resolve the nearly singular solution, we develop specially designed adaptive (moving) meshes with a maximum effective resolution of order $10^12$ in each direction. A careful local analysis also suggests that the blowing-up solution is highly anisotropic and is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. Very recently we prove rigorously that the 1D model develops finite time singularity. Using a very delicate method of analysis which involves computer assisted proof, we prove the existence of a discrete family of self-similar profiles for a variant of this model. Moreover, we show that the self-similar profile enjoys some stability property.