Optimal control and optimal stopping/switching are two related topics in stochastic analysis. Karatzas and Shreve (1984) considered a singular stochastic control problem of the monotone follower type, and established its equivalence to a related optimal-stopping problem. Furthermore, they proved that both problems admit optimal solutions. Building upon their work, Guo and Tomecek (2008) explored the connection between singular control and optimal switching problems to provide a method for solving high-dimensional singular control problems. Recently, Choi, Sirbu and Zitkovic studied the control problem of optimal consumption with transaction costs and reduced it to a free boundary problem for a one-dimensional first-order singular ODE. It is interesting to ask whether there is a stochastic interpretation to this free boundary problem and if a connection (in the spirit of Karatzas and Shreve) to an optimal stopping problem exists. (This is a candidacy talk)

This semester we will go through Kevin Costello's book "Renormalization and Effective Field Theory". It gives a mathematically rigorous treatment of perturbation theory in quantum field theory, including a proof of the renormalizability of Yang-Mills theory in four dimensions. The book is available at http://www.ams.org/bookstore-getitem/item=SURV-170 You can read the introduction there to get an idea of the approach. In the first meeting of the seminar we will organize the subsequent lectures.

The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the prob- lem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.

The classical Torelli map is the moduli map from the moduli space of smooth projective curves to the moduli space of principally polarized abelian varieties sending a curve to its Jacobian endowed with the polarization induced by the theta divisor. The classical Torelli theorem asserts that the Torelli map is injective and the Schottky problem asks for a characterization of its image. There are two known compactifications of the Torelli map from the moduli space of Deligne-Mumford stable curves to two toroidal compactifications of A_g, namely the perfect cone compactification (Alexeev-Brunyate) and the Voronoi compactification (Mumford-Namikawa). The first compactification is known to be the canonical model of A_g for g 11 (Shepherd-Barron) while the second one has a modular interpretation in terms of stable semiabelic pairs (Alexeev). In a joint work with Caporaso, we study the fibers of the compatified Torelli map towards the Alexeev?s moduli space of stable semiabelic pairs. Moreover, in a joint work with Melo, we study the biggest open subset where the perfect compactification coincides with the Voronoi?s compactification. Finally, I will outline the construction of the tropical Torelli map (joint work with Brannetti and Melo) and I will discuss a tropical Torelli theorem (joint work with Caporaso) and a tropical Schottky problem. Emphasis will be given to the connection between the compactified Torelli map and the tropical Torelli map.

It is known that any two simply connected homotopy equivalent closed smooth 4-manifolds differ by a surgery along a contractible sub-manifold so-called cork. It is a challenging problem to find corks that change the smooth structures of 4-manifolds. In this talk, I'll present a recent work with S. Akbulut showing that a large family of 2-handlebodies arising from symmetric links are corks of their symplectic compactifications. The main tool used in the proof is the Ozsvath-Szabo contact invariant.

Abstract: We introduce a new class of algebras associated to a quiver and its dimension vector , which yield a flat (PBW) -deformation of the algebra of differential operators on the space of matrices associated to . This algebra admits a -deformed moment map from the quantum group , acting by base change at each vertex. The quantum Hamiltonian reduction, , of by at the character is simultaneously a quantization of the multiplicative quiver variety, and a -deformation of the quantized quiver variety, associated to . Specific examples of the data yield -deformations of important algebras in representation theory: for example, the spherical DAHA of type may be obtained in this way as a deformation of a construction of the spherical rational Cherednik algebra by Gan and Ginzburg, and the construction gives an explicit quantization of the symplectic structure on the moduli space of flat connections on a punctured Riemann surface.

Speakers will schedule talks for the semester.

We will pick topics for the semester.

The BMV conjecture states that for any two Hermitian square matrices A and B of the same dimension, the function f(z)=tr[exp(A-zB)] is the Laplace transform of a positive measure on the real line. After 36 years it is still open, even for 3x3 matrices. In this talk I describe some of the known results related to this conjecture, current approaches, reformulations, things that have been tried, etc.

The GST seminar this semester is studying Kevin Costello's book "Renormalization and Effective Field Theory."

Let's start things off easy. I'll talk about some relatively simple results on surfaces in 3--manifolds, especially knot exteriors and handlebodies. These surfaces give us information about the manifold and can be useful for all sorts of arguments. In particular, I'll give some examples of how I've used these results in my work.

In this talk, we will give results on the quadratic twist of the modular curve X_0(N) through the Atkin-Lehner involution w_N and a quadratic extension K/Q. Given (N,d,p) we give necessary and sufficient conditions for the existence of a Q_p-rational point on the twisted curve whenever p is not simultaneously ramified in K and Q(\sqrt{-N}). The main result yields a population of curves which have local points everywhere but no points over Q; in several cases we show that this obstruction to the Hasse Principle is explained by the Brauer-Manin obstruction. If time permits, we will mention some generalizations of this question as well.

Abstract: I will report on a joint work with G. Bini and F. Viviani in which we study the GIT problem for the Hilbert and Chow scheme of curves of genus g embedded via a complete linear system of low degree. We give a characterization of GIT-semistable, polistable and stable points for both the Hilbert and Chow scheme for degree 2(2g-2)d3.5(2g-2) and d4(2g-2). Our description extends the work of L. Caporaso in ?A compactification of the universal Picard variety over the moduli space of stable curves?, which deals with the case d\geq 10(2g-2) for the Hilbert scheme. As a corollary of our analysis, we obtain a new modular compactification of the universal Picard variety over the moduli space of pseudo-stable curves. This construction should also play an important role in the study of the birational geometry of the universal Picard variety.

I will introduce a few Banach spaces of entire functions, describe their duals and (some of) their properties, and discuss their applications to some extremal problems in Fourier analysis.

I'll informally play little games with the gamma function and indicate some of it's possibly endless uses in number theory. No techniques more advanced than calculus are required for most of the talk, but being tricky is a big part of many of the proofs (if you like that sort of thing as I do) and exposure to infinite products and partial fractions is good for understanding some of the more fascinating results.

Let be a number field and an elliptic curve. Iwasawa theory for is concerned with (among other things) the behavior of the Mordell-Weil and Tate-Shafarevich groups of in the layers of infinite abelian extensions of (such as -extensions). If is such an extension and an intermediate field of , finite over , then the Mordell-Weil theorem states that is a finitely generated abelian group. One of the motivating questions in Iwasawa theory for is whether or not is finitely generated. In this talk we will give an overview of what is known regarding this and other related questions.

We show the existence of stationary solutions for some reaction- diffusion type equations in the appropriate H^2 spaces using the fixed point technique when the elliptic problem contains second order differential operators with and without Fredholm property.

The GST seminar this semester is studying Kevin Costello's book "Renormalization and Effective Field Theory."

This talk has two goals. The first is to discuss the unknotting number of algebraic knots, with special attention to 3-stranded pretzel knots. The second is to discuss the phenomenon known as cosmetic surgery. Surgery which results in both compact and non-compact manifolds will be considered. The author will make no attempt to connect the two topics.

In this talk we outline the proof of a recent result stating that for every space X not admitting an l_1 spreading model, there is a space Y not admitting any l_p or c_0 as a spreading model and having X as a quotient. Related results and possible extensions will also be discussed. This is joint work with Spiros Argyros.

It follows from Margulis' proof of the Oppenheim conjecture that, using a suitable normalization, there are only finitely many indefinite quadratic forms in 3 or more variables whose minima is bigger than \epsilon. We will discuss an effective version of this theorem.

Point estimates for a solution of a homogeneous equation allows by rescaling to prove a geometric decay of the oscillation. Here we will show several variations on the subject of point estimates. We will start with the mean value property of the laplatian, then to the already classical theory of fully non linear equations and finally to the world of the non local problems. In this respect, point estimates are almost natural for the non local operators if you allow the estimates to depend on the order of the equation. The bad new is that such estimate degenerates if you try to go to the classical problem with second order derivatives. The last part of the talk consists then in getting such an improvement in the estimates by making them independent of the order of the equation. References: L. Caffarelli, X. Cabre. Fully nonlinear elliptic equations. L. Silvestre. Hölder estimates for solutions of integro-differential equations like the fractional Laplace. L. Caffarelli, L. Silvestre. Regularity theory for fully nonlinear integro differential equations.

Degenerate elliptic and parabolic obstacle problems arise in mathematical finance when valuing American-style options on an underlying asset modeled by a degenerate diffusion process. We will describe our work on existence, uniqueness, and regularity for stationary and evolutionary variational inequalities and associated obstacle problems when the underlying asset is modeled by a degenerate diffusion process, such as the two-dimensional Heston stochastic volatility process and, more generally, diffusion processes in higher-dimensions or variable coefficients and comparable degeneracies. This is joint work with Panagiota Daskalopoulos (Department of Mathematics, Columbia University) and Camelia Pop (Department of Mathematics, Rutgers University).

Given a linear operator bounded from Lp to Lq, and also from Lp' to Lq', we will look at what it does to functions in Lr for r between p and p'. I will state and prove the Riesz-Thorin interpolation theorem, which addresses this question. The proof, surprisingly, comes from complex analysis. I will then give some simple applications and mention other results of this nature.

A genus g surface gives rise to two interesting graphs: the pants complex, defined by Hatcher and Thurston, and the dual curve complex, defined by Jesse Johnson.  Recently, Johnson has used paths in the pants and dual curve complexes of Heegaard surfaces for a 3-manifold M to define invariants of M.  Analogously, we will show how paths in the pants and dual curve complexes of bridge surfaces for a knot K can be used to produce invariants of K.  We will compute some of these invariants for a class of knots and discuss intriguing connections between the pants complex and hyperbolic geometry.

Topological Field Theories (TFTs) were introduced by Atiyah to axiomatize Witten's work on supersymmetric quantum mechanics and the Chern-Simons theory. They take a form of a functor from a geometric category (cobordisms) to a more tractable category of linear objects (vector spaces). This talk will serve as a very basic introduction to a series of lectures on TFTs. In the end I will prove a classification theorem for TFTs in 1 and 2 dimensions.

We'll provide an introduction to a type of Floer cohomology for non-compact manifolds, and discuss what can be said about it in the case that the manifold is the complement of a divisor in a projective variety.

"What is arithmetic dynamics?" you may find yourself saying as you cock your head to the side and sip from a drinking vessel. This talk is intended to be an introduction to the basic objects of study in the relatively young field of arithmetic dynamics together with a brief survey of active research problems.

We consider graphs with a large but fixed number of vertices, in which the edges are (dependent) random variables. Extremal graph theory considers pairs of features of the graphs (such as edges and triangles) and optimizes the count of one while the count of the other is fixed at some natural value, such as maximizing the number of edges in graphs which have no subtriangles. I will discuss a generalization in which the counts of both features are parameters, studying the interaction of the features through the number of graphs satisfying both constraints. Such interactions produce phase transitions in the strict, traditional sense.

The GST seminar this semester is studying Kevin Costello's book "Renormalization and Effective Field Theory."

Many notable theorems and conjectures in low-dimensional topology concern the presence of surface subgroups within a given group, G. We will address a few of these questions and discuss their significance to 3-manifolds, surface bundles over surfaces, and the mapping class group. In particular, we will show that, up to conjugacy, there are only finitely many surface subgroups of a given genus within the fundamental groups of a closed hyperbolic manifold.

This is an introductory talk on Lie groups and Lie algebras, combining definitions and examples with explanations of basic properties. Most prominently, we discuss the assignment of a Lie algebra to a Lie group and some consequences of this association. No previous knowledge of Lie groups and Lie algebras is necessary, and a basic differential topology background is helpful.

I will describe some generalities of the motives of the title and then focus on those of weight zero. These give rise to Artin L-functions and I will further concentrate on the case where the corresponding Galois group is the (a subgroup of the) Weyl group of F_4. This group has order 1152 and an irreducible, primitive representation of dimension 4. I will discuss how we can explicitly compute some of the associated degree four L-functions and their relation to the lines in certain affine cubic surfaces.

We will discuss estimates for certain variational operators arising in harmonic analysis. We will focus on endpoint cases where the desired estimates fail, but where the rate of failure gives rise to interesting phenomena. For instance, we will show that the standard ordering of the trigonometric system is not optimal for certain problems involving these operators. Some applications of our results to number theory will also be discussed. This is joint work with Allison Lewko.

We define Clifford algebras and construct the Lie group Spin(n) as a subgroup of a Clifford algebra. We use this construction to consider equivalences between filtrations on Clifford algebras. This talk will assume only assume a familiarity with basic algebraic structures.

Two-dimensional gauge invariant TQFTs with finite gauge group are a toy model for TQFTs since there are only finitely many fields. We will outline the construction of this theory and see how to do some computations. We will then explore some surprising applications to group theory.

The computation of the first n terms of an expansion into orthogonal polynomials in O(n log n) operations is a long-standing challenge in computational mathematics. In the first part of the talk we describe such algorithms for the computation of expansions in ultrasperical (a.k.a. Gegenbauer) polynomials. This proceeds in three somewhat counterintuitive steps. Firstly, expansion coefficients are expressed as an infinite linear combination of derivatives. Secondly, using the Cauchy theorem, this is converted into an integral transform with a hypergeometric kernel. Finally, using a serendipitous transformation of the kernel into a rapidly-convergent function, the integral transform is computed as a finite linear combination of a discrete Fourier transform of the underlying function along a Bernstein ellipse. In the second part of the talk we generalize the first two stages to arbitrary orthogonal polynomial systems supported by compact real intervals, as well as to polynomials and Laurent polynomials orthogonal on the unit circle. This is joint work with María José Cantero.

In general the completions of an infinite degree algebraic number field at its nonarchimedean places are not local fields. However, some have the property that all of their completions are local fields, and furthermore their local degrees are uniformly bounded. We will discuss examples of such fields, a few recent results on the properties they enjoy, and some open questions about them.

The GST seminar this semester is studying Kevin Costello's book "Renormalization and Effective Field Theory."

We give a complete calculation (mod 2) of the infinity flavor of Heegaard Floer homology for all three-manifolds and torsion Spin^c structures. This agrees with the conjectured computation of Ozsvath-Szabo and is thus determined by the cohomology ring of the underlying three-manifold.

Abstract: Let T be a bounded linear operator on a Banach space of inf dim. A closed linear subspace X is called almost invariant for T if it is of inf dimension and co-dimension, and there exists a finite rank perturbation S of T so that X is invariant under S. Let T be the bilateral shift, acting on L2 [0, 2π] as multiplication by eit . We will discuss progress on the Conjecture: If X is almost invariant for T and T −1 , then there are a finite co-dimensional subspace X ′ of X and a set of positive measure on which members of X ′ v vanish ae.

Students often look to the teacher for confirmation of their correct answer. Furthermore, when students see an incorrect solution presented, they will often accept it without question. We plan a group discussion on strategies for tackling these two problems.

I will discuss my work at Oak Ridge National Laboratory last summer. I will consider local and nonlocal heat transport along magnetic field lines in a tokamak, a device used to confine plasma undergoing fusion. I will give a background on issues like anisotropic heat flow and nonlocality. Then, as time permits, I will discuss results that relate certain dynamical features of the magnetic field, i.e. resonance islands, chaotic regions, transport barriers, etc. relate to the asymptotic temperature profiles for heat flow along magnetic field lines.

I'll say what Cohen-macauley rings are, draw a couple of cute simplicial complexes on the blackboard, compute their Stanley/Reisner ring, check if it is Cohen/Macaulay, say how the Stanley/Reisner Ring is related to the face vector.

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Consider a hyperbolic knot in and an exceptional filling . There is a one dimensional family of hyperbolic cone structures on with cone singularity along . These structures can be deformed until the cone angle reaches some critical value (possibly ) at which point the hyperbolic geometry collapses. In the special case that the hyperbolic structures collapse onto a hyperbolic plane, we show how to construct a natural continuation of this path into AdS structures (with cone singularities). We also discuss combinatorial/algebraic tools for constructing transitions using ideal tetrahedra.

I will discuss a definition of an open-closed 2d TFT. It is a version of a topological field theory, where we assign "boundary conditions" to some boundaries. My goal will be to explain when a 2d TFT can be extended to an open-closed one, and, conversely, how to reconstruct the "closed" part from the open-closed data.

One of the early successes of quantum groups was the construction by Reshetikhin and Turaev of knot and 3-manifold invariants, using a braid group action on certain tensor products of representations of the quantum group. The braid group of an arbitrary surface is the fundamental group of the configuration space of distinct points on ; when , this identifies with the usual braid group. In this talk, I will explain that also acts naturally on certain tensor products of representations of the quantum group. The new ingredient in this construction is a module over the q-deformed algebra of differential operators on .

The main goal of the talk is to present a proof of Banaszczyk's 1992 result on a bound for the product of certain successive minima associated to a lattice in N-dimensional Euclidean space and its dual lattice. This result improves on a classical theorem of Mahler by replacing an upper bound for the expression of interest from the order of (2N)! to order N. I will be following some notes of Vaaler, which include some modifications and generalizations of Banaszczyk's work.

I will report on a recent joint project with A. Eskin and M. Mirzakhani where we prove orbit closures for the above action are locally linear.

Developed by Gabai in the 1980's, the theory of taut sutured manifolds has had considerable success in proving some long-standing conjectures in low-dimensional topology. While Gabai's work is based on foliation of a 3-manifold by surfaces, shortly after its development, Scharlemann reproved the foundational theorems (and many of the major results) without the use of foliations. I'll give a brief outline of the foundational theorems from Scharlemann's viewpoint, and give a few sample applications.

What should we expect from random chance? The answer is full of surprises. Whether we flip a coin, look at data, or ask for birthdays, we find that our intuition about what to expect is often far from what actually happens by chance alone. When dealing with chance, we have to learn to expect the unexpected.

The space of initial conditions of the KdV equation has a description in terms of a certain infinite-dimensional Grassmannian. My goal will be to describe a certain subset of the Grassmannian in terms of line bundles on Riemann surfaces. This gives a simple description of non-linear KdV flows as linear flows on the Jacobian.

We will discuss a proof of a well known joint result of Beale, Kato, and Majda for the 3D Euler equations, which states that the norm of vorticity controls the smoothness of solutions. If time permits we will also look at a generalization of the central estimate in the context of pseudo-differential operator theory.

In this talk, I would like to introduce the "Cheng-Yau gradient estimate"(1975) on a manifold with Ricci curvature bounded from below, and I will give a proof on the simpler case for R^n. Because the proof requires a lot of calculation(merely differentiation) and some small tricks(maximum principle and so on), I think I am going to talk more on the method and its applications than the calculation itself, skip some steps in tedious differentiation work.

It is a natural question to ask which knots in handlebodies have nontrivial handlebody surgeries. Based on work of Berge and Gabai, Wu conjectured that such knots should be relatively simple in the sense that they are 1--bridge. In this talk I'll give an example of a knot in a genus 2 handlebody which has a nontrivial handlebody surgery, and then I'll show that this knot is not 1--bridge.

We are going to present a study about the fundamental solutions for a special type of fully nonlinear elliptic operators, the extremal Pucci operators, and see how they are related with the fundamental solutions for the Laplace operator. As an application and example, we will prove the existence of Barriers functions that can be used to control the behavior of a viscosity solution for these operators due to the comparison principle.

Circle packing is the study of the possible configurations of closed disks, for example in the plane, satisfying given tangency relations. I will give an introduction to circle packing, state currently known results, discuss applications to complex analysis, and state some open questions. The talk should be accessible to everyone, and the theorems and applications are in the flavor of geometry and in particular complex analysis.

We will define group cohomology for an arbitrary group G. We will then shift our focus to group extensions and show how they are related to group cohomology.  Then we will compute the cohomology of certain groups and use them to talk about extensions of those groups.

In this talk, we present a recent joint work with Natasa Pavlovic (UT = Austin). We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimensions , from an -body Schrodinger equation describing a gas of interacting bosons in the GP scaling. In particular, we prove that the GP hierarchy obtained from the limit of the corresponding BBGKY hierarchy, is contained in solution spaces introduced by Klainerman and Machedon, which we used and generalized in our earlier work on the well-posedness of the Cauchy problem for GP hierarchies. We note that in , this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, the special case of factorized solutions provides us with a new derivation of the cubic, defocusing nonlinear Schr\"odinger equation (NLS) in .

In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal s=t, so the integral formally defining the operator is not well defined (the integrand is not integrable) even on `nice' functions. However, as it turned out, the situation with interpretation of singular integral operators is much simpler, than it seems; to investigate the boundedness one only needs to study an elementary and well defined `restricted' bilinear form. While our result is not a replacement for the hard analysis still necessary to investigate the boundedness of a singular integral operator, it simplifies the setup significantly. This is joint work with Sergei Treil.

An exceptional surgery is a Dehn surgery on a hyperbolic knot in the 3-sphere that results in a non-hyperbolic manifold. After giving a general overview of the theory, we'll look at some examples of when such surgeries exist, and when they do not. We'll also investigate the famed Montesinos trick and get an idea of how it can help us better understand surgery on strongly invertible knots in the 3-sphere.

I plan to give a brief introduction of the GP hierarchy. I'll mainly present Klainerman and Machedon's proof of uniqueness of the GP for cubic NLS in dimension 3. Also talk about the related work of lower dimensional cases in Chen and Pavlovic's paper on quintic NLS.

In this talk, I will cover work done applying the Mimetic Finite Difference method (MFD) to the area of subsurface modeling. MFD can solve Elliptic PDEs over general polyhedral meshes with rough coefficient. One of the main challenges of using MFD is the need to solve a saddle-point problem. We identify a generalization of the MFD method that allows for a reduction to a symmetric positive definite system with fewer unknowns for certain meshes. We also demonstrate the use of MFD in resolving non-matching grids and reservoirs with special features.

I will discuss the connections between exponential sums on finite reductive Lie algebras, characters of finite reductive groups and the topology of moduli spaces of representations of the fundamental groups of punctured Riemann surfaces. This is joint work with T. Hausel and F. Rodriguez-Villegas.

Kohn-Sham density functional theory (KSDFT) is by far the most widely used electronic structure theory for condensed matter systems. However, the discretization cost, i.e. the number of basis functions to discretize the Kohn-Sham Hamiltonian operator is generally large, which is an important factor that hinders the application of KSDFT to systems of large size. Recently we have developed two techniques to reduce the discretization cost effectively and systematically. In the first part of the talk, we propose the adaptive local basis functions, which achieve high accuracy in the total energy calculation with a small number of basis functions. The adaptive local basis functions are strictly localized in the real space, and the Kohn-Sham orbitals are reconstructed from the adaptive local basis functions under the discontinuous Galerkin (DG) framework. In the second part of the talk, we propose the optimized local basis functions which further improve the adaptive local basis functions, and can be used for the force calculation with application to structure optimization and molecular dynamics. We show that using the optimized local basis functions, the atomic force can be accurately evaluated by the Hellman-Feynman force with systematically controlled Pulay force. (Joint work with Weinan E, Jianfeng Lu and Lexing Ying).

We will talk about higher derivative estimates of 3D Navier-Stokes equations. Roughly, it tells us that third derivatives lie in weak L^1 in the space-time variable. This can be proven by making a quantitative version of the famous partial regularity theorem, which was proved by L. Caffarelli, R. Kohn and L. Nirenberg(1982). We will combine this theorem with Galilean invariance of transport part of the equations. This is joint work with A. Vasseur.

In this talk, a study of random times on filtered probability spaces is undertaken. One of the main messages is that, as long as distributional properties of adapted processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomized stopping time. This perspective sheds an intuitive light on results in the theory of progressive enlargement of filtrations, as is the semimartingale decomposition result of Jeulin and Yor. Financial applications of the previous theory include the role of the numeraire portfolio in stochastic finance as an indicator of overall market performance, as well as the problem of expected utility maximization from terminal wealth with a random time-horizon. Further applications in distributional properties of one-dimensional transient diffusions up to certain random times will be discussed.

My talk will be an introduction to algebraic geometry intended for people who have not taken a course in the field, assuming only a familiarity with simple algebraic notions (rings, ideals, etc). I will go over basic algebraic geometric definitions and do some examples, with the goal of describing some of the interactions between the geometry and the algebra.

The first half of the talk will be a review of how to solve, following Conway, Pell's equations such as over the integers. Roughly speaking, one enters a number (value of the form) in a few complementary components of the spine (= topograph) in the hyperbolic plane, and the solutions appear. Positive and negative values are separated by a line (= river), which is the axis of the symmetry group. The second half of the talk will sketch a generalization of this to hyperbolic 3-space, corresponding to integral binary hermitian forms. The topograph becomes the spine of the suitable Bianchi group, and the river becomes an ocean (a copy of hyperbolic plane). This is joint work with Gordan Savin.

Augmented links have played an important role in low dimensional topology in recent years. Using Agol's techniques Chesebro-DeBlois-Wilton proved that hyperbolic augmented links virtually fiber. I will provide combinatorial conditions on the diagram of such links which imply they are fibered. I will also prove that certain surgeries on these links yield fibered manifolds. This is then used to prove that links within another very large class are fibered

A symplectic resolution is a resolution of singularities of a singular variety by a symplectic algebraic variety. Examples include symmetric powers of Kleinian (or du Val) singularities, resolved by Hilbert schemes of the minimal resolutions of Kleinian singularities, and the Springer resolution of the nilpotent cone of semisimple Lie algebras. Based on joint work with P. Etingof, I define a new homology theory on the singular variety, called Poisson-de Rham homology, which conjecturally coincides with the de Rham cohomology of the symplectic resolution. Its definition is based on "derived solutions" of Hamiltonian flow, using the algebraic theory of D-modules. I will give applications to the representation theory of noncommutative deformations of the algebra of functions of the singular variety. In the examples above, these are the spherical symplectic reflection algebras and finite W-algebras modulo their center. [Note: The speaker is an Assistant Professor candidate.]

Starting with counting curves passing through given points on the plane, we generalize this elementary question step by step until reaching the general definition of Kontsevich moduli space and Gromov-Witten invariants. This talk aims at providing background and intuition, and preliminaries for a more formal introduction to Gromov-Witten on Jr Geometry by Travis next week.

The talk will be centered on the use of ergodic theory to study the dynamics of two classes of stochastic networks involving both queuing and routing, with a special focus on their infinite dimensional versions. We will first consider the class of monotone, finite dimensional stochastic queuing/routing networks. We will review the ergodic theory tools allowing one to 1) analyze their 'stability' phase transition, which separates the region where there exists a finite stationary regime from that where there is no such regime; 2) give pathwise representations of the stationary regime(s). The main focus will be on recent advances extending this approach to a classe of infinite dimensional queuing networks without routing. We will then discuss joint queuing and routing dynamics on random graphs. We will focus on the class of SINR random graphs. In the infinite graph case, we will show how to use first passage percolation and ergodic theory in order to analyze the phase transition separating the ballistic regime, where the asymptotic velocity on a route is positive, from the sub-ballistic regime, where it is 0. We will conclude by listing a few open problems and research directions along these lines and by discussing related stochastic networks where such tools could be used.

In this talk we will prove Dirichlet's classical theorem on primes in arithmetic progression. We will introduce Dirichlet characters, the zeta function, Dirichlet L-functions, and show how to analytically continue these L-functions to the complex plane. Time permitting we will also mention a few corollaries. This talk is meant to serve as preparation for Gil's talk the following week on Chebotarev's density theorem which generalizes this theorem.

In this talk we will discuss sharp regularity estimates for non-homogeneous elliptic equations , when source function lies in weak borderline spaces of integrability. All the results are optimal. If time permits, we will also discuss applications of these estimates to a class of singular free boundary problems. 

Thurston demonstrated that every link in is a torus link, a satellite link or a hyperbolic link and these three categories are mutually exclusive. It also follows from work of Menasco that an alternating link represented by a prime diagram is either hyperbolic or a --torus link. A new method for computing the hyperbolic structure of the complement of a hyperbolic link, based on ideal polygons bounding the regions of a diagram of the link rather than decomposition of the complement into ideal tetrahedra, was suggested by M. Thistlethwaite. Although the method is applicable to all diagrams of hyperbolic links under a few mild restrictions, it works particularly well for alternating (non-torus) links. The talk will introduce the basics of the method. Some applications will be discussed, including a surprising rigidity property of certain tangles, a new numerical invariant for tangles, and formulas that allow one to calculate the volume of 2--bridged links directly from the diagram.

This paper introduces a fast algorithm for computing multilinear operators in harmonic analysis. The dificulty of fast computation of multilinear operators is due to the multiplicative function. This fast algorithms is based on hierachical decomposition of matrices and a fast random method to get the low-rank approximation of matrices, so that FFT can be used to reduce the time complexity from O(N^3)for direct computation to O(Nlog(N)^2) for fast algorithm.

There have been many computational investigations into modularity of elliptic curves over number fields other than Q, especially for imaginary quadratic fields (Cremona and his students, Dieulefait--Guerberoff--Pacetti) and real quadratic fields (Dembele). In this talk we present work that treats two very different examples: the CM complex quartic field of fifth roots of unity and the non-Galois cubic field of discriminant -23. We will discuss our computational techniques and will give examples of elliptic curves over this field whose L-series apparently match those given by the Hecke data of eigenforms. This is joint work with Farshid Hajir and Dan Yasaki.

I will present material from a sequence of papers by Pierre Louis Lions. Problems in the calculus of variations which are straightforward when posed in a bounded domain can present new difficulties when posed in R^n, coming from a loss of compactness due to translation (and possibly dilation). Lions presents a method for analyzing such problems by looking more closely at how the compactness breaks down. I will sketch the proof of the main lemma and apply it to an example.

We'll give a brief history of the classical program of geometry starting with Klein's Erlangen program. We'll then talk about the synthetic geometry approach and show how from a few basic axioms for a projective plane we can construct a division ring F over which the given geometry is FP^2. If we have time, we'll talk about the symmetries of projective space.

Following Yuan's talk on Gromov-Witten invariants from last week, I will briefly give some more details on their definition and work a couple more examples. I will then introduce quantum cohomology, which is an extension of usual cohomology, with coefficients in a Novikov ring and multipliation defined in terms of Gromov-Witten invariants. Quantum cohomology plays important roles in mirror symmetry and enumerative geometry.

I will explain in as elementary terms as possible my conjecture (and theorem in dim 2), joint with Hacking and Gross, which gives a synthetic construction of the mirror to an affine CY manifold, by writing down directly its ring of polynomial functions. The main object is a natural Mori theoretic generalisation of Thurston's space of integer laminations on a punctured Riemann surface. I believe if you know what is meant by the order of zero or pole of a meromorphic function along a complex codim one submanifold, then you know enough to understand the construction. Then I will indicate some surprising implications for classical algebraic geometry: Namely, that every affine CY manifold (e.g. the complement in the plane to an elliptic curve) comes with a canonical basis of global functions, generalizing the natural trace functions on a character variety.

Last week we discussed Dirichlet's Theorem on Primes in Arithmetic Progression and answered a question about the density of primes that reduce mod m to a certain number in (Z/mZ)*. That is, given an element in the Galois group of a cyclotomic extension, we found the density of its preimage in the map "reduce mod m" from the set of rational primes to the group. This week we will replace "cyclotomic extension" with "arbitrary extension of number fields," describe what new language is needed in this context, and state Chebotarev's Density Theorem. I'll then use the theorem to compute, given a cubic polynomial, the probability that it will split a certain way modulo a random prime. With what time remains, I'll discuss questions the theorem raises that are answered by Global Class Field Theory, and how Class Field Theory can be used to generalize Dirichlet L-functions for the proof of Chebotarev's Theorem.

In this presentation we will talk about the Gross-Pitaevskii(GP) Hierarchy associated to the cubic nonlinear Schrödinger equation (NLS). After we introduce the hierarchy, we will focus on Klainerman and Machedon's proof of uniqueness of solutions to the hierarchy. Then we will look at the quantum mechanic background for the GP hierarchy and Chen and Pavlovic's work on derivation of the quintic NLS. Time permitting, we will give a brief outlook of our current work on derivation of the septic NLS. [This is a candidacy talk.]

Simplicial complexes are commonly used to represent spaces in topological data analysis. Although simplicial complexes are easy to implement, they frequently produce extremely complex and inefficient models. In the context of filtered spaces, this complexity is often prohibitive. We'll take a brief look at a how simplicial sets can be used to model spaces more effectively, and how we might go about extending this approach to the setting of filtered spaces.

The classical local Langlands correspondence associates to every complex representation of the Weil group of a local field F a corresponding representation of GL_n(F) for some n. We describe several results conjectures on the behavior of this correspondence when the Weil group representation varies in a family, and reinterpret these ideas in terms of the Bernstein center of the category of smooth GL_n(F)-modules.

I will recall the definition of cluster algebra, and then reformulate the theory in terms of birational geometry of affine CY manifolds. Then I'll explain how the Fock-Goncharov conjectures on canonical dual bases are a special case of the general mirror conjecture from my GADGET talk, and deduce from our theorem the rank 2 case of their conjecture.

We study the continuous time utility maximization of consumption together with addictive habit formation in a general incomplete semimartingale market. By introducing the properly defined auxiliary state process and modified dual space, we embed our original path dependent problem into an abstract time separable utility maximization problem with some shadow random endowment. We establish existence and uniqueness of the optimal solution using convex duality approach by defining the primal value function both on initial wealth and initial habit. At last, We will also discuss some recent progress in the utility maximization with general nonlinear habit formation.

I will introduce a discretization of the configuration space of n points on a graph. Using a combinatorial argument I will show which simple graphs have discretized configuration spaces that also turn out to be manifolds. I will then describe the structure of two distinct 2-manifolds from configuration spaces of 2 points on a graph.

In this talk, a study on -string free tangle decompositions of knots with tunnel number two will be presented. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one: , for and prime knots.

A quiver is a directed graph, and a representation of a quiver consists of a vector space at each vertex, and a linear transformation at each edge. Many problems in mathematics can be phrased in terms of the representation theory of quivers, the most direct examples being those relating to canonical forms of linear transformations. I will sketch a geometric proof (due to Bernstein, Gel'fand, and Ponomarev) of Gabriel's Theorem, which states that the quivers with finitely many indecomposable representations are precisely (orientated copies of) the ADE Dynkin diagrams.

Three dimensional cryo-electron microscopy (3D cryo-EM, for short) is the problem of determining the three dimensional structure of a large molecule from the set of images, taken by an electron microscope, of randomly oriented and positioned identical molecular particles which are frozen in a thin layer of ice. A solution to this problem is of particular interest, since it promises to be an entirely general technique which does not require crystallization or other special preparation stages. Present approaches to the problem fail with particles that are too small, cryo-EM images that are too noisy or at resolutions where the signal-to-noise ratio becomes too small.

The focus of my talk is the intrinsic reconstitution algorithm, due to Singer and Shkolnisky, which constitutes a basic step for the solution of the 3D cryo-EM problem and whose main appealing property is its remarkable numerical stability to noise. My goal is to give an introductory explanation of the mathematical principles underlying this novel algorithmic approach, while hinting about how they apply to other fundamental problems in cryo-EM and beyond. Along the way, I will describe the mathematical model underlying the experimental set-up, specifying the main computational problems/technical di¢ culties that should be resolved as part of three dimensional structure determination from cryo-EM images.

Finally, to put things in a broader mathematical perspective, I will brieáy mention the general picture: explaining how the intrinsic reconstitution algo- rithm can be recasted in the (yet to be fully developed) framework of categorical optimization, which is a novel paradigm for solving certain types of non-linear optimization problems by characterizing the solution as an object of a category instead of as an element of a set.

We will start with a couple equivalent definitions of the cohomology of a profinite group with coefficients in a discrete -module. Next we'll introduce the important functorial maps on cohomology (inflation, restriction, etc.) and give some of their properties. We'll finish with a brief summary of the cohomological part of local class field theory in characteristic zero. This talk should be vastly more accessible than my previous one.

In this presentation we will talk about the Gross-Pitaevskii(GP) Hierarchy associated to the cubic nonlinear Schrödinger equation (NLS). After we introduce the hierarchy, we will focus on Klainerman and Machedon's proof of uniqueness of solutions to the hierarchy. Then we will look at the quantum mechanic background for the GP hierarchy and Chen and Pavlovic's work on derivation of the quintic NLS. Time permitting, we will give a brief outlook of our current work on derivation of the septic NLS. [This is a candidacy talk.]

Various stability estimates in isoperimetric and Plateau-type problems are presented.

This talk is about the interface between 3-manifold topology and 4-manifold topology. Many important well-known theorems about 3-manifolds can be proven by viewing the 3-manifold as the boundary of a 4-manifold. By equating surgery on a 3-manifold with handle attachment on a 4-manifold, we can use techniques in 4-manifold topology to prove 3-manifold results (showing a quantity defined using a surgery diagram for a 3-manifold is actually an invariant of the manifold). On the other hand, computations in 3-manifolds are easier if we do not need to keep track of a 4-manifold that it bounds. We will show how to move between more general Dehn surgery on 3-manifolds and integral surgery which has meaning in 4-manifolds, and introduce some nifty surgery diagram moves along the way.

Given an elliptic curve defined over a number field , we can ask what proportion of quadratic twists of have 2-Selmer rank for any non-negative integer . The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over with . We present new results for elliptic curves with and with . I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with . Additionally, I will present some new results of my own for curves with , including some surprising results that conflict with the conjecture.

Mathematics teachers know the "official" definitions and concepts that comprise school mathematics, but our students may see things differently. Where might their creative mathematical thinking lead us? Join us for a new look at some familiar mathematics, and for extensions that will stretch your own creative mathematical thinking!

This talk is about a connection between combinatorial topology and probability models of a gas in equilibrium. A cluster expansion for such a model is an expansion in a small parameter with terms that are indexed by connected graphs. What makes this work is a remarkable cancellation that reduces the convergence problem to another expansion with terms that are indexed by tree graphs. This cancellation in turn depends on topological properties of the set of connected graphs on a given vertex set.

In quantum mechanics, a resonance is a pole of an analytic continuation of the resolvent of the Hamiltonian. Physically, resonances correspond to a states that initially behave like bound states but eventually decay. In this talk, I'll carefully define resonances and provide some examples that come up in physics. Time permitting, I'll prove a decay estimate.

We consider a class of finite dimensional Ito processes for which we build simpler processes (mimicking processes) such that their marginal probability distributions match at each time . The mimicking process is the unique weak solution of a stochastic differential equation whose coefficients are allow to be degenerate and be unbounded. To achieve this, we show existence, uniqueness and regularity of solutions in weighted Holder spaces to a suitable initial value problem.

I will describe a project to create various tables of elliptic curves over K=, and explain some of my motivation for studying such curves. This talk will also touch on explicit computation of Hilbert modular forms and the Birch and Swinnerton-Dyer conjecture.

Twenty years ago Scharlemann and Thompson used deep results from sutured manifold theory to prove that a genus reducing crossing change on a knot maybe be realized as untwisting a Hopf band plumbed onto a minimal genus Seifert surface. This gives a hint that understanding genus reducing crossing changes is closely related to understanding how a compact surface in S^3 changes when it is twisted. In this talk we use modern technology from the theory of Heegaard splittings to show that understanding when two surfaces are related by a single twist implies the existence of an algorithm to determine when two (hyperbolic or fibered) knots of different genus are related by a single crossing change.

I'll give an introduction to quiver varieties a la Lusztig and Nakajima, focusing on many examples: flag varieties, Kleinian singularities, Hilbert schemes, Calogero-Moser spaces, commuting varieties.

Mathematical models of phase transitions in statistical physics lead to random fractal structures appearing at the critical value of a parameter. I will introduce some of these models (self-avoiding walk, loop-erased walk) and then focus on some of the mathematical tools that are used to analyze such models. I will emphasize two dimensions where complex analysis plays a major role. As time permits, I will discuss some more recent results in this area. This talk is intended for a general mathematical audience.