Please click here to register for this event. We have space for up to 30 registrants on a first come, first serve basis.
We may be able to provide some financial support for grad students and postdocs interested in this event. If you are interested in funding, please send a letter of support from your mentor to Hansol Hong at hansol84@gmail.com.
Confirmed Speakers:
The schedule is as follows:
Thursday 4/5/2018
Time  Speaker  Title/Abstract 
12:001:30pm  Lunch  
1:302:30pm  Tristan Collins  Title: BPS Bbranes and stability
Abstract: I will give a short introduction to the deformed HermitianYangMills equation and discuss the (conjectural/motivational) relationship with stability in the sense of Bridgeland.
This talk will cover joint work with A. Jacob, D. Xie, and S.T. Yau.

2:302:45pm  Break  
2:453:45pm  Dimitry Vaintrob  Title: Operads and circle actions
Abstract: Cohomology of the topological operad FLD of framed little disks (a.k.a. the BV operad) acts on the Hochschild homology of any CalabiYau algebra. Cohomology of the related topological operad of marked nodal genus zero curves acts on a deformation of the cohomology of any symplectic manifold, and this action is responsible for all quantum product operations. It was proven by Bruno Vallette and Drummond Cole that an action of $\mathbb{Q}$homology of the operad of marked nodal curves is equivalent, in genus zero, to an action of the homology of the operad of framed little disks together with a trivialization, in a homotopytheoretic sense, of the BV operator $\Delta$. In a later paper DrummondCole showed that this result holds in a certain category of topological operads, so that in particular it is also true (on the dg level) for cohomology with coefficients in $\mathbb{Z}$ (or in an arbitrary field). In work with Alex Oancea we give a higher genus version of this result, using Segal moduli spaces of curves with parametrized boundary and their compactifications. Time permitting, I will also mention certain motivic enhancements of our result (based on more recent work) which give compatibility with Galois actions and de Rham lattices on the two sides, a result already new in genus zero. 
3:454:15pm  Break  
4:155:15pm  Mandy Cheung  Title: Counting tropical curves by quiver representation
Abstract: The work of GrossHackingKeelKontsevich tell the relation between scattering diagrams and cluster algebras. In the talk, we will describe those objects with quiver representations. After that, we will give a expression of tropical curves counting by quiver representations. This is a joint work in progress with Travis Mandel. 
Friday 4/6/2018
Time  Speaker  Title/Abstract 
9:00 – 9:30am  Breakfast  
9:3010:30 am  Zack Sylvan  
10:3011:00am  Break  
11:0012:00pm  Yu Pan  Title: Augmentations categories and exact Lagrangian cobordisms.
Abstract: To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrianknots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomologyof morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

12:001:30pm  Lunch  
1:302:30pm  CheukYu Mak  Title: Tropically constructed Lagrangians in mirror quintic threefolds
Abstract: In this talk, we will explain how to construct closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toricdegeneration technique. As an example, we will illustrate how the corresponding Lagrangians look like for tropical curves that contribute to the Gromov–Witteninvariant of the line class of the quintic threefold. We will also show that multiplicity of a tropical curve, in this symplectic setting, will be realized as the order of the torsion the first homology group of the Lagrangian. This is a joint work with Helge Ruddat.

2:302:45pm  Break  
2:453:45pm  YuShen Lin  
3:454:15pm  Break  
4:155:15pm  Yoosik Kim  Title: Mirror construction of Grassmannians via immersed Lagrangian Floer theory
Abstract: A partial flag manifold admits a completely integrable system, socalled a GelfandCetlin system, constructed by GuilleminSternberg. The fibers of the system are almost like toric fibers. However, as the big torus action does not extend to boundary strata, nontoric Lagrangian fibers may appear at a boundary stratum. In the first part of the talk, we classify all Lagrangian fibers on partial flag manifolds of various types. After discussing it, we exhibit a construction of a mirror of some low dimensional Grassmannians using StromingerYauZaslow mirror symmetry. To incorporate nontoric Lagrangian fibers, which are sometimes nonzero objects in the Fukaya category, we produce immersed Lagrangians arising from smoothing faces containing a face having nontoric Lagrangian. We then glue deformation spaces of Lagrangians to obtain the Rietsch mirror. This talk is based on joint work with Yunhyung Cho and YongGeun Oh, and ongoing joint work with Hansol Hong and SiuCheong Lau. 
Saturday 4/7/2018
Time  Speaker  Title/Abstract 
8:309:00am  Breakfast  
9:0010:00am  Jacob Bourjaily 
Title: Stratifying OnShell Cluster Varieties
Abstract: There exists a deep correspondence between a class of physically important functions—called “onshell functions”—and certain (cluster variety) subspaces of Grassmannian manifolds, endowed with a volume form that is left invariant under cluster coordinate transformations. These are called “onshell varieties” (which may or may not include all cluster varieties). It is easy to prove that the number of onshell varieties is finite, from which it follows that the same is true for onshell functions. This is powerful and surprising for physics, because these onshell functions encode complete information about perturbative quantum field theory.
In this talk, I describe the details of this correspondence and how it is constructed and give the broad physics motivations for obtaining a more systematic understanding of onshell cluster varieties. I outline a general, bruteforce strategy for classifying these spaces; and describe the results found by applying this strategy to the case of Gr(3,6).

10:0010:15am  Break  
10:1511:15am  ShuHeng Shao  Title: Vertex Operator Algebra, WallCrossing Invariants, and Physics
Abstract: Motivated by fourdimensional conformal field theory with N=2 supersymmetry, we discuss an interesting relation between vertex operator algebras (VOAs) and KontsevichSoibelman wallcrossing. We discuss a conjectured formula for the vacuum character of this VOA from the associated KontsevichSoibelman wallcrossing invariant of the fourdimensional field theory. We further generalize this proposal to include extended supersymmetric objects, known as line defects and surface defects, into the fourdimensional field theory. Each such defect gives rise to a module of the associated VOA and we propose a formula for the character of this module. 
11:1511:30am  Break  
11:3012:30pm  Mauricio Romo  Title: Aspects of Btwisted (2,2) and (0,2) hybrid models
Abstract: I will talk about properties and definition of certain sphere correlators for elements on the chiral ring of Btwisted hybrid models for the case they posses (2,2) and (0,2) supersymmetry. I will review these models and their Bchiral ring. I will present some interesting analytic properties of these correlators and some sufficient criteria for the absence of instanton corrections in the (0,2) case.

The organizing committee consists of Yang Wang (HKUST), Ronald Lui (CUHK), David Gu (Stony Brook), and ShingTung Yau (Harvard).
Please click here to register for the event.
Confirmed Speakers:
This event is supported by the CMSA and the NSF.
Schedule:
Saturday, March 24
Time………….  Speaker  Title/Abstract 
9:009:30am  Breakfast & opening speech  
9:3010:30am  Stephen Wong
Houston Methodist/Weill Cornell Medicine 
Title: Applications of deep learning in pathologic image diagnosis and high content screening 
10:3011:00am  Break  
11:0012:00pm  Lakshminarayanan Mahadevan
Harvard University 
Title: Programming Shape 
12:001:30pm  Lunch  
1:302:30pm  Monica Hurdal
Florida State University 
Title: Geometry, Computation, and Modeling the Folding Patterns of the Human Brain
Abstract: The folding patterns of each brain are unique. There is much controversy in the biological community as to how the folding patterns of the brain develop and if the folding patterns can be used to identify and diagnose disease. In this presentation, I will discuss some of the mathematical and modeling approaches my research group is using to investigate these topics. Conformal mapping, topology, and Turing patterns are some of the methods we are using to characterize and model the folding patterns of the human brain in development, health, and disease. 
2:302:45pm  Break  
2:453:45pm  Allen Tannenbaum
Stony Brook University 
Title: Optimal Mass Transport for MatrixValued Densities: A Quantum Mechanical Approach
Abstract: Optimal mass transport is a rich area of research with applications to numerous disciplines including econometrics, fluid dynamics, automatic control, transportation, statistical physics, shape optimization, expert systems, image processing, and meteorology. In this talk, we describe a noncommutative counterpart of optimal transport where density matrices (i.e., Hermitian matrices that are positivedefinite and have unit trace) replace probability distributions, and where “transport” corresponds to a flow on the space of such matrices that minimizes a corresponding action integral. We employ generalizations of the seminal approach of Benamou and Brenier. In particular, we utilize ideas from quantum mechanics in a Benamou–Brenier framework. Our version of noncommutative optimal mass transport allows us to define geodesics on the space of positivedensities. Applications are given to diffusion tensor MR data. This is joint work with Yongxin Chen and Tryphon Georgiou. 
3:454:15pm  Break  
4:155:15pm  David Gu
Stony Brook University 
Title: A Geometric View to Generative Models
Abstract: In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This leads to a geometric interpretation to generative models, and leads to a novel framework for generative models. By using the optimal transportation view of GAN model, we show that the discriminator computes the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a closeform formula. Therefore, it is sufficient to solely optimize the discriminator. This shows the adversarial competition can be avoided, and the computational architecture can be simplified. Preliminary experimental results show the geometric method outperforms WGAN for approximating probability measures with multiple clusters in low dimensional space. 
5:156:15pm  Shikui Chen
Stony Brook University 
Title: Design for Discovery: Generative Design of Conformal Structures using LevelSetBased Topology Optimization and Conformal Geometry Theory
In this method, a manifold (or freeform surface) is conformally mapped onto a 2D rectangular domain, where the level set functions are defined. With conformal mapping, the corresponding covariant derivatives on a manifold can be represented by the Euclidean differential operators multiplied by a scalar. Therefore, the TO problem on a freeform surface can be formulated as a 2D problem in the Euclidean space. To evolve the boundaries on a freeform surface, we propose a modified HamiltonJacobi Equation and solve it on a 2D plane following the Conformal Geometry Theory. In this way, we can fully utilize the conventional levelsetbased computational framework. Compared with other established approaches which need to project the Euclidean differential operators to the manifold, the computational difficulty of our method is highly reduced while all the advantages of conventional level set methods are well preserved. The proposed computational framework provides a solution to increasing applications involving innovative structural designs on freeform surfaces for different fields of interests. 
Sunday, March 25
Time………….  Speaker………..  Title/Abstract 
9:009:30am  Breakfast  
9:3010:30am  Hongkai Zhao
University of California, Irvine 
Title: A scaling law for the intrinsic complexity of high frequency wave fields, random fields and random matrices
Abstract: We characterize the intrinsic complexity of a set S in a metric space W by the least dimension N of a linear space V ⊂ W that can approximate S to an error. We show a scaling law for N for high frequency wave fields in term of the wavelength, for random fields in term of the correlation length, and for a set of random vectors in term of the dimension. 
10:3011:00am  Break  
11:0012:00pm  Guowei Wei
Michigan State University 
Title: Is it time for a great chemistry between mathematics and biology?
Abstract: In the history of science, mathematics has been a great partner of natural science, except for biology. The Hodgkin–Huxley model and Turing model indicate the union between mathematics and biology in the old days. However, in 1960s, biology became microscopic, i.e., molecular, while mathematics became abstract. They have been on two divergent paths since then. Modern biology, including molecular biology, structural biology, cell biology, evolutionary biology, biochemistry, biophysics, genetics, etc. are intimidating to mathematicians, while advanced mathematics, such as algebra, topology, geometry, graph theory, and analysis are equally frightening to biologists. Fortunately, biology assumed an omics dimension around the dawn of the millennium. The exponential growth of biological data has paved the way for biology to undertake a historic transition from being qualitative, phenomenological and descriptive to being quantitative, analytical and predictive. Such a transition offers both unprecedented opportunities and formidable challenges for mathematicians, just as quantum physics did a century ago. I will discuss how deep learning and mathematics, including algebraic topology, differential geometry, graph theory, and partial differential equation, lead my team to be a top performer in recent two D3R Grand Challenges, a worldwide competition series in computeraided drug design and discovery. It is time for mathematics to embrace modern biology. 
12:001:30pm  Lunch  
1:302:30pm  Laurent Demanet
MIT 
Title: 1930s Analysis for 2010s Signal Processing: Recent Progress on the Superresolution Question
Abstract: The ability to access signal features below the diffraction limit of an imaging system is a delicate nonlinear phenomenon called superresolution. The main theoretical question in this area is still mostly open: it concerns the precise balance of noise, bandwidth, and signal structure that enables superresolved recovery. When structure is understood as sparsity on a grid, we show that there is a precise scaling law that extends ShannonNyquist theory, and which governs the asymptotic performance of a class of simple “subspacebased” algorithms. This law is universal in the minimax sense that no statistical estimator can outperform it significantly. By contrast, compressed sensing is in many cases suboptimal for the same task. Joint work with Nam Nguyen. 
2:302:45pm  Break  
2:453:45pm  Yue Lu
Harvard University 
Title: Understanding Nonconvex Statistical Estimation via Sharp Asymptotic Methods: Phase Transitions, Scaling Limits, and Mapping Optimization Landscapes
Abstract: We are in the age of ubiquitous collection and processing of data of all kinds on unprecedented scales. Extracting meaningful information from the massive datasets being compiled by our society presents challenges and opportunities to signal and information processing research. For many modern statistical estimation problems, the new highdimensional settings allow one to apply powerful asymptotic methods from probability theory and statistical physics to obtain precise characterizations that would otherwise be too complicated in moderate dimensions. I will present three vignettes of our work on exploiting such blessings of dimensionality to understand nonconvex statistical estimation via sharp asymptotic methods. In particular, I will show (1) the exact characterization of a widelyused spectral method for nonconvex signal recoveries; (2) how to use scaling and meanfield limits to analyze nonconvex optimization algorithms for highdimensional inference and learning; and (3) how to precisely characterize the optimization landscape of a highdimensional binary regression problem by exactly counting and mapping local minima. In all these problems, asymptotic methods not only clarify some of the fascinating phenomena that emerge with highdimensional data, they also lead to optimal designs that significantly outperform commonly used heuristic choices. 
3:454:15pm  Break  
4:155:15pm  Jianfeng Cai
HKUST 
Title: NonConvex Methods for LowRank Matrix Reconstruction
Abstract: We present a framework of nonconvex methods for reconstructing a low rank matrix from its limited information, which arises from numerous practical applications in machine learning, imaging, signal processing, computer vision, etc. Our framework uses the geometry of the Riemannian manifold of all rankr matrices. The methods will be applied to several concrete example problems such as matrix completion, phase retrieval, and robust principle component analysis. We will also provide theoretical guarantee of our methods for the convergence to the correct lowrank matrix. 
5:156:15pm  Lixin Shen
Syracuse University 
Title: Overcomplete Tensor Decomposition via Convex Optimization Abstract: Tensors provide natural representations for massive multimode datasets and tensor methods also form the backbone of many machine learning, signal processing, and statistical algorithms. The utility of tensors is mainly due to the ability to identify overcomplete, nonorthogonal factors from tensor data, which is known as tensor decomposition. I will talk about our recent theories and computational methods for guaranteed overcomplete, nonorthogonal tensor decomposition using convex optimization. By viewing tensor decomposition as a problem of measure estimation from moments, we developed the theory for guaranteed decomposition under three assumptions: (i) Incoherence; (ii) Bounded spectral norm; and (iii) Gram isometry. Under these three assumptions, one can retrieve tensor decomposition by solving a convex, infinitedimensional analog of l1 minimization on the space of measures. The optimal value of this optimization defines the tensor nuclear norm that can be used to regularize tensor inverse problems, including tensor completion, denoising, and robust tensor principal component analysis. Remarkably, all the three assumptions are satisfied with high probability if the rankone tensor factors are uniformly distributed on the unit spheres, implying exact decomposition for tensors with random factors. I will also present and numerically test two computational methods based respectively on BurerMonteiro lowrank factorization 
Monday, March 26
Superfast 3D imaging techniques and applications
Time………….  Speaker……..  Title/Abstract 
9:009:30am  Breakfast  
9:3010:30am  Jun Zhang
University of Michigan, Ann Arbor 
Title: Geometry of Maximum Entropy Inference
Abstract: We revisit classic framework of maximum entropy inference. It is wellknown that the MaxEnt solution is the exponential family, which can be characterized by the duallyflat Hessian geometry. Here, we provide a generalization to the classic formulation by using a general form of entropy function, which leads to the deformedexponential family as its solution. The resulting geometry may still be Hessian, but there is an extra degree of freedom in specifying the underlying geometry. Our framework can cover various generalized entropy functions, such as Tsallis entropy, Renyi entropy, phientropy, and crossentropy functions widely used in machine learning and information sciences. It is an elementary application of concepts from Information Geometry. 
10:3011:00am  Break  
11:0012:00pm  Eric Miller
Tufts University 
Title: A Totally Tubular Talk
Abstract: The objective of this talk is to provide an overview of work over the past decade or so within my group at Tufts related to the quantification of tubular structures arising in a variety of medical imaging applications. We describe methods for detailed segmentation of dense neuronal networks from electron microscopy data as well as the identification of the bare connectivity structure of the murine cerebral microvasculature network given farfromideal fluorescence microcopy data sets. Beyond mapping, we have addressed problems of anomaly detection when considering the localization of intracranial aneurysms and developed graphbased methods as the basis for capturing differences in cerebral vascular networks across subjects. Though the problems and the data types are quite diverse, the mathematical and algorithmic methods share a number of commonalities. With the exception of the anomaly detection work, the neuronal segmentation, vascular network connectivity analysis, and network registration methods are all cast as binary integer programming problems. Though not in this class of solutions, the anomaly detection problem makes use of a geometric feature associated with curves, the writhe number, but applied to tubular surfaces and thus seems appropriate given the subject matter of this workshop. 
12:001:30pm  Lunch  
1:302:30pm  Jerome Darbon
Brown University 
Title: On convex finitedimensional variational methods in imaging sciences, and HamiltonJacobi equations
Abstract: We consider standard finitedimensional variational models used in signal/image processing that consist in minimizing an energy involving a data fidelity term and a regularization term. We propose new remarks from a theoretical perspective which give a precise description on how the solutions of the optimization problem depend on the amount of smoothing effects and the data itself. The dependence of the minimal values of the energy is shown to be ruled by HamiltonJacobi equations, while the minimizers u(x,t) for the observed images x and smoothing parameters t are given by u(x,t)=x – \nabla H(\nabla E(x,t)) where E(x,t)is the minimal value of the energy and H is a Hamiltonian related to the data fidelity term. Various vanishing smoothing parameter results are derived illustrating the role played by the prior in such limits. Finally, we briefly present an efficient numerical numerical method for solving certain HamiltonJacobi equations in high dimension and some applications in optimal control. 
2:302:45pm  Break  
2:453:45pm  Rongjie Lai
RPI 
Title: Understanding ManifoldStructure Data via Geometric Modeling and Learning.
Abstract: Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. Different from image and signal processing which handle functions on flat domains with welldeveloped tools for processing and learning, manifoldstructured data sets are far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higherlevel tasks in manifoldstructured data analysis and understanding even more challenging. To overcome these ambiguities, I will first discuss modeling based methods. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data and extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds and demonstrate its potential to conduct geometric deep learning on manifolds. This geometric way of defining convolution provides a natural combination of modeling and learning on manifolds. It enables further applications of comparing, classifying and understanding manifoldstructured data by combing with recent advances in machine learning theory. If time permits, I will also discuss extensions of these methods to understand manifoldstructured data represented as incomplete interpoint distance information by combining with lowrank matrix completion theory. 
3:454:15pm  Break  
4:155:15pm  Song Zhang
Purdue University 
Title: Superfast 3D imaging techniques and applications
Abstract: Advances in optical imaging and machine/computer vision have provided integrated smart sensing systems for the manufacturing industry; and advanced 3D imaging could have profound impact on numerous fields, with broader applications including manufacturing, biomedical engineering, homeland security, and entertainment. Our research addresses the challenges in highspeed 3D imaging and optical information processing. For example, we have developed a system that simultaneously captures, processes and displays 3D geometries at 30 Hz with over 300,000 measurement points per frame, which was unprecedented at that time (a decade ago). Our current research focuses on achieving speed breakthroughs by developing the binary defocusing techniques and the mechanical projection method. The binary defocusing methods coincide with the inherent operation mechanism of the digitallightprocessing (DLP) technology, permitting tens of kHz 3D imaging speed at camera pixel spatial resolution; and utilizing the mechanical projection system further broaden the light spectrum usage. In this talk, I will present two platform technologies that we have developed as well as some of the applications that we have been exploring including cardiac mechanics, forensic science, as well as bioinspired robotics. 
5:156:15pm  Ronald Lok Ming Lui
CUHK 
Title: Mathematical models for restoration of turbulencedegraded images
Abstract: Turbulencedegraded image frames are distorted by both turbulent deformations and spacetimevarying blurs. To suppress these effects, a multiframe reconstruction scheme is usually considered to recover a latent image from the observed distorted image sequence. Recent approaches are commonly based on registering each frame to a reference image, by which geometric turbulent deformations can be estimated and a sharp image can be restored. A major challenge is that a fine reference image is usually unavailable, as every turbulencedegraded frame is distorted. A highquality reference image is crucial for the accurate estimation of geometric deformations and fusion of frames. Besides, it is unlikely that all frames from the image sequence are useful, and thus frame selection is necessary and highly beneficial. In this talk, we will describe several mathematical models to restore turbulencedistorted images. Extensive experimental results will also be shown to demonstrate the efficacy of different models 
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The main focus of the workshop is the application of algebraic method to study problems in combinatorics. In recent years there has been a large number of results in which the use of algebraic technique has resulted in significant improvements to long standing open problems. Such problems include the finite field Kakeya problem, the distinct distance problem of Erdos and, more recently, the capset problem. The workshop will include talks on all of the above mentioned problem as well as on recent development in related areas combining combinatorics and algebra.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
Coorganizers of this workshop include Zeev Dvir, Larry Guth, and Shubhangi Saraf.
Click here for a list of registrants.
Monday, Nov. 13
Time  Speaker  Title/Abstract 
9:009:30am  Breakfast  
9:3010:30am  Jozsef Solymosi

On the unit distance problem Abstract: Erdos’ Unit Distances conjecture states that the maximum number of unit distances determined by n points in the plane is almost linear, it is O(n^{1+c}) where c goes to zero as n goes to infinity. In this talk I will survey the relevant results and propose some questions which would imply that the maximum number of unit distances is o(n^{4/3}). 
10:3011:00am  Coffee Break  
11:0012:00pm

Orit Raz  Intersection of linear subspaces in R^d and instances of the PIT problem
Abstract: In the talk I will tell about a new deterministic, strongly polynomial time algorithm which can be viewed in two ways. The first is as solving a derandomization problem, providing a deterministic algorithm to a new special case of the PIT (Polynomial Identity Testing) problem. The second is as computing the dimension of the span of a collection of flats in high dimensional space. The talk is based on a joint work with Avi Wigderson. 
12:001:30pm  Lunch  
1:302:30pm  Andrew Hoon Suk 
Ramsey numbers: combinatorial and geometric Abstract: In this talk, I will discuss several results on determining the tower growth rate of Ramsey numbers arising in combinatorics and in geometry. These results are joint work with David Conlon, Jacob Fox, Dhruv Mubayi, Janos Pach, and Benny Sudakov. 
2:303:00pm  Coffee Break  
3:004:00pm  Josh Zahl 
Cutting curves into segments and incidence geometry 
4:006:00pm  Welcome Reception 
Tuesday, Nov. 14
Time  Speaker  Title/Abstract 
9:009:30am  Breakfast  
9:3010:30am  Péter Pál Pach 
Polynomials, rank and cap sets Abstract: In this talk we will look at a new variant of the polynomial method which was first used to prove that sets avoiding 3term arithmetic progressions in groups like $\mathbb{Z}_4^n$ and $\mathbb{F}_q^n$ are exponentially small (compared to the size of the group). We will discuss lower and upper bounds for the size of the extremal subsets and mention further applications of the method. 
10:3011:00am  Coffee Break  
11:0012:00pm  Jordan Ellenberg 
The Degeneration Method Abstract: In algebraic geometry, a very popular way to study (nice, innocent, nonsingular) varieties is to degenerate them to (weirdlooking, badly singular, nonreduced) varieties (which are actually not even varieties but schemes.) I will talk about some results in combinatorics using this approach (joint with Daniel Erman) and some ideas for future applications of the method. 
12:001:30pm  Lunch  
1:302:30pm  Larry Guth  The polynomial method in Fourier analysis
Abstract: This will be a survey talk about how the polynomial method helps to understand problems in Fourier analysis. We will review some applications of the polynomial method to problems in combinatorial geometry. Then we’ll discuss some problems in Fourier analysis, explain the analogy with combinatorial problems, and discuss how to adapt the polynomial method to the Fourier analysis setting. 
2:303:00pm 
Coffee Break  
3:004:00pm  Open Problem 
Wednesday, Nov. 15
Time  Speaker  Title/Abstract 
9:009:30am  Breakfast  
9:3010:30am

Avi Wigderson 
The “rank method” in arithmetic complexity: Lower bounds and barriers to lower bounds Abstract: Why is it so hard to find a hard function? No one has a clue! In despair, we turn to excuses called barriers. A barrier is a collection of lower bound techniques, encompassing as much as possible from those in use, together with a proof that these techniques cannot prove any lower bound better than the stateofart (which is often pathetic, and always very far from what we expect for complexity of random functions). In the setting of Boolean computation of Boolean functions (where P vs. NP is the central open problem), there are several famous barriers which provide satisfactory excuses, and point to directions in which techniques may be strengthened. In the setting of Arithmetic computation of polynomials and tensors (where VP vs. VNP is the central open problem) we have no satisfactory barriers, despite some recent interesting attempts. This talk will describe a new barrier for the Rank Method in arithmetic complexity, which encompass most lower bounds in this field. It also encompass most lower bounds on tensor rank in algebraic geometry (where the the rank method is called Flattening). I will describe the rank method, explain how it is used to prove lower bounds, and then explain its limits via the new barrier result. As an example, it shows that while the best lower bound on the tensor rank of any explicit 3dimensional tensor of side n (which is achieved by a rank method) is 2n, no rank method can prove a lower bound which exceeds 8n (despite the fact that a random such tensor has rank quadratic in n). No special background knowledge is assumed. The audience is expected to come up with new lower bounds, or else, with new excuses for their absence. 
10:3011:00am  Coffee Break  
11:0012:00pm  Venkat Guruswami 
Subspace evasion, list decoding, and dimension expanders Abstract: A subspace design is a collection of subspaces of F^n (F = finite field) most of which are disjoint from every lowdimensional subspace of F^n. This notion was put forth in the context of algebraic list decoding where it enabled the construction of optimal redundancy listdecodable codes over small alphabets as well as for errorcorrection in the rankmetric. Explicit subspace designs with nearoptimal parameters have been constructed over large fields based on polynomials with structured roots. (Over small fields, a construction via cyclotomic function fields with slightly worse parameters is known.) Both the analysis of the list decoding algorithm as well as the subspace designs crucially rely on the *polynomial method*. Subspace designs have since enabled progress on linearalgebraic analogs of Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In particular, they yield an explicit construction of constantdegree dimension expanders over large fields. While constructions of such dimension expanders are known over any field, they are based on a reduction to a highly nontrivial form of vertex expanders called monotone expanders. In contrast, the subspace design approach is simpler and works entirely within the linearalgebraic realm. Further, in recent (ongoing) work, their combination with rankmetric codes yields dimension expanders with expansion proportional to the degree. This talk will survey these developments revolving around subspace designs, their motivation, construction, analysis, and connections. (Based on several joint works whose coauthors include Chaoping Xing, Swastik Kopparty, Michael Forbes, Nicolas Resch, and Chen Yuan.) 
12:001:30pm  Lunch  
1:302:30pm

David Conlon 
Finite reflection groups and graph norms Abstract: For any given graph $H$, we may define a natural corresponding functional $\.\_H$. We then say that $H$ is norming if $\.\_H$ is a seminorm. A similar notion $\.\_{r(H)}$ is defined by $\ f \_{r(H)} := \  f  \_H$ and $H$ is said to be weakly norming if $\.\_{r(H)}$ is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we identify a much larger class of weakly norming graphs. This result includes all previous examples of weakly norming graphs and adds many more. We also discuss several applications of our results. In particular, we define and compare a number of generalisations of Gowers’ octahedral norms and we prove some new instances of Sidorenko’s conjecture. Joint work with Joonkyung Lee.

2:303:00pm  Coffee Break  
3:004:00pm  Laszlo Miklós Lovasz 
Removal lemmas for triangles and kcycles. Abstract: Let p be a fixed prime. A kcycle in F_p^n is an ordered ktuple of points that sum to zero; we also call a 3cycle a triangle. Let N=p^n, (the size of F_p^n). Green proved an arithmetic removal lemma which says that for every k, epsilon>0 and prime p, there is a delta>0 such that if we have a collection of k sets in F_p^n, and the number of kcycles in their cross product is at most a delta fraction of all possible kcycles in F_p^n, then we can delete epsilon times N elements from the sets and remove all kcycles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to our work, the best known bound for any k, due to Fox, showed that 1/delta can be taken to be an exponential tower of twos of height logarithmic in 1/epsilon (for a fixed k). In this talk, we will discuss recent work on Green’s problem. For triangles, we prove an essentially tight bound for Green’s arithmetic triangle removal lemma in F_p^n, using the recent breakthroughs with the polynomial method. For kcycles, we also prove a polynomial bound, however, the question of the optimal exponent is still open. The triangle case is joint work with Jacob Fox, and the kcycle case with Jacob Fox and Lisa Sauermann. 
Thursday, Nov. 16
Time  Speaker  Title/Abstract 
9:009:30am  Breakfast  
9:3010:30am  Janos Pach  Let’s talk about multiple crossings
Abstract: Let k>1 be a fixed integer. It is conjectured that any graph on n vertices that can be drawn in the plane without k pairwise crossing edges has O(n) edges. Two edges of a hypergraph cross each other if neither of them contains the other, they have a nonempty intersection, and their union is not the whole vertex set. It is conjectured that any hypergraph on n vertices that contains no k pairwise crossing edges has at most O(n) edges. We discuss the relationship between the above conjectures and explain some partial answers, including a recent result of Kupavskii, Tomon, and the speaker, improving a 40 years old bound of Lomonosov. 
10:3011:00am  Coffee Break  
11:0012:00pm  Misha Rudnev 
Few products, many sums Abstract: This is what I like calling “weak Erd\H osSzemer\’edi conjecture”, still wide open over the reals and in positive characteristic. The talk will focus on some recent progress, largely based on the ideas of I. D. Shkredov over the past 56 years of how to use linear algebra to get the best out of the Szemer\’ediTrotter theorem for its sumproduct applications. One of the new results is strengthening (modulo the log term hidden in the $\lesssim$ symbol) the textbook Elekes inequality $$ A^{10} \ll AA^4AA^4 $$ to $$A^{10}\lesssim AA^3AA^5.$$ The other is the bound $$E(H) \lesssim H^{2+\frac{9}{20}}$$ for additive energy of sufficiently small multiplicative subgroups in $\mathbb F_p$. 
12:001:30pm  Lunch  
1:302:30pm  Adam Sheffer 
Geometric Energies: Between Discrete Geometry and Additive Combinatorics Abstract: We will discuss the rise of geometric variants of the concept of Additive energy. In recent years such variants are becoming more common in the study of Discrete Geometry problems. We will survey this development and then focus on a recent work with Cosmin Pohoata. This work studies geometric variants of additive higher moment energies, and uses those to derive new bounds for several problems in Discrete Geometry. 
2:303:00pm  Coffee Break  
3:004:00pm  Boris Bukh 
Ranks of matrices with few distinct entries Abstract: Many applications of linear algebra method to combinatorics rely on the bounds on ranks of matrices with few distinct entries and constant diagonal. In this talk, I will explain some of these application. I will also present a classification of sets L for which no lowrank matrix with entries in L exists. 
Friday, Nov. 17
Time  Speaker  Title/Abstract 
9:009:30am  Breakfast  
9:3010:30am  Benny Sudakov 
Submodular minimization and setsystems with restricted intersections Abstract: Submodular function minimization is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint types under which it remains efficiently solvable. The arguably most relevant nontrivial constraint class for which polynomial algorithms are known are parity constraints, i.e., optimizing submodular function only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the oddcut problem, and they are a key tool in a recent technique to efficiently solve integer programs with a constraint matrix whose subdeterminants are bounded by two in absolute value. We show that efficient submodular function minimization is possible even for a significantly larger class than parity constraints, i.e., over all sets (of any given lattice) of cardinality r mod m, as long as m is a constant prime power. To obtain our results, we combine tools from Combinatorial Optimization, Combinatorics, and Number Theory. In particular, we establish an interesting connection between the correctness of a natural algorithm, and the nonexistence of set systems with specific intersection properties. Joint work with M. Nagele and R. Zenklusen 
10:3011:00am  Coffee Break  
11:0012:00pm  Robert Kleinberg 
Explicit sumofsquares lower bounds via the polynomial method Abstract: The sumofsquares (a.k.a. Positivstellensatz) proof system is a powerful method for refuting systems of multivariate polynomial inequalities, i.e. proving that they have no solutions. These refutations themselves involve sumofsquares (sos) polynomials, and while any unsatisfiable system of inequalities has a sumofsquares refutation, the sos polynomials involved might have arbitrarily high degree. However, if a system admits a refutation where all polynomials involved have degree at most d, then the refutation can be found by an algorithm with running time polynomial in N^d, where N is the combined number of variables and inequalities in the system. Lowdegree sumofsquares refutations appear throughout mathematics. For example, the above proof search algorithm captures as a special case many a priori unrelated algorithms from theoretical computer science; one example is Goemans and Williamson’s algorithm to approximate the maximum cut in a graph. Specialized to extremal graph theory, they become equivalent to flag algebras. They have also seen practical use in robotics and optimal control. Therefore, it is of interest to identify “hard” systems of lowdegree polynomial inequalities that have no solutions but also have no lowdegree sumofsquares refutations. Until recently, the only known examples were either not explicit (i.e., known to exist by nonconstructive means such as the probabilistic method) or not robust (i.e., a system is constructed which is not refutable by degree d sos polynomials, but becomes refutable when perturbed by an amount tending to zero with d). We present a new family of instances derived from the capset problem, and we show a superconstant lower bound on the degree of its sumofsquares refutations. Our instances are both explicit and robust. This is joint work with Sam Hopkins. 
12:001:30pm  Lunch 
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Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
Seminar on Evolution Equations
Concluding Conference on Nonlinear Equations Program
MiniSchool on Nonlinear Equations, Dec. 2016
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Name  Home Institution  Tentative Visiting Dates 

Stefano Bianchini  SISSA  04/01/2016 – 05/31/2016 
Lydia Bieri  University of Michigan  02/01/2016 – 04/30/2016 
Albert Chau  University of British Columbia  02/26/2016 – 05/26/2016 
Binglong Chen  Sun Yatsen University  09/01/2015 – 11/30/2015 
Qingtao Chen  ETHZ (Swiss Federal Institute of Technology in Zurich)  03/17/2016 – 04/04/2016 
Piotr Chrusciel  University of Vienna  03/01/2016 – 05/30/2016 
Fernando Coda Marques  Princeton University  04/25/2016 – 04/29/2016 05/23/2016 – 05/27/2016 
Mihalis Dafermos  Princeton University  04/01/2016 – 04/30/2016 
Camillo De Lellis  University of Zurich  02/01/2016 – 4/30/2016 
Michael Eichmair  University of Vienna  03/21/2016 – 04/01/2016 
Felix Finster  Universitat Regensburg  09/20/2015 – 10/20/2015 03/20/2016 – 04/20/2016 
Xianfeng David Gu  SUNY at Stony Brook  04/01/2016 – 04/30/2016 
ZhengCheng Gu  Perimeter Institute for Theoretical Physics  08/15/2015 – 09/15/2015 
Pengfei Guan  McGill University  10/10/2015 – 10/17/2015 
Xiaoli Han  Tsinghua University  01/20/2016 – 04/19/2016 
Thomas Hou  California Institute of Technology  11/01/2016 – 11/30/2016 
Feimin Huang  Chinese Academy of Sciences  02/15/2016 – 04/15/2016 
Xiangdi Huang  Chinese Academy of Sciences  09/10/2015 – 12/10/2015 
Tom Ilmanen  ETH Zurich  10/19/2015 – 12/18/2015 
Niky Kamran  McGill Univeristy  04/04/2016 – 04/08/2016 
Nicolai Krylov  University of Minnesota  11/01/2015 – 11/30/2015 
Junbin Li  Sun Yatsen University  02/01/2016 – 04/30/2016 
Yong Lin  Renmin University of China  02/01/2016 – 03/31/2016 
Andre Neves  Imperial College London  4/25/2016 – 4/29/2016; 5/23/2016 – 5/27/2016 
Duong H. Phong  Columbia University  04/08/2016 – 04/10/2016 
Ovidiu Savin  Columbia University  10/15/2015 – 12/14/2015 
Richard Schoen  Stanford University  03/21/2016 – 03/25/2016 
Mao Sheng  University of Science and Technology of China  01/15/2016 – 01/28/2016 
Valentino Tosatti  Northwestern University  02/01/2016 – 04/15/2016 
John Toth  McGill University  04/04/2016 – 04/08/2016 
ChungJun Tsai  National Taiwan University  05/01/2016 – 05/08/2016 
TaiPeng Tsai  University of British Columbia  03/20/2016 – 05/31/2016 
LiSheng Tseng  UC Irvine  02/08/2016 – 02/19/2016; 04/27/2016 – 05/11/2016 
Chun Peng Wang  Jilin University  02/01/2016 – 04/30/2016 
XuJia Wang  Australian National University  04/01/2016 – 05/31/2016 
Ben Weinkove  Northwestern University  02/28/2016 – 03/18/2016 
Sijue Wu  University of Michigan  04/01/2016 – 04/30/2016 
Chunjing Xie  Shanghai Jiao Tong University  09/08/2015 – 12/07/2015 
Zhou Ping Xin  The Chinese University of Hong Kong  10/01/2015 – 11/30/2015 
Hongwei Xu  Zhejiang University  09/01/2015 – 11/30/2015 
Peng Ye  University of Illinois at UrbanaChampaign  11/15/2015 – 11/22/2015 
Pin Yu  Tshinghua University  09/07/2015 – 12/10/2015 
Yi Zhang  Fudan University  01/18/2016 – 05/31/2016 