Co-organizers:
Vladimir Retakh (vretakh {at} math [dot] rutgers [dot] edu)
Yukun Yao (yao {at} math [dot] rutgers [dot] edu)
The character variety of a smooth projective complex variety parametrized isomorphism classes of irreducible complex local systems. There are special subloci, the study of which was started by Carlos Simpson at the beginning of the 90s. He predicted that the more special, the more geometric. We will give an overview of some theorems one can prove, using in particular the pendant of the construction over finite fields and the Langlands program, notably for rigid local systems.
Let C be a smooth projective curve of genus at least 2 defined over the rational numbers. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points. In the case when the Jacobian of C has rank less than its genus, the Chabauty--Coleman method can often be used to find the rational points of C, using the construction of p-adic line integrals. In certain cases of higher rank, p-adic heights can often be used to find rational or integral points on C. I will describe these "quadratic Chabauty" techniques (part of Kim's nonabelian Chabauty program) and will highlight some recent examples where the techniques have been used: this includes a 1700-year old problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This talk is based on joint work with Amnon Besser, Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.
It is a classical result by L. Boutet de Movel that any compact strictly pseudoconvex (hypersurface type) CR manifold of dimension strictly greater than 3 is embeddable in $\mathbb{C}^n$ for some $n$. For strictly pseudoconvex CR manifold of dimension 3, the situation is more subtle. It is known that “most” are not embeddable. A characterization of embeddability in terms of a closed range property of $\bar\partial$ was given by J. Kohn. In this talk, we shall discuss the embeddability problem for strictly pseudoconvex CR 3-folds in more geometric terms. The approach will be to realize the embeddable deformations of an already embedded CR 3-fold as a geometric flow in complex space.
In this talk I will survey recent developments on the existence theory of closed minimal hypersurfaces in Riemannian manifolds, including a Morse-theoretic existence result for the generic case.
Cluster algebras were introduced by Fomin and Zelevinsky almost 20 years ago and have since found exciting applications in many areas including algebraic geometry, representation theory, integrable systems, theoretical physics and Poisson geometry. The latter connection proved instrumental in uncovering cluster algebra structures in coordinate rings of Poisson varieties such as Grassmannians and double Bruhat cells in semisimple Lie groups. In this talk, based on the joint work with M. Shapiro and A. Vainshtein, I will describe how a Poisson geometric point of view leads to a construction of multiple nonequivalent cluster structures in GL(n).
In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity had only been proved for degrees d=1,2,3. We prove the hyperbolicity of all (but possibly finitely many) the Jensen polynomials of every degree d. Moreover, we establish the outright hyperbolicity for all degrees d<10^26. These results follow from a proof of the "derivative aspect" GUE distribution for zeros. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.
One of the most fundamental results of mathematical logic is the celebrated Godel completeness theorem, which asserts that every consistent first-order theory T admits a model. In the 1980s, Makkai proved a much sharper result: any first-order theory T can be recovered, up to a suitable notion of equivalence, from its category of models Mod(T) together with some additional structure (supplied by the theory of ultraproducts). In this talk, I'll explain the statement of Makkai's theorem and sketch a new proof of it, inspired by the theory of "pro-etale sheaves" studied by Scholze and Bhatt - Scholze.
The Diederich-Fornaess worm domain is a domain in complex space with very special geometric and analytic properties. Anyone with a basic course in complex analysis can understand the worm. But working with the worm is a challenge. It is subtle and usually quite difficult. In joint work with Marco Peloso and Caterina Stoppato we have been able to study the Bergman kernel for the worm and to learn some of its fundamental properties.
This will be an expository lecture, and will be accessible to a broad audience.
I will survey and describe very recent new results on the most classical problems in Ramsey theory. For example, I will exhibit a new connection between pseudorandom graphs and classical Ramsey numbers (joint work with Verstraete). This connection provides a novel perspective on these questions and enables one to quickly provide new and currently best known quantitative estimates for basic graph Ramsey problems. I will also survey the current state of the art regarding classical hypergraph Ramsey numbers, and describe a solution to a longstanding open question posed by Erdos and Hajnal in hypergraph Ramsey theory for which Erdos offered a $500 prize (joint work with Razborov).
Symplectic capacities are measurements of symplectic size. They are often defined as the lengths of certain periodic trajectories of dynamical systems, and so they connect symplectic embedding problems with dynamics. I will explain joint work showing how to recover the volume of many symplectic 4-manifolds from the asymptotics of a family of symplectic capacities, called "ECH" capacities. I will then explain how this asymptotic formula was used by Asaoka and Irie to prove the following dynamical result: for a C^{\infty} generic diffeomorphism of S^2 preserving an area form, the union of periodic points is dense.
In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe my recent proof, joint with Mahesh Kakde, of the Brumer-Stark conjecture away from p=2. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. Next I will state a conjectural exact formula for these Brumer-Stark units that has been developed over the last 15 years. I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a solution to Hilbert's 12th problem.
Piecewise linear homeomorphisms of the unit interval have been a source of groups with interesting properties in which calculations are practical. Recently a program has been initiated to understand the collective structure of these groups under the embeddability relation. I will discuss a number of recent results and conjectures along these lines. Remarkably the ordinal $\epsilon_0$ makes an appearance, suggesting that logic may provide important tools and insight into this program.
The Atlas project brings computational methods to the study of Lie groups and their representations. Of particular interest is the problem of the unitary dual: the classification of the irreducible unitary (norm-preserving) representations of a Lie group. I will give an overview of the project, and discuss some of the computational aspects of it.
The "classical" Langlands correspondence relates n-dimensional Galois representations to representations of the p-adic general linear group GL_n(Q_p), both over the field of complex numbers. The mod p (resp. p-adic) Langlands correspondence is an analogue of this correspondence over a field of characteristic p (resp. over a p-adic field). So far it has only been established in dimension n \le 2, with powerful applications to the modularity of Galois representations. In this talk I will give motivation and discuss some earlier developments as well as some recent progress for n > 2. No background in the Langlands program will be assumed.
More than a century ago, Ramanujan discovered remarkable formulas for $1/\pi$. Inspired by these discoveries, similar Ramanujan-like expressions for $1/\pi^2$ have been uncovered recently by Guillera. We explain the provenance of these formulas: we recognize certain special hypergeometric motives as arising from Hilbert modular forms in an explicit way. This is joint work with Lassina Demb\'el\'e, Alexei Panchishkin, and Wadim Zudilin.
Negative-index metamaterials are engineered structures whose refractive index has a negative value over some frequency range. Their existence was postulated by Veselago in 1964 and confirmed experimentally by Shelby, Smith, and Schultz in 2001. Negative-index metamaterial research has been a very active topic of investigation not only because of potentially interesting applications but also because of challenges in understanding their surprising properties. From a mathematical point of view, the subtlety and the challenges in the study of negative-index metamaterials are from the sign-changing coefficients in the modeling equations, hence the ellipticity and the compactness are lost in general. Moreover, localized resonance, i.e., a phenomenon for which the field explodes in some regions and remains bounded in some others as the loss (the damping/viscosity coefficient) goes to 0, might occur. In this talk, I discuss superlensing and cloaking applications of negative-index metamaterials, and the stability of associated fields in the frequency domain. Mathematical ideas/techniques/tools used to analyse these phenomena are mentioned. These involves deriving/analysing Cauchy's problems, applying/establising three-sphere inequalities (with partial data), and introducing the removing localized singularity technique.
The Willmore Lagrangian is a functional that shows up in many areas of sciences such as conformal geometry, general relativity, cell biology, optics... We will try first to shed some lights on the universality of this Lagrangian. We shall then present the project of using the Willmore energy as a Morse function for studying the fascinating space of immersed 2-spheres in the Euclidian 3 space and relate topological obstructions in this space to integer values and minimal surfaces.
Metamaterials are artificial materials with properties that go well beyond what offered by nature, providing unprecedented opportunities to tailor and enhance the control of waves. In this talk, I discuss our recent activity in electromagnetics and acoustics, showing how suitably tailored meta-atoms and their arrangements open exciting venues for enhanced wave-matter interactions. I will discuss unusual scattering, absorption and waveguiding responses, from cloaking and scattering suppression, to nonreciprocity and topological phenomena, enhanced nonlinear effects at subwavelength scales, and bound states in the continuum. Physical insights into the underlying phenomena and new devices based on these concepts will be presented.
I will start with an introduction to the classical random matrix models. Then I will explain our results on two classical topics: the extreme gap problem and the normality of C\betaE. In the end, I will talk about mathematical aspects of the Sachdev-Ye-Kitaev model which originated in physics. Some open problems and conjectures will be discussed.
Let X be a smooth variety (e.g., affine space) over a finite field (e.g., the integers modulo a prime). In the course of proving the last of Weil's conjectures on zeta functions of varieties over finite field, Deligne studied a certain category of representations of the fundamental group of X which carry information about these zeta functions. He also made a far-reaching conjecture to the effect that such objects always look as if they "come from geometry". We will state the conjecture, describe some of its more concrete consequences, and discuss some results of various authors (L. Lafforgue, V. Lafforgue, Deligne, Drinfeld, T. Abe, Abe-Esnault, and the speaker) which very recently have led to a resolution of this 40-year-old open problem.
A rainbow set for a family of sets is the range of a partial choice function (i.e., choosing one element from each member of a subfamily). Problems on the existence of choice functions whose range satisfies some desirable property lend themselves nicely to topological methods. A major part of the talk is about results of the form "(quantifiably) many sets satisfying some property have a rainbow set satisfying this property".
Graphene is a two-dimensional material made up of a single atomic layer of carbon atoms arranged in honeycomb pattern. Many of its remarkable electronic properties, e.g. quasi-particles (wave-packets) that propagate as massless relativistic particles and topologically protected edge states, are closely related to the spectral properties of the underlying single-electron Hamiltonian: -Laplacian + V(x), where V(x) is a potential with the symmetries of a hexagonal tiling of the plane. Taking inspiration from graphene, there has been a great deal of activity in the fundamental and applied physics communities related to the properties of waves (photonic, acoustic, elastic,…) in media whose material properties have honeycomb symmetry. In this talk I will review progress on the mathematical theory.
Complex dynamics is centered around studying rational functions on the Riemann sphere from the point of view of iteration. In this talk, we begin with the family of quadratic polynomials, a great success story that involves the famous Mandelbrot set. The family of quadratic polynomials naturally lives inside the moduli space of quadratic rational maps, which is isomorphic to $\C^2$. We will introduce the Milnor curves inside this moduli space; these are dynamically-defined algebraic curves. A major open problem in the subject is to determine if these curves are connected. We present the first result establishing that an infinite collection of them are irreducible over complex numbers. Our proof uses arithmetic techniques. This is joint work with Xavier Buff and Adam Epstein.
In recent research on the Riemann zeta function and the Riemann Hypothesis, it is important to calculate certain integrals involving the characteristic functions of N x N unitary matrices and to develop asymptotic expansions of these integrals as N goes to infinity. In this talk, I will evaluate many of these integrals exactly, verify that the leading coefficients in their asymptotic expansions are non-zero, and relate these results to conjectures about the distribution of the zeros of the Riemann zeta function on the critical line. Finally, I will explain how these calculations are related to mathematical statistics and to the hypergeometric functions of Hermitian matrix argument.
Collective dynamics is driven by different rules for alignment that self-organize the crowd, and by different external forces that keep the crowd together. Different emerging equilibria are self-organized into clusters, flocks, tissues, parties, etc. I will overview recent results on the hydrodynamics of large-time, large-crowd collective behavior, driven by different “rules of engagement”. In particular, I address the question how short-range interactions lead, over time, to the emergence of long-range patterns, comparing geometric vs. topological interactions.
Integro-differential equations have been a very active area of research in recent years. In this talk we will explain what they are and in what sense they are similar to more classical elliptic and parabolic partial differential equations. The Boltzmann equation is a nonlinear model from statistical mechanics that represents the evolution of particle densities in dilute gases. We will discuss how the techniques learned in the study of general parabolic integro-differential equations are used to obtain regularity estimates for solutions to the Boltzmann equation in the "non-cutoff" case.
Two central themes in logic are how much the universe of sets resembles Gödel's constructible universe $L$ versus what is possible from forcing and large cardinals. Both are addressed by using infinite combinatorics to investigate how much compactness can be obtained in the universe. Compactness is the phenomenon when a given property holding for every smaller substructure of some object implies that property holds for the object itself. This is usually a consequence of large cardinals, and tends to fail in $L$. A key instance of compactness is the tree property, which states that every tree of height $\kappa$ and levels of size less than $\kappa$ has a cofinal branch. Informally, this principle is a generalization of König's infinity lemma to uncountable cardinals. It turns out that the tree property and certain strengthenings capture the combinatorial essence of large cardinals. An old project in set theory is to force the tree property (and some strengthenings) at every regular cardinal greater than $\aleph_1$. I will go over the background and then discuss some recent results giving the state of the art of this project.
Knot Floer homology provides a great set of tools for studying questions about knots in the standard 3-sphere. Since the homology theory extends to knots in arbitrary 3-manifolds, the study of invariants of the double branched cover of the 3-sphere along a given knot provides further ways for deriving information about the knot concordance group, an infinitely generated Abelian group of central importance in low dimensional topology. In the talk we plan to review this group, together with the construction of knot Floer homology, and describe the adaptation of the method of Hendricks-Manolescu to the present context to derive invariants through the double branched cover construction.
The representation theory of finite groups of the last decades has been focused on some deep conjectures that relate in a spectacular way global and local invariants of finite groups. We give an overview of these conjectures and some recent advances.
Symbolic dynamics is a way to study general dynamical systems, by coding iterates of points with some alphabet. The symmetries of a symbolic dynamical system form a countable group, and can range from being quite complicated to very simple, reflecting the underlying dynamics. I will give an overview of what is known about these groups and highlight numerous open questions in the area.
A basic problem in mechanics is to understand ways that periodic behavior can occur in a physical system. In many cases this corresponds to finding periodic orbits of the Reeb vector field on a contact manifold. We discuss some new results on the existence of such periodic orbits in the three dimensional case. In particular, in the three dimensional case, under some mild assumptions, we can show that there are either two or infinitely many periodic orbits. This result is joint work with Dan Cristofaro-Gardiner and Dan Pomerleano, based on earlier joint work with Dan C-G and Vinicius Ramos.
A notable appeal of Thurston's work on the geometrization of 3-manifolds lies in its hands-on nature: synthetic and combinatorial constructions involving geodesic loops, simplicial surfaces, and coarse "quasi-geodesics" gain their power from rigidity theorems due to Mostow and Sullivan that guarantee that rough geometric estimates ensure explicit control. But considerable analytic work of Ahflors and Bers lies at the foundation, and only recently has work of Graham and Witten been found to give clues to an analytic framework for understanding Thurston's intuition and conjectures. In this talk I will describe history context and recent developments that use Graham and Witten's notion of "renormalized volume", as elaborated by Krasnov and Schlenker, to provide a satisfying analytic explanation for the connection between volumes of hyperbolic 3-manifolds that fiber over the circle and Weil-Petersson lengths of closed geodesics on moduli space. I'll discuss an array of applications to Weil-Petersson geometry as well as some new results. This talk describes joint work with Ken Bromberg and Martin Bridgeman.
We offer a gentle introduction to one aspect of modular representation theory, accompanied by photographs giving a bit of history. We emphasize the surprising nature of actions on vector spaces over a field of positive characteristic. We touch upon issues of extensions and the role played by cohomology.
A venerable technique to study manifolds of dimension n is to consider invariants of submanifolds of dimension n-1, dual to a 1-dimensional cohomology class. A classic instance of this technique in dimension 4 arises from Rochlin’s theorem about smooth spin 4-manifolds. I will describe what 4-dimensional gauge theory (Seiberg-Witten theory and Yang-Mills theory) tells us about such invariants, and give some applications of this new point of view.
When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this talk we will give an overview of progress that has been made for surfaces with boundary, and contrast this with some recent results in higher dimensions. This is joint work with R. Schoen.
If a group G acts on a vector space V by linear transformations, then the invariant polynomial functions on V form a ring. In this talk I will give an overview of known upper and lower bounds for the degrees of generators of this invariant ring. This includes recent results by Visu Makam and the speaker on some cases where polynomial upper bounds exists, and some cases where we have exponential lower bounds.
Spacetime and Quantum Mechanics form the pillars of our understanding of modern physics, but there are several indications that these concepts are approximate and must emerge from deeper principles, undoubtedly involving new mathematics. In this talk I will describe some emerging ideas along these lines, and present a new formulation of some very basic physics-- fundamental to particle scattering and to cosmology--not following from quantum evolution in space-time, but associated with simple new mathematical structures in "positive geometry". In these examples we can concretely see how the usual rules of space-time and quantum mechanics can arise, joined at the hip, from primitive geometric and combinatorial origins.
At the end of his monograph Almgren addresses the question of regularity of solutions at the boundary. Full regularity was proved by Allard in his Ph.D. thesis when the ambient manifold is the Euclidean space and the boundary surface lies in the boundary of uniformly convex open set. The general case in codimension one was then settled by Hardt and Simon in the early 80's. But in codimension higher than one and in general ambient manifolds the current state of the art does not even give guarantee the existence of a single boundary regular point. This prevents the understanding of seemingly innocent questions like the following: does the connectedness of the boundary imply the connectedness of the minimizer? In a joint work with Guido de Philippis, Jonas Hirsch and Annalisa Massaccesi we give a first general boundary regularity theory which allows us to answer positively to the question above.
It is a classical result of Jordan that any finite group acting transitively on a set of size greater than one contains a derangement (i.e. an element with no fixed points). We will discuss variations of this result about counting derangements and applications of these results to various topics in number theory, algebra and arithmetic geometry.
I will discuss two basic analytic problems about the Cauchy -Riemann geometry in 3-dimensions. The first is concerned with optimal two Sobolev inequalities on such manifolds related to the CR version of the Yamabe problem, and the Q-prime curvature equation. The second one is related to the isoperimetric problem.
We'll look at three separate problems in math: Explaining the analytic continuation of the Riemann zeta function, a topological approach to the inscribed rectangle problem, and explaining the Uncertainty Principle. For each, we'll look at how the process of creating visuals aimed at making one aspect of these topics clearer can offer unexpected insight into related problems. The aim is to make a case that taking the time to create such visuals can be beneficial not just for expository purposes, but also for one's own understanding, and potentially for research.
This talk will be about some classical and some new results concerning class numbers of number fields. I will eventually focus on Iwasawa theory. This has to do with the growth of the p-parts of class numbers as one moves up various towers of number fields. By the end of the talk I will discuss some new results concerning the leading terms in these growth rates, rather than just the first order terms which are the focus of classical Iwasawa theory. This is joint work with F. Bleher, R. Greenberg, M. Kakde, R. Sharifi and M. J. Taylor.
Abstract: Grand unified theories envision the Standard Model of Particle Physics as a piece of a larger system. However, in this talk we will ask the opposite question: Could the Standard Model result from a set of algebras much smaller than itself?By the late 1930s, Arthur Conway knew that the complex quaternions (just a 4 complex-dimensional algebra) could single-handedly encode the notion of rotations and boosts, in addition to the degrees of freedom of electric and magnetic fields, energy and momentum, fermionic spin and chirality. Here we will demonstrate hints that the octonions might be capable of similar feats in efficiency.
Already one-dimensional model equations relevant for fluid flows can present serious challenges for the PDE theory. After a brief update on an open problem concerning the full Navier-Stokes equations discussed by the speaker at Rutgers some years ago, the lecture will describe results on certain model equations.
Lennart Carleson's celebrated theorem of 1966 asserts the pointwise convergence of the partial Fourier sums of square integrable functions, giving the positive answer to Luzin's conjecture from 1915. The aim of this talk is to provide yet another proof of this fact. In particular, we will see a new simplified approach to this result, which can be presented in a brief self-contained manner. A number of related results can be seen by variants of the same argument. We survey the historical background and some complements to Carleson's theorem, as well as open problems.
The so-called Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase change, for example ice passing to water, and one aims to describe the regularity of the interface separating the two phases. In its stationary version, the Stefan problem can be reduced to the classical obstacle problem, which consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed, and that is constrained to lie above a given obstacle. The aim of this talk is to give a general overview of the classical theory of the obstacle problem, and then discuss some very recent developments on the optimal regularity of the free boundary both in the static and the parabolic setting.
The ultraproduct construction gives a way of averaging an infinite sequence of mathematical structures, such as fields, graphs, or linear orders. The talk will be about the strength of such a construction.
The first part of the talk will be general background on 7-dimensional Riemannian manifolds with the exceptional holonomy group G2, going back to Berger's classification from the 1950's. Then we will explain that there is a natural boundary value problem for these structures, involving fixing a 3-form on the boundary, and discuss some of the existence questions that arise. We will consider various reductions of the equations, imposing symmetry, to lower dimensions which lead to interesting PDE problems—some of which are familiar and some new.
What is the minimum possible number of vertices of a graph that contains every k-vertex graph as an induced subgraph? What is the minimum possible number of edges in a graph that contains every k-vertex graph with maximum degree 3 as a subgraph? These questions and related ones were initiated by Rado in the 60s, and received a considerable amount of attention over the years, partly motivated by algorithmic applications. The study of the subject combines probabilistic arguments and explicit, structured constructions. I will survey the topic focusing on a recent asymptotic solution of the first question, where an asymptotic formula, improving earlier estimates by several researchers, is obtained by combining combinatorial and probabilistic arguments with group theoretic tools.
This talk will be a (hopefully) gentle introduction to applications of gauge theory to some questions in low dimensional topology. I will focus on some methods with origins in the mathematics behind gauge theory for detecting the unknot and hint at how extensions of these might give a route to a new proof of the four-color map theorem.
A complex variety is rational if it can be obtained from projective space by modifications, i.e., algebraic surgeries like blow-ups. Is rationality a deformation invariant for smooth projective varieties? This is the case for curves and surfaces but not when the dimension is at least four. The case of threefolds remains mysterious but we now know that stable rationality—rationality after taking products with projective spaces—is not a deformation invariant. (joint with Kresch, Pirutka, and Tschinkel)
Contact geometry is a beautiful subject that has important interactions with topology in dimension three. In this talk I will give a brief introduction to contact geometry and discuss its interactions with Riemannian geometry. In particular I will discuss a contact geometry analog of the famous sphere theorem and more generally indicate how the curvature of a Riemannian metric can influence properties of a contact structure adapted to it. This is joint work with Rafal Komendarczyk and Patrick Massot.
To understand a finitely presented group it is natural to explore its finite quotients. If the groups at hand are residually finite, a natural question is the extent to which the groups are determined by the totality of their finite quotients. This talk will discuss recent progress on constructing residually finite groups completely determined by their finite quotients.