The moduli spaces of curves are useful and fascinating objects of study in algebraic geometry. One way to understand them is to consider their birational geometry. In this talk, we will construct rational maps from the moduli space to other spaces that have interesting geometric consequences.

In 2005 Ozsvath and Szabo developed an unknotting number one obstruction using Heegaard Floer Homology. In this talk I will explain how this technique can be used to classify a certain infinite class of 3-stranded pretzel knots with unknotting number greater than 1.

We will discuss how the algebraic K-theory of fields can be studied using only information from the representation theory of its absolute Galois group. In particular, we will discuss a context in which K-theoretic "descent" holds on the nose, and also relate the ideas with representation varieties of infinite discrete groups.

How do mathematicians, artists, and philosophers try to comprehend geometry beyond our third dimension? Modern computer graphics approaches make it possible to see and interact with four-dimensional phenomena in ways never available before. With the aid of computer graphics images and animations I will discuss the 4th dimension.

We will discuss a Brauer-Manin obstruction to a local-global principle for integral points on affine curves over number fields.

We are interested in studying geometric methods for studying high dimensional data sets. These data sets typically come equipped with a notion of distance, and can be thought of as finite metric spaces. We will describe how to apply standard methods to obtain homological signatures for finite metric spaces, as well as how to construct diagrams which help to analyze various stability properties for them. We will also discuss topological visualization techniques. All the topics will include examples.

We will go over some of the background for Dusa McDuff's November 2nd talk. Subjects include symplectic geomtry and a refresher on continued fractions.

For a decreasing, positive, divergent series {a_n} we will look at the effects of "weighing" the series with a sequence {b_n} converging to zero. Will the new series {a_n*b_n} converge or diverge?

The family of multidimensional diffusions named Volatility-stabilized market models (VSM) was introduced by Fernholz and Karatzas in 2005 as a toy model that nevertheless reflects some of the traits of a real-world equity market. These models reflect the fact that in real markets the smaller stocks tend to have a greater volatility and a greater rate of growth than the larger ones. A central question left open was to analyze the behavior of market weights, i.e., the vector of the ratios of stock capitals of individual firms over the entire market capital. In this talk we will show that the law of the market weights is actually a well-known object in another area of probability, namely, the Wright-Fisher model of population genetics. We analyze various dynamic and equilibrium properties of the market weights. In particular, we show that these weights do not exhibit a power law decay, as opposed to another class of models introduced by the same authors called the rank-based models. Power law decay is an empirically observed fact and has been a center of interest for decades of economic literature.

In this talk I will describe a new method to obtain short-time asymptotics for the heat kernel of parabolic linear operator with non-constant coefficients. Our method is based on dilation at a point z = z(x,y), Dyson and Taylor series expansions, and the Baker-Campbell-Hausdorff commutator formula. I will relate this method and the resulting expansion to other classical short-time expansions. Finally, I will present some new closed form asymptotic expansions of certain securities derivative prices using our method, and discuss some possible extensions. This based on joint works with W. Cheng, R. Constantinescu, A. Mazzucato, and V. Nistor.

We study volume-preserving maps with collections of continuous symmetries and invariants. Unlike in the Hamiltonian case, there are examples of maps with a symmetry that do not have an invariant. Conversely, a map with an invariant need not have a symmetry, and the dynamics on level surfaces of the invariant can be chaotic. We explore the consequences of symmetries in the context of volume preserving maps and exact incompressible, or Liouville vector fields. In order to simplify certain cases, we define a symmetry as a flow that has a global section. This has many consequences for the discrete map and leads to a coordinate system that simplifies the form of the map.

Continuing with the proof of the Density Conjecture, we explain the last part of the proof, exploring the notion of non-realizability of laminations. We describe how the problem can be reduced to a purely topological problem and then how we use hyperbolic geometry to solve this problem.

I will give an introduction to free boundary problems through a specific problem known as the obstacle problem. Continuity-related tools for studying the problem will be discussed. For the technical part of the talk I will show that solutions to the obstacle problem are harmonic away from the contact set. Many regularity theorems will be stated without proof.

I will talk about the derivation on a function field over a finite field, and how it helps me solve my research problem.

Schubert Calculus: what it is and what it's good for.

This work is motivated by a recent series of papers by Degond, M¨¦hats and Ringhofer about the formal derivation of Quantum Hydrodynamics models from the entropy principle. Starting from the Quantum Liouville equation, they obtain a non-closed system for the first moments of the density operator. Then, as in the Levermore moment method for kinetic equations, the system is closed by introducing the solution to the quantum moment problem which consists in minimizing the quantum free energy under some constraints of density, current or energy. The first step towards the rigorous derivation of such Quantum Hydrodynamics models is the mathematical analysis of the quantum moment problem. We will present in the talk a result of existence for the quantum moment problem under a density constraint and give some elements of the proof. The solution will also be characterized in a simplified setting. This is joint work with Florian M¨¦hats (Universit¨¦ de Rennes, France).

I'll describe new invariants of a framed link in a 3-manifold, which arise as the pages of a spectral sequence generalizing the surgery exact triangle in monopole Floer homology. The construction introduces a connection between the topology of link surgeries and the combinatorics of polytopes called graph associahedra. It also sheds light on the relationship between gauge theory and symplectic geometry. For a classical link L in S^3, we obtain a sequence of bigraded vector spaces, interpolating between the reduced, Z/2Z Khovanov homology of L and a version of the monopole Floer homology of the branched double cover. In addition, this perspective yields a simple, topological proof that odd Khovanov homology is mutation invariant. Paper reference: arxiv.org/abs/0903.3746, arxiv.org/abs/0909.0816

The study of shock statistics in Burgers equation arises as a problem of fundamental interest in mathematical physics, among other areas. In statistics, it appears in the context of limit theorems with `cube-root' asymptotics, culminating in a remarkable analysis of the parabolic hull of Brownian motion by Groeneboom (1989). More recent work in stochastic coalescence (by Bertoin, Duchon, and others) has provided another viewpoint on this problem, from which we show a Lax pair is naturally derived. This fact, in addition to the form of some exact solutions, suggests that the model is completely integrable. If time permits I will also touch upon some interesting connections with random matrices. This is joint work with Govind Menon (Brown).

The asymptotic dimension of the mapping class group dimension. A key piece of the proof is the construction of a quasi-tree that mapping class acts on. This construction works in a quite general setting for groups that have some aspect of negative curvature. We will describe this construction and explain how it relates to the asymptotic dimension of the mapping class group. This is joint work with K. Fujiwara and M. Bestvina.

We will look at the minimization of the functional J(u) = \int |\nabla u|^2 + Q^2\chi_{u0} over non negative functions with prescribed value at the boundary. We will show Lipschitz regularity for the minimizers and non degeneracy in order to prove that the free boundary has finite (n-1)-Hausdorff measure and derive the free boundary condition. Alt, H. W.; Caffarelli, L. A. Existence and Regularity for a Minimun Problem with Free Boundary, J. Reine Angew. Math., 1981 Cafarelli, L. A.; Salsa S. A Geometric approach to Free Boundary Problems, AMS 2005

I will talk about the derivation on a function field over a finite field, and how it helps me solve my research problem.

On this talk we'll talk about some curvature conditions that are preserved by Ricci flow and some that are not. We later show that non-negative Ricci curvature is not preserved in dimensions four and above.

We study the high energy limit of the scattering operator for the Schrodinger equation in the presence of an obstacle that is a disjoint union of handlebodies. From the high energy limit of the scattering operator, we recover the magnetic field and the electric potential. Our work proves that the approximate solution of the Schrodinger equation proposed by Aharanov and Bohm in their classical paper from 1959 is close to the exact solution that describes the experimental setup of Tonomura et al by which the Aharanov-Bohm effect was confirmed. This is a joint work with Ricardo Weder.

The largeness and positive virtual betti number conjectures are two active areas of research in 3-manifold topology. It is not difficult to see that largeness implies positive , but an interesting question is how much stronger the largeness conjecture is. In my talk I will discuss some characterizations of largeness and positive , and then talk about some conditions under which these two conjectures are equivalent. This is a practice candidacy talk.

The purpose of this talk is to discuss how the classical maximum value principles in PDEs be extended to their numerical approximations arising from finite element methods. In particular, the discussion will be primarily focused on the second order elliptic problems, and the finite element methods shall include standard Galerkin, P1 non-conforming, and mixed elements.

We will discuss several methods for reliable computation of invariant dynamics in conservative discrete dynamical systems. This will include some numerical techniques based on computational topology which are used to automatically compute topological horseshoe factors; and some quasi-analytical methods for computing high order expansions of stable and unstable manifolds and their intersections. This is a Ph.D. thesis defense.

Let E be an elliptic curve over Q. For a prime p of good reduction for E, let E_p be the reduction of E modulo p. In 1988, Neal Koblitz formulated a conjecture regarding the distribution of the primes p such that the number of F_p-points on E_p is also a prime. In many ways, this conjecture may be viewed as a 2-dimensional analogue of the classical twin prime conjecture. I will discuss a result saying that, for most elliptic curves in a two-dimensional family, Koblitz's conjecture is true. This is joint work with Antal Balog and Chantal David.

The cochains of a simplicial complex or singular cochains of a space can be viewed as a homotopy theoretic version of the algebra of functions. The "spectrum" or "variety" of this algebra is a homotopy theoretic completion of the space.

In this talk, we shall discuss our recent progress on acoustic and electromagnetic cloaking with transformation media. In wave scattering, a cloaking device is an artificially designed device which makes the target object invisible to wave detections. We shall present our study on both perfect cloaking with singular transformation media and near-invisibility cloaking from a regularization viewpoint.

This talk will discuss hyperbolic knots, some reasons why they're interesting, and how the hyperbolic structure given to such knot complements can be used to get topological information about the knot. In particular, we will consider how the geometry of such knot complements provides some conditions for determining when a given arc is an unknotting tunnel.

The most likely potential counterexamples to the 4D smooth Poincare Conjecture, as viewed by many 4-manifold topologists in recent years, were constructed by Cappell and Shaneson in the 1970's. These are indexed by an ifninite family of matrices. It took years of hard work with Kirby calculus in the 1980's to show that even the simplest example was standard (Akbulut, Kirby and then myself). Using further work from that decade and a clever observation, Akbulut showed last summer that a subfamily indexed by the integers is standard. This talk will discuss how to show a much larger family is standard, by a vastly simpler method that requires essentially no Kirby calculus. Instead, we exploit symmetries of a simple embedded 3-manifold.

In this seminar we will present an alternative proof of regularity for solutions of fully non-linear convex elliptic operators using the Method of Continuity. Moreover, we will compare this proof with the standard way of using this method in the case of linear second order elliptic equations.

We will describe some recent, elementary results in the theory of electromagnetic scattering. There are two classical approaches that we will review - one based on the vector and scalar potential and applicable in arbitrary geometry, and one based on two scalar potentials (due to Lorenz, Debye and Mie), valid only in the exterior of a sphere. In extending the Lorenz-Debye-Mie approach to arbitrary geometry, we have encountered some new mathematical questions involving differential geometry, partial differential equations and numerical analysis. This is joint work with Charlie Epstein.

Although the cohomology of a space is a commutative algebra, the cochains form only an "E-infinity" algebra. I will explain what this structure is, how it relates to the Steenrod operations, why it reduces to a commutative differential graded algebra in characteristic zero, and why it does not in other characteristics.

The theorem of Gross, Kohnen, and Zagier says approximately that the Heegner divisors for a weight two Hecke eigenform lie on a line in the Jacobian, and that their positions on this line are given by the coefficients of certain half-integer weight modular forms. In this talk we formulate two conjectures which partially generalize their theorem to higher weight modular forms. This is done by the construction of what we call `higher weight Heegner points'. Examples and results will be discussed. We will try to motivate this topic from the angle of the celebrated Riemann Hypothesis.

Following Davi's talk from last week, I'll discuss curvature conditions that are preserved under the Ricci flow. The focus will be a recent result of Wilking that puts the analysis of curvature conditions into a ridiculously simple algebraic framework.

Sea ice is both an indicator and agent of climate change. It also hosts extensive microbial communities which sustain life in the polar oceans. Fluid flow through porous sea ice mediates a broad range of processes such as the growth and decay of seasonal ice, the evolution of melt ponds and sea ice reflectance, and biomass build-up. We'll discuss recent mathematical advances using percolation theory, hierarchical models, and diffusion processes in understanding the fluid permeability of sea ice and the thermal evolution of its microstructure. Our work will help in predicting how global warming may affect Earth's sea ice packs and how polar ecosystems may respond. Related results on electromagnetic properties will help in monitoring ice thickness and in remote sensing of the polar marine environment. Video from a 2007 Antarctic expedition where we measured fluid and electrical transport in sea ice will be shown.

Polycrystalline materials like metals and ceramics are formed by a mesoscopic network of grains. Grain boundaries evolve by mean curvature, and the network coarsens over time. We are interested in modeling certain statistics of the network, in particular the Grain Boundary Character Distribution. I will discuss a simple birth / death model for the GBCD and progress towards a more detailed Boltzmann equation approach.

We consider the Navier-Stokes equations for a heat-conducting, compressible, viscous fluid in a bounded domain in R^3. We will discuss the existence of weak solutions to this system of equations due to E. Feireisl. We will also describe a bound from below on the temperature for these solutions, recently obtained by Mellet and Vasseur. This bound makes use of techniques related to De Giorgi's argument for the regularity of elliptic equations. This is a candidacy exam.

The largeness and positive virtual betti number conjectures are two active areas of research in 3-manifold topology. It is not difficult to see that largeness implies positive , but an interesting question is how much stronger the largeness conjecture is. In my talk I will discuss some characterizations of largeness and positive , and then talk about some conditions under which these two conjectures are equivalent. This is a PhD candidacy exam.

Constant-power loads are popular in applications that utilize distributed power architectures. They present interesting control problems. Boundary control is one method that can achieve stabilization of dc-dc converters loaded with these loads.

This is a thesis defense.

In characteristic zero, a space is said to be formal if its rational De Rham complex is equivalent to its cohomology as a commutative differential graded algebra; such spaces have all rational homotopy theory determined by their cohomology. I will explain why the analogue never holds in characteristic p and outline an approach that may work for partial information about p-adic homotopy theory.

Argumentation and experimentation are the two basic methods for pursuing truth. The importance and applicability of each method varies with the complexity and accessibility of the system under consideration. For instance, the stationarity of a physical system allows for considerable weight to be placed upon empiricism when searching for truth. In contrast, mathematics relies heavily upon argumentation when establishing truth, since it lacks a physical existence. Finding truth in mathematical finance is particularly difficult because financial systems do lend themselves to experimentation or axiomatization. Nonetheless, we often look upon finance as a science, albeit a "soft science." I suggest that even this characterization is misleading and that finance and economics would be better characterized as decision making tools. Note: this talk will be more philosophical than mathematical.

We will introduce the nonlinear Schrodinger equation, define criticality, and then survey recent progress.  We will discuss some of the technology involved using the energy-supercritical case as our model.

We improve Kolyvagin's upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and Swinnerton-Dyer conjectural formula.

In many of the modern biomedical imaging modalities, the measurable signal can be described as the solution of a partial differential equation that depends nonlinearly on the tissue properties (the "parameters") one would like to image. Consequently, there are typically no explicit solution formulas for these so-called "inverse problems" that can recover the parameters from the measurements, and the only way to generate body images from measurements is through numerical approximation. The resulting parameter estimation schemes have the underlying partial differential equations as side-constraints, and the solution of these optimization problems often requires solving the partial differential equation thousands or hundred of thousands of times. The development of efficient schemes is therefore of great interest for the practical use of such imaging modalities in clinical settings. In this talk, the formulation and efficient solution strategies for such inverse problems will be discussed, and we will demonstrate its efficacy using examples from our work on Optical Tomography, a novel way of imaging tumors in humans and animals. The talk will conclude with an outlook to even more complex problems that attempt to automatically optimize experimental setups to obtain better images.

With the perpetual trend towards smaller and smaller scales in science and engineering, fluid-flow problems in the transitional molecular/continuum regime have rapidly gained prominence over the past years. Accurate numerical simulation of flow problems in the transitional regime poses a fundamental challenge, on account of the large difference between the molecular free path and a typical length scale of observation. To bridge the gap between the molecular length scale and the continuum length scale in numerical simulations, many heuristic approaches have recently appeared in the literature to couple molecular-dynamics models to continuum models, such as the Navier-Stokes equations. The appropriateness of such a direct connection between a molecular model and a continuum model is arguable, however, because the range of validity of the models is highly disparate. A suitable model for transitional molecular/continuum flows is provided by the Boltzmann equation. In the Boltzmann equation, the flow is characterized by a one-particle probability-density function, which measures the probability that a molecule resides in a certain subset of the position/momentum space. The Boltzmann equation itself is an integro-differential equation that governs the evolution of the one-particle probability-density. The Boltzmann equation is in principle valid downto the molecular scale, while on the other hand it encapsulates all conventional continuum models, such as the compressible and incompressible Navier-Stokes equations, in the sense that with appropriate scalings of the macroscopic length and time scales, limit solutions of the Boltzmann equation correspond to solutions of these continuum equations. Essentially, the Boltzmann equation is connected to the continuum equations by the fact that solutions of the Boltzmann equation converge to a particular class of solutions, the so-called Maxwell-Boltzmann equilibrium distributions. Direct numerical simulation of the Boltzmann equation is prohibited by its high-dimensional setting: for a problem in d spatial dimensions, the corresponding position/momentum domain is 2d dimensional. However, it is anticipated that in many cases the computational complexity can be significantly reduced by means of adaptive low (d) dimensional approximations based on Boltzmann moment closures. In many applications, interest is in fact restricted to one particular goal functional. For instance, in micro-scale heat-transfer problems, it is ultimately only the heat flux across a certain part of the boundary that is of interest. This class of problems provides fertile ground for goal-oriented adaptive-refinement strategies. An essential impediment in the development of goal-oriented adaptive-refinement techniques for the Boltzmann equation, is the fact that the convergence-to-equilibrium property which forms the basis of the hierarchical modeling process only occurs for d=2,3. Hence, to test our ideas, we would have to consider problems in 4 or 6 dimensions. To bypass this complication, we have developed a 1D prototype of the Boltzmann equation, which exhibits all the characteristic features of the Boltzmann equation, including the weak-convergence-to-equilibrium property. The underlying molecular model is based on random collisions, which conserve energy but not momentum. In the presentation, I will give an overview of transitional molecular/continuum flows, from the perspective of hierarchical modeling and model adaptivity. I will then elaborate the 1D prototype of the Boltzmann equation that we have developed, and derive its characteristic properties, such as an entropy inequality. Finally, I will present numerical result obtained by a discontinuous Galerkin finite-element discretization of the prototype, including recent results for goal-oriented error estimation.

We will discuss the proofs of the various classification theorems for real division algebras, including Frobenius' Theorem, Hopf's Theorem and Hurwitz's Theorem, along with the rich history underlying them.

We study the Hamilton-Jacobi equation with fractional diffusion. If the power of the Laplacian in the diffusion is larger than 1/2, the effects of the diffusion is stronger than the first order term and the equation is well posed. If the power of the Laplacian is less than 1/2, the solution is Lipschitz, but may not be differentiable. For the critical case of the Laplacian to the 1/2, we prove that the equation is also well posed in the classical sense. The result is derived using a new Holder estimate for reaction-(fractional)diffusion equations similar to the one recently obtained by Caffarelli and Vasseur, but for bounded vector fields that are not necessarily divergence free. We will also discuss its application to the parabolic Isaacs-Bellman equation of order one.

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in the 3-sphere. Additionally we show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that for Heegaard genus greater than or equal to 2 the coarse mapping class group of a Heegaard splitting is isomorphic to the genuine mapping class group of the Heegaard splitting. This is joint work with Matt Rathbun.

This talk will give a survey of the restriction phenomena in Fourier analysis. I will discuss connections with PDE, combinatorics and number theory. Michael Kelly will continue discussing this topic in his talk next week.

We present a survey of results on the quadratic forms induced by the Bezoutians of two polynomials over the real and p-adic fields, with emphasis on invariants. On $\R$, we generalize a classical result on interlacing of roots and study the cases of other patterns over the real line.

Fractional diffusion is formulated via integral operators of fractional order and its natural energy space is thus a fractional Sobolev space. We show how to extend the rather rich theory of a posteriori error analysis for differential operators to this setting. We discuss the following key issues: o Localization of fractional Sobolev norms; o Evaluation of residuals; o Localization of the space estimators to the non-contact set in case of obstacle constraints; o Derivation of time estimators which exhibit the correct order because they exploit the gradient flow structure.

This talk will continue the discussion of the restriction phenomena that began in Mark Lewko's talk last week. First we will give a brief overview of the restriction conjecture for the sphere and then prove it for 2 dimensions (we'll give a complete proof for Fourier series, then a detailed sketch for Fourier transforms).

We consider 3D Navier Stokes Eq. After brief presentation for known results about existence, regularity and uniqueness, we will describe Prof.Vasseur's work about higher derivatives. Finally, we will discuss about possible improvements. This is a candidacy exam.

In this talk, I discuss several vector space norms on the space of algebraic numbers modulo torsion which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We will then evaluate these norms on certain classes of algebraic numbers and prove that the infimum in the construction is achieved in a certain finite dimensional space. This is joint work with Paul Fili.

In this talk we present some recent results obtained conjointly with C. Mouhot and B. Wennberg concerning the study of mean-field limit for systems of indistinguables particles undergoing collision processes. As formulated by Kac (1956) this limit is based on the chaos propagation, and our goal is to prove and quantify this property for Boltzmann collision processes with unbounded collision rates. We introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting non-linear equation (stability estimates). Our approach yields (1) the first uniform in time chaos propagation result for the spatially homogeneous Boltzmann equation for true (without cut-off) Maxwell molecules whose ``Master equation'' shares similarities with the one of a Levy process, and (2) the first quantitative chaos propagation result for the spatially homogeneous Boltzmann equation for hard spheres (improvement of the non-contructive convergence result of Sznitman (1984)). Our approach is inspired from the tremendous but wrong paper by F. Grunbaum (1971).

In this talk, we discuss some results related to the question: let K be a bounded, convex subset of a Banach space X and f: K - K a continuous mapping. Is there a sequence (x_n) in K so that the sequence (x_n - f(x_n)) converges weakly to zero in X? Positive answers are given when K is weakly compact and when X is an Asplund space. We also discuss some consequences and applications.

Given a prime, twist-reduced diagram D(K) of a link K in S^3, I will show how to construct the augmented link L corresponding to this diagram. These augmented links have simple structure and can be used to obtain information about the original K.

There is considerable interest in the development of optical methods for biomedical imaging. The physical problem consists of recovering the optical properties of a medium in which light propagates by multiple scattering. This talk will review recent work on related inverse scattering problems for the radiative transport equation and fast image reconstruction algorithms for large data sets. Numerical simulations and experimental data from model systems are used to illustrate the results.

According to legend, the god Indra had a collar of pearls. Each of the pearls saw a reflection of all the other pearls, including the reflection of itself in the other pearls.

I will begin by introducing Deligne's conjecture for the special values of L-functions, in the context of symmetric power L-functions attached to a modular form. In the second half of my talk I will show how one can use Langlands' principle of functoriality to prove theorems about special values of odd symmetric power L-functions.

Classical Brill-Noether theory addresses the existence of special divisors on a general curve of genus g. In particular, Griffiths, Harris, Kleiman and Laksov combined to show that a general curve of genus g has a divisor of degree d and rank r if and only if g is at least (r+1)(g-d+r). Baker and Norine introduced a theory of ranks of divisors on graphs and metric graphs that is closely linked to ranks of divisors on algebraic curves, through specialization. Furthermore, Baker conjectured that suitable versions of Brill-Noether theorems should exist in tropical geometry and showed that some Brill-Noether type theorems on algebraic curves could be reproved tropically. In this talk, I will present recent joint work with Filip Cools, Jan Draisma, and Elina Robeva exploring these topics. The speaker is an assistant professor candidate.

We formulate an explicit conjecture that relates the degree of the Jones polynomial of a knot (and its parallels) with the slopes of incompressible surfaces in the knot complement. We will give ample motivation, and a proof in some botanically chosen classes of knots: alternating, 9 crossing or less, (-2,3,n) pretzel, and torus knots.

Analogies between the integers and the ring F_q[t] have been long known. However, from an arithmetic combinatorics perspective, these analogies have been little and only recently explored. As it turns out, in many cases existing methods can be transfered directly to F_q[t], while at times extra difficulties will arise. In this talk, I will discuss about analogs of some well-known results in this setting, including: -Green-Tao theorem for F_q[t]: The irreducible polynomials in F_q[t] contain affine spaces of abitrarily high dimension. -Sarkozy's theorem for F_q[t]: In any subset of positive density in F_q[t], we can find polynomials f,g such that f-g = h^2 for some nonzero polynomial h in F_q[t].

In this talk we will formulate the Monge-Kantorovich problem, mention some applications to economics and probability, and prove a few basic results.