Dehn surgery along a knot in a 3-manifold is a procedure that you drill out a tubular neighborhood of a knot and glue it back in a possibly different way. A well-known result of Lickorish–Wallace states that any closed oriented connected 3-manifold is obtained by surgery along a link in S^3. However, surgery on which knots yields which manifolds is still a widely open question which has been actively studied until recent years. Two among those are cabling conjecture and Berge conjecture. I will be talking about the settings of those conjectures, some recent results on them, and their variations on Lens spaces and the Poincaré homology sphere.

Time-harmonic scattering in variable media can often be modeled by linear elliptic partial differential equations. Such equations pose several numerical challenges. For example, they can be intrinsically ill-conditioned, necessitate the imposition of radiation conditions, and produce pollution errors when discretized with standard finite difference or finite element methods. To avoid these issues, it is often convenient to first reformulate the differential equation as an integral equation. The tradeoffs are that an integral operator with a singular kernel must be discretized and that the resulting linear system that must be inverted is dense. Sometimes, the latter issue can be handled using a fast matrix-vector product algorithm (e.g., the fast Fourier transform or the fast multipole method) paired with an iterative solver (e.g., GMRES). However, this approach can be prohibitively slow when there is a large amount of backscattering and when multiple incident fields are of interest. In these cases, it is better to use direct solvers. In this talk, I describe some recent projects on developing direct solvers in this regime with applications to acoustic scattering and metasurface design.

One of the most fundamental unsolved problems in representation theory is to classify the set of irreducible unitary representations of a semisimple Lie group. In this talk, I will define a class of such representations coming from filtered quantizations of certain Poisson varieties. The representations I construct are expected to form the "building blocks" of all unitary representations.

The study of closed geodesics on hyperbolic surfaces is a subject that shares deep connections with many areas of mathematics. We survey classic and recent results in the subject, emphasizing the relations between three different perspectives: topology, geometry, and arithmetic. In particular, we discuss recently discovered connections with mapping class groups and Teichmüller dynamics. The talk assumes no previous knowledge on the subject and is aimed at a wide mathematical audience.

In this talk we will try to explain the connections between Fourier summation formulas, quasicrystals, crystalline measures, almost periodic functions and holomorphic modular forms. The idea is to culminate in a new recent classification result by the author arXiv:2312.11185 and discuss implications, and future work, in the description of physical quasicrystals and their diffraction patterns.

This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics. My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress in the field. If time permits, I will propose several important open questions.

In this talk, we will discuss some recents developments around two classical problems in discrete geometry, where projection theory plays an important role: the Erdős-Szekeres problem and the Heilbronn triangle problem. The former concerns the question of determining the size of largest convex subset in a given finite set of points, whereas the latter asks for the size of the smallest area triangle (also as a function of the number of points), if the point set is constrained to lie in a bounded region. Along the way, we will also be discussing several other old and new connections between these problems and various different parts of mathematics.

We present a general framework for approximating non-linear maps between infinite dimensional Banach spaces from observations. Our approach follows the "discretize last" philosophy by designing approximation architectures directly on the function spaces of interest without tying parameters to any finite dimensional discretization. Such architectures exhibit an approximation error that is independent of the training data discretization and can utilize data sources with diverse discretization common to many engineering problems. We review the infinite-dimensional approximation theory for such architectures, showing the universal approximation property and the manifestation of the curse of dimensionality translating algebraic rates in finite dimensions to exponential rates in infinite dimensions. We discuss efficient approximation of certain operators arising from parametric partial differential equations (PDEs) and show that efficient parametric approximation implies efficient data approximation. We demonstrate the utility of our framework numerically on a variety of large-scale problems arising in fluid dynamics, porous media flow, weather modeling, and crystal plasticity. Our results show that data-driven methods can provide orders of magnitude in computational speed-up at a fixed accuracy compared to classical numerical methods and hold immense promise in modeling complex physical phenomena across multiple scales.

An important object in the study of high-dimensional non-convex functions is the so-called complexity, defined as the logarithm of the number of critical points. For random functions, this is typically obtained by the Kac-Rice formula, which requires in particular the computation of certain random Hessian determinants. In this talk I will present some recent progresses concerning the complexity of the Thouless-Anderson-Palmer (TAP) energy associated to the celebrated Sherrington-Kirkpatrick model for spin glasses. A joint work with D. Belius, F. Concetti.

Systems of interacting particles describing notable physical phenomena, such as Bose-Einstein condensation, superconductivity or superfluidity, exhibit a daunting complexity. This complexity renders the exact many-body theory computationally non-approachable, even for physicists conducting computer experiments and simulations. Therefore, an approximate description using effective macroscopic models is highly useful, and the rigorous study of the regime of validity of such approximations is of primary importance in mathematical physics. In this talk we will focus on the dynamics of systems made of quantum particles in the mean-field regime, where weakly interacting particles exhibit a collective behavior approximated by an averaged potential in convolution form. We study time scales where the semiclassical description becomes relevant. Through a novel technique based on weak-strong stability principles for partial differential equations, we show that the many-body dynamics is well approximated by a Vlasov equation describing the evolution of the effective probability density of particles on the phase space.

In the 19th century, German Mathematician Hermann Schubert was interested in questions in enumerative geometry, like "given 4 generic lines in 3-space, how many lines intersect all four lines?" The modern treatment of such questions centers around the study of the cohomology ring of the complete flag variety, with a basis given by the classes of Schubert varieties. While this topic and its variations have been extensively studied, a central question, which seeks a combinatorial interpretation of the structure constants, remains open. However, the same question has been fully solved using many different methods in the special case when the classes are pullbacks from the Schubert classes in Grassmannians, in which case the Schubert structure constants are known as the Littlewood-Richardson coefficients. The ``lifting dream'' of Schubert calculus hopes to find ways to lift the Littlewood-Richardson rules to the general case. In this talk, I will talk about potential venues to lift the combinatorial and geometric techniques for the Grassmannian case using bumpless pipe dreams and positroid varieties, and some success stories so far.

Modular forms are classical mathematical objects dating back to the 19th century. Because of its connections to and appearances in many different areas of math and physics, it remains a popular subject today. Since the work of Hans Maass in 1949, real-analytic modular forms have found important applications in arithmetic geometry and number theory. In this talk, I will discuss the amazing works in this area over the past 20 years, and give a glimpse of its fascinating future directions.

We study subgroups of the Lie group SL(3,R) which are discrete in the subspace topology and isomorphic to the abstract group generated by reflections along the sides of a hyperbolic triangle. If we require a slightly stronger property (Anosov representations), we can find all such groups by studying the dynamics of their action on the projective plane. More precisely, we can show that a representation is Anosov by finding special "boxes" in the projective plane which are mapped into each other by certain group elements. Iterating this, the boxes converge to a fractal curve called the limit curve. We hope that this provides a simple model to help understand the space of Anosov representations in other contexts like surface group representations. All of it is joint work with Gye-Seon Lee and Jaejeong Lee.

We will discuss some computations of ribbon concordances between knots and talk about the methods and the results. This is a joint work with Nathan Dunfield.

In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to dynamical systems, and I will end with some results following from viewing the periodic nonlinear Schrödinger equation as an infinite dimensional Hamiltonian system.

Serre's modularity conjecture (1973) was proven by Khare-Wintenberger building on work of Kisin in the mid-2000s. The methods and problems surrounding the conjecture have played an important role in the field of modularity lifting and the p-adic Langlands program. One such development is the recent construction of moduli stacks of p-adic Galois representation by Emerton--Gee. I will explain the connection between Emerton--Gee's space and Serre's conjecture in the simplest case of two-dimensional representation of the Galois group of Q_p. I will then go on to describe how this geometry helps us understand higher dimensional generalizations of Serre's conjecture based on joint work with Daniel Le, Bao V. Le Hung, and Stefano Morra..

Stacking a few layers of 2D materials such as graphene or molybdenum disulfide at controlled twist angle has opened the possibility of tuning the electronic properties of 2D materials. One of the main issues encountered in the mathematical and computational modeling of twisted 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. We have developed a novel variational model for elastic relaxation and a new mathematical formulation for the electronic density of states, band structure, and transport for the incommensurate Hamiltonian. New computational approaches will be presented, and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.

In this series of two talks talk I will survey the construction of canonical (intrinsic) mirror families via the canonical wall structure, obtained in joint work with Mark Gross. The construction is based on punctured Gromov-Witten invariants previously defined in joint work with Abramovich and Qile Chen, that I will also comment on. The first talk is targeted at a general audience with an algebraic geometric background.

Whether the Landau equation can develop a finite time singularity is an important open problem in kinetic equations. In this talk, we will discuss the slightly perturbed homogeneous Landau equation with very soft potentials, where we increase the nonlinearity from $ c(f) f$ in the Landau equation to $\alpha c(f) f$ with . For $\alpha 1 $ and close to , we establish finite time nearly self-similar blowup from some smooth nonnegative initial data, which can be radially symmetric or non-radially symmetric. The blowup results are sharp as the homogeneous Landau equation $(\alpha=1)$ is globally well-posed, which was recently established by Guillen and Silvestre. The proof builds on our previous framework on sharp blowup results of the De Gregorio model with nearly self-similar singularity to overcome the diffusion. Our results shed light on potential singularity formation in the inhomogeneous setting.

We develop algebraic geometry for the coupled cluster (CC) theory of quantum many-body systems. Coupled cluster theory provides a large class of high-accuracy methods that are widely employed in the computational many-body community. In this ansatz, the high-dimensional eigenvalue problem that encodes the electronic Schrödinger equation is approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Plücker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations. The talk is on Zoom - https://utexas.zoom.us/j/91039414253 - and will also be broadcast in PMA 12.166.

Neural operators are a class of deep learning data-driven architectures that learn maps between function spaces. Typically, an operator like that is capable of learning how to solve a family of a particular PDE problem by learning the operator that maps the known function(s) of an equation to the unknown solution. One of the challenges I would like to address in this talk is how to obtain data in order to train such models. Classical numerical method (such as finite element/differences) have been used to solve instances of PDEs for training and testing models. However, if we wish to outperform classical solvers, how can we really still depend on them to generate training data? This talk will be based on a recent preprint with Prof. Rachel Ward (see https://arxiv.org/pdf/2401.02398.pdf).

The Levine-Tristram signature of a link is a classical invariant with many definitions and applications – it is closely related to the Alexander polynomial, provides lower bounds on topological invariants such as the slice and doubly slice genus, and is almost-everywhere a concordance invariant. In 2018 Kashaev introduced a link invariant using a simple algorithm on link diagrams which he conjectured also computes the Levine-Tristram signature. In this talk we give a proof of Kashaev’s conjecture using the original Seifert surface definition of the Levine-Tristram signature. The proof also gives another way of computing the Alexander polynomial.

I will speak about the paper by June Huh and Botong Wang with the same title, where the Bellinson-Bernstein-Deligne decomposition theorem is used to prove Dowling and Wilson’s ‘top-heavy’ conjecture in combinatorial geometry.

In this series of two talks talk I will survey the construction of canonical (intrinsic) mirror families via the canonical wall structure, obtained in joint work with Mark Gross. The construction is based on punctured Gromov-Witten invariants previously defined in joint work with Abramovich and Qile Chen, that I will also comment on. The first talk is targeted at a general audience with an algebraic geometric background.

The talk originally scheduled at this time will have to be rescheduled.

The (Hilbert metric) geometry of properly convex domains generalize real hyperbolic geometry. This generalization is far from the Riemannian notion of non-positive curvature, but they have some intriguing similarities. I will explore this connection from a coarse geometry viewpoint. The focus will be on Morse geodesics ("negatively curved directions", in a coarse sense) in properly convex domains. I will show that Morse-ness can be characterized entirely using linear algebraic data (i.e. singular values of matrices that track the geodesic). Further, I will discuss how this coarse geometric notion of Morse is related to the symmetric space geometry as well as the smoothness of boundary points. This is joint work with Theodore Weisman.

The reconstruction of flexible proteins is one of the most critical challenges in structural biology, allowing us to gain insights into the functions and mechanisms of biomolecules by observing their motion. Cryogenic electron microscopy (cryo-EM) stands out as an ideal technique for studying the dynamic conformational landscape (i.e., range of motions) as it can capture a snapshot of the entire conformational ensemble. However, this reconstruction task comes with notable mathematical and computational challenges due to massive datasets, sometimes exceeding terabytes, high dimensionality, and substantial noise. After delving into the basics of the cryo-EM reconstruction problem, I will introduce a framework for reconstructing the protein distribution in a dataset by representing it in a linear subspace. The initial step can be viewed as a linear algebra problem: how can one compute a basis for proteins (3D volumes) from only incomplete and noisy measurements (2D images)? I will propose a method based on a Nyström extension of a regularized estimator of the covariance of the volumes. In subsequent steps, we will use this low-dimensional basis to reconstruct individual volumes using standard statistical methods and infer motions using elements of optimal control. I will conclude by discussing remaining challenges and open problems.

There are two perspectives we often get on a ribbon disk for a knot K in the 3-sphere: A ribbon disk is defined as either an immersed disk in S^3 with only ribbon self-intersections, or an embedded disk in B^4 with no maxima with respect to the radial Morse function. These definitions are often called "equivalent," but more technically, the difference between them is like the difference between a knot and a knot diagram; one is an embedding, and one is a lower-dimensional projection of an embedding. The "ribbon number" r(K) for a ribbon knot K is the smallest number of ribbon intersections among any projection of any ribbon disk bounded by K, and r(K) can then be seen as a sort of crossing number for ribbon disks. The Alexander polynomial of K can be used with rather surprising accuracy to give lower bounds for r(K), and we discuss our work in understanding and cataloguing ribbon numbers for low-crossing ribbon knots. Parts of this talk are joint with Stefan Friedl, Jeffrey Meier, Filip Misev, and lots of undergrads.

Let F be a p-adic field and G the F-points of a reductive group over F. If U is a compact open subgroup of G, and R is a ring, then the Hecke algebra attached to U, G, and R is the space of compactly supported R-valued functions on G that are left and right U-invariant; convolution gives this space the structure of an R-algebra. When R is a field of characteristic zero, a celebrated result of Bernstein shows that such Hecke algebras have good finiteness properties; in particular they are Noetherian. For R a more complicated ring such as Z_{\ell}, the question has been open for over thirty years. I will explain how recent results of Fargues and Scholze can be used to resolve this question.

Chern-Weil theory is, in my opinion, one of the all-time great mathematical theories. It unites the hard differential geometry concepts of connections and curvature with the soft algebraic topological idea of a characteristic class. The prerequisites are a familiarity with smooth manifolds, vector bundles, and de Rham cohomology.

Phase transitions of interacting particles in statistical mechanics can be seen in computer simulations, but cannot be mathematically proven. A few ‘spin’ models, chiefly the 2-dimensional Ising model, can be proven to show a transition. We show the underlying difficulty, and discuss a new class of models in which some examples can be proven to show transitions. Relevance to scissors congruence and the Banach Tarski paradox will be discussed if there is time.

The Boltzmann hierarchy is an infinite system of coupled equations that are instrumental for the rigorous derivation of the Boltzmann equation from many particles. In this talk we will first show uniqueness of the mild solutions to the Boltzmann hierarchy by combining, for the first time, a combinatorial technique known as the Klainerman-Machedon board game argument together with an -based estimate. Then we will show existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data. The proof of existence is constructive, and employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.

In this talk we will focus on the derivation and the analysis of the inhomogeneous kinetic wave equation (IKWE). Although there is an extended amount of literature for the space homogeneous KWE, the inhomogeneous problem has barely been addressed. In the first part we examine the validity of the IKWE for quadratic nonlinearities and dispersion relations close to Laplacian, based on joint work with Collot, Germain. In the second part of the talk, we investigate the global well-posedness and stability of the 4-wave inhomogeneous kinetic wave equation near vacuum, employing techniques from classical kinetic theory, such as the Kaniel-Shinbrot iteration.

Two knots are called n-friends if they share the same n-surgery. Manolescu and Piccirillo defined RBG links, a kind of three-component framed links in S^3, which provide a systematic way to look for 0-friends. In this talk, we will explore how to generalize their RBG-link technique to find n-friends and how certain n-friends have the potential to create exotic simply-connected definite 4-manifolds. Finally, we will focus on a special subclass of RBG links, called k-special RBG links, which can be used to obstruct knots from being (-k)-slice (i.e. bounding a disk with self-intersection number k) in mCP^2.

We consider an abelian variety A defined over various base fields F, and discuss its arithmetic over the cyclotomic Z_p-extension and more general p-adic Lie extensions. After reviewing some known results over number fields, we shift our focus to the case of global function fields. In this context, we compare the arithmetic of A over different p-adic Lie extensions without assuming the finiteness of the Selmer group of A over the base field F.

We will show that a conic in P^2 has a solution in Q if and only if it has solutions in Q_p for all primes p. If you don’t know what some of these words mean it’s no problem, everything will be introduced. The proof will rely on the volume of an ellipsoid and the fact that pi 3. Along the way we will be introduced to the p-adic numbers and the Minkowski bound.

This is the first of a two-talk series about Atiyah Bott localization in enumerative geometry. We will first (quickly) review equivariant cohomology and then discuss Atiyah Bott localization for smooth complex varieties with a torus action and finitely many fixed points. Anyone with a basic background in algebraic geometry or topology should be able to follow this talk.

We study an optimal transport problem with a backward martingale constraint in a pseudo-Euclidean space S. We show that the dual problem consists in the minimization of the expected values of the Fitzpatrick functions associated with maximal S-monotone sets. An optimal plan γ and an optimal maximal S-monotone set G are characterized by the condition that the support of γ is contained in the graph of the S-projection on G. For a Gaussian random variable Y, we get a unique decomposition: Y = X +Z, where X and Z are independent Gaussian random variables taking values, respectively, in complementary positive and negative linear subspaces of the S-space. Based on joint work with Dmitry Kramkov: https://arxiv.org/abs/2209.04664.

Far-from-equilibrium behavior in physical systems is widespread. A statistical description of these events is provided by macroscopic fluctuation theory, a framework for non-equilibrium statistical mechanics that postulates a formula for the probability of a space-time fluctuation based on the constitutive equations of the system. This formula is formally obtained via a zero noise large deviations principle for the associated fluctuating hydrodynamics, which postulates a conservative, singular stochastic PDE to describe the system out-of-equilibrium. In this talk, we will focus particularly on the fluctuations of certain interacting particle processes about their hydrodynamic limits. We will show how the associated MFT and fluctuating hydrodynamics lead to a class of conservative SPDEs with irregular coefficients, and how the study of large deviations principles for the particles processes and SPDEs leads to the analysis of parabolic-hyperbolic PDEs in energy critical spaces. The analysis makes rigorous the connection between MFT and fluctuating hydrodynamics in this setting, and provides a positive answer to a long-standing open problem for the large deviations of the zero range process.

As many a depressed undergraduate learning ring theory has been unlucky to learn, not all rings are UFDs. Primary decomposition allows for a different type of factorization in a more general class of rings. In my talk, I will introduce the primary decomposition of an ideal and discuss some of its properties. I will then describe the classification of rings where primary decomposition holds (called Lasker rings). Finally, I will end with a discussion of the geometric interpretation of primary decomposition.

The Hofstadter Hamiltonian describes an electron moving on a two-dimensional lattice under the influence of a perpendicular magnetic field. It has been important for the theory of integer quantum Hall effect. This model, and the related almost-Mathieu maps, are a popular playground in the area of Schrödinger operators, dynamical systems, quantum groups, etc. A famous plot, known as the Hofstadter butterfly, displays striking self-similarity properties of the spectrum. The appropriate approach to such phenomena, developed first in the theory of critical phenomena, is renormalization. In this talk, besides introducing the models, describing phenomena, and showing interesting plots, I will describe a "universality" conjecture and proofs in special cases. Some of these proofs are computer-assisted.

We consider axisymmetric incompressible inviscid flows without swirl. When the axial vorticity is non-positive in the upper half space and odd in the last coordinate, we call the flow anti-parallel and we may expect a head-on collision of anti-parallel vortex rings. By establishing monotonicity and infinite growth of the vorticity impulse on the upper half-space, we obtain infinite growth of vorticity maximum at infinite time for certain classical vorticity. On the other hand, a finite but faster growth for some smooth vorticity is obtained thanks to the stability of Hill's vortex. This talk is based on joint work with In-Jee Jeong(SNU).

Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on the preprint arXiv:2301.13227.

This is joint work with Abert and Fraczyk. Given a percolation E of the Cayley graph of a group G (thought of as a random subgraph), Burton-Keane and Lyons-Schramm showed the existence of the trajectory density, which is the almost sure limit (1/n)|\&ob;1\leq j\leq n:g_&ob;1&cb;\cdots g_&ob;j&cb; is connected to the identity in E&cb;|. for a random walk g_&ob;1&cb;\cdots g_&ob;j&cb; in the Cayley graph of G. Moreover, this limit is a.s. independent of the random walk. We give a separate proof of this result in the more general context of inclusion of equivalence relations S inside of R (where R is given by a free action of G). Moreover, we give a complete computation of this quantity in terms of the index of R restricted to the ergodic components of the subrelation. When G has Property (T) we show that zero trajectory density implies a uniform upper bound (independent of S) on the co-spectral radius (in the sense of our previous work) of S inside of R.

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin—Selberg, with applications to the Bloch—Kato conjecture and to Iwasawa theory. For this talk, I'll consider Galois representations attached to automorphic forms on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). I'll discuss a new "first explicit reciprocity law" in this setting and its application to the corresponding Bloch—Kato conjecture, focusing on new obstacles which arise from the lack of local multiplicity one.

We know them from algebra class, but we never see them around and we just can’t multiply them very well. To bring them out of hiding, I will explain some four-dimensional geometry and algebra using the quaternions.

In the third and final talk in a series on Atiyah-Bott localization, we will count some lines. We'll discover that there are 27 lines on a general cubic surface in P^3 and, time permitting, will see that there is a unique line through two general points of P^2. It seems Euclid must have known about torus actions.

This will be an informal talk about a class of equations that arise in various areas of applied probability, such as optimal stochastic control, stochastic differential games, and mathematical economics/finance, but are also related to stochastic analysis on manifolds and Karcher means. Formally, they resemble classical stochastic differential equations but come with a terminal condition instead of an initial one. This 'minor' difference leads to a class of significantly different problems, which require distinct techniques. Unlike SDEs, BSDEs are an active area of research. Indeed, while the one-dimensional case has been well understood for over 20 years, basic questions of existence and uniqueness about systems of BSDEs are still wide open.

The breakthrough of Mouhot and Villani (2011) on the so called Landau Damping for the collision-less Vlasov equation \begin{eqnarray} & \partial_t F +v\cdot \nabla_x F \pm \nabla_x V\cdot\nabla_v F=0 \; \hbox{with} \; (x,v) \in (\Omega \subset (\mathbf R_v)^d\times \mathbf R^d)\\ & -\Delta V =G+f = F- \langle F\rangle, \; \langle F \rangle =\int_{\Omega} F(x,v,t) dx \end{eqnarray} has generated in the mathematical community many activities around the qualitative behavior (in particular for large time) of their solutions and for the behavior of the space independent average: \begin{equation} G(v,t) = \int_{\Omega} G(t,v,x)dx\,. \end{equation} As I intend to show, starting from physical motivation, such behavior depends on the spectra of the linearized operator and on the size of the perturbation. Existing mathematical results are in line with a very rich and diverse behavior.

Schubert calculus involves studying intersection problems among linear subspaces of C^n. A classical example of a Schubert problem is to find all 2-dimensional subspaces of C^4 which intersect 4 given 2-dimensional subspaces nontrivially (it turns out there are 2 of them). In the 1990's, B. and M. Shapiro conjectured that a certain family of Schubert problems has the remarkable property that all of its complex solutions are real. This conjecture inspired a lot of work in the area, including its proof by Mukhin-Tarasov-Varchenko in 2009. I will present a strengthening of this result which resolves some conjectures of Sottile, Eremenko, Mukhin-Tarasov, and myself, based on surprising connections with total positivity, the representation theory of symmetric groups, symmetric functions, and the KP hierarchy. This is joint work with Kevin Purbhoo.

Computing transport is a fundamental problem in engineering and a fertile area of applied mathematics methods and ideas. Although significant progress has been made, the complexity and importance of the problem call for novel methods to overcome limitations of current approaches. The models of interest in this presentation are Fokker-Planck type equations in which local (e.g., diffusive) transport is represented by differential operators and nonlocal transport by integrodifferential operators (e.g., non-diffusive turbulent transport closures and fractional diffusion). Computational approaches for these problems can be roughly classified as deterministic methods (e.g., finite-difference) and stochastic methods (e.g., Monte-Carlo). Although extensively used, continuum deterministic methods face stability and scalability challenges specially in the case of nonlocal operators that result in dense (non-sparse) matrices. On the other hand, particle-based stochastic methods face poor convergence due to statistical sampling. Here we present an alternative approach based on the Feynman-Kac theory that establishes a link between the Fokker-Planck equation and the stochastic differential equation of the underlying stochastic process. The proposed method reduces the computation to the deterministic evaluation of mathematical expectations (i.e., integrals) bypassing the need of discrete sampling of individual stochastic trajectories. The resulting algorithm is unconditionally stable and parallelizable. We present applications of the method to the study of the exit time problem and the initial value problem for local and nonlocal Fokker-Planck equations modeling transport in fluid dynamics and plasma physics.