In this talk, I will introduce a general variational framework for nonlinear evolution equations with a gradient flow structure, which arise in material science, animal swarms, chemotaxis, and deep learning, among many others. Building upon this framework, we develop numerical methods that have built-in properties such as positivity preserving and entropy decreasing, and resolve stability issues due to the strong nonlinearity. Two specific applications will be discussed. One is the Wasserstein gradient flow, where the major challenge is to compute the Wasserstein distance and resulting optimization problem. I will show techniques to overcome these difficulties. The other is to simulate crystal surface evolution, which suffers from significant stiffness and therefore prevents simulation with traditional methods on fine spatial grids. On the contrary, our method resolves this issue and is proved to converge at a rate independent of the grid size.
"The definition of angular momentum in general relativity has been a subtle issue since the 1960's, due to the discovery of "supertranslation ambiguity": the angular momentums recorded by two distant observers of the same system may not be the same. In this talk, I shall show how the mathematical theory of optimal isometric embedding and quasilocal angular momentum identifies a correction term, and leads to a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.
The problem of community detection is one of the fundamental problems in machine learning, with applications from recommending systems to matrix completion and biogenomics. While random graph and hypergraph models are being widely and successfully used to build and benchmark algorithms for clustering and community detection, analyzing these algorithms relies crucially on tools from random matrix and spectral graph theory. In this talk, I will give an overview of existing threshold bounds for various detection regimes in graph and hypergraph models, and mention some of the important tools used. This talk will touch upon joint work with Gerandy Brito, Christopher Hoffman, Shirshendu Ganguly, Kameron Harris, Linh Tran, Haixiao Wang, and Yizhe Zhu.
Real world fluid flows are often turbulent. However, remarkably, certain flow patterns, such as two dimensional shear flows and vortices, can be quite stable even at high Reynolds numbers. These flow patterns can be observed in large scale flows, e.g. the great red spot on Jupiter, and the analysis of their stability is a fascinating mathematical problem.
The stability of fluid flows is an old problem, considered already in the late 19th and early 20th century by Kelvin, Rayleigh, Orr, among many others. The classical works concern discrete eigenvalues of the linearized problem whose existence often indicate violent (exponential) instability. The continuous spectrum was only studied recently, and leads to a new phenomenon called "inviscid damping": the perturbation converges, weakly but not strongly as time approaches infinity. This weak convergence is consistent with 2d turbulence theory, which predicts a partial transfer of energy to high frequencies and a reverse cascade of energy to low frequencies.
In this talk we will consider the two dimensional Euler equation focusing on shear flows and vortices, review recent progresses and discuss some important open problems on both linear and nonlinear inviscid damping. This is based on joint work with A. Ionescu.
Consider the space whose points consist of n-tuples of distinct complex numbers. The Betti numbers of this space are called Stirling numbers, and they form a log concave sequence by a theorem of Isaac Newton. I will state a conjectural generalization of this result that takes into account the action of the symmetric group by permuting the points. The full conjecture is open, but I will explain how to leverage the theory of representation stability to prove infinitely many cases.
Abelian and Non-Abelian X-ray transforms are examples of integral-geometric transforms with applications to X-ray Computerized Tomography and the imaging of magnetic fields inside of materials (Polarimetric Neutron Tomography). Their study uses tools from classical inverse problems (assessments of injectivity, stability and inversions), and mathematical statistics to deal with cases with noisy data.
After giving a brief introduction to the topic, I plan on covering the following recent results:
(1). We will first discuss a sharp description of the mapping properties of the X-ray transform (and its associated normal operator I*I) on the Euclidean disk, associated with a special L2 topology on its co-domain.
(2). We will then focus on how to use this framework to show that attenuated X-ray transforms (with skew-hermitian attenuation matrix), more specifically their normal operators, satisfy similar mapping properties.
(3). Finally, we will discuss an important application of these results to the Bayesian inversion of the problem of reconstructing an attenuation matrix (or Higgs field) from its scattering data corrupted with additive Gaussian noise. Specifically, I will discuss a Bernstein-VonMises theorem on the 'local asymptotic normality' of the posterior distribution as the number of measurement points tends to infinity, useful for uncertainty quantification purposes. Numerical illustrations will be given throughout.
(2) and (3) are joint work with R. Nickl and G.P.Paternain (Cambridge).
We will introduce wave maps and discuss some of their basic properties, leading to recent works on the soliton resolution conjecture for critical wave maps and related equations.
The simplest manifolds are arguably the disc, but their diffeomorphism groups remain mysterious. I will discuss why they are fundamental objects of geometric topology, what is known about them, as well as joint work with Oscar Randal-Williams, which has the goal of understanding the rational homotopy type of the group of diffeomorphisms of even-dimensional discs.
The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question:
Which multiplicities are nonzero?
In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's nonvanishing results for the Newell-Littlewood numbers.
This is joint work with Shiliang Gao (UIUC), Gidon Orelowitz (UIUC), and Nicolas Ressayre (Universite Claude Bernard Lyon I). The presentation is based on arXiv:2005.09012, arXiv:2009.09904, and arXiv:2107.03152.
Derived categories are a construction used across many mathematical disciplines, including algebraic geometry, topology, and non-commutative algebra. I will introduce derived categories and illustrate how to use them to recover a well-known topological invariant. By discussing some open conjectures revolving around derived categories in algebraic geometry, I will then explain how to decompose derived categories into smaller, more manageable pieces and compare them to one another through my research in geometric invariant theory. To conclude, I will outline my program to solve these conjectures and discuss some works in progress.
Evolutionary dynamics are at the core of carcinogenesis. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to shed light into cancer origins, progression, and mechanisms of treatment. I will present two very general types of evolutionary patterns, loss-of-function and gain-of-function mutations, and discuss scenarios of population dynamics -- including stochastic tunneling and calculating the rate of evolution. Applications include origins of cancer, and development of resistance against treatment. I will also talk about evolution in random environments. The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Of particular interest are the dynamics of non-selected mutants, which exhibit counterintuitive properties.
The representations of Lie algebras and their associated quantum groups are known to have symmetries captured by the braid group. When considering their combinatorial shadows known as crystals, an action of a closely related group—the cactus group—emerges. I will describe how this action has surprising appearances throughout representation theory: it can be realized both geometrically as a monodromy action coming from a family of "shift of argument" algebras, as well as categorically by studying the structure of certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint work with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.
In this talk, I will discuss how incorporating geometric information into classical learning algorithms can improve their performance. The main focus will be on optimal mass transport (OMT), which has evolved as a major method to analyze distributional data. In particular, I will show how embeddings can be used to build OMT-based classifiers, both in supervised and unsupervised learning setting settings. The proposed framework significantly reduces the computational effort and the required training data.
Using OMT and other geometric data analysis tools, I will demonstrate applications in cancer research, focusing on the analysis of gene expression data and on protein dynamics.
From networks to genomics, large amounts of data are abundant and play critical roles in helping us understand complex systems. In many such settings, these data take the form of large discrete structures with important combinatorial properties. The interplay between structure and randomness in these systems presents unique mathematical challenges. In this talk I will highlight these through two vignettes on inferring latent structure in random graphs: (1) inference of latent high-dimensional geometry, and (2) improved recovery of communities using multiple correlated graphs.
First, I will talk about a canonical random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels. We are particularly interested in the high-dimensional case when d is large and, in the dense regime, we determine the phase transition for when geometry is detectable/lost. The proofs highlight novel graph statistics, as well as connections to random matrices.
Next, I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph.
When a sheet of paper is crumpled, it spontaneously develops a network of creases. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Recent experiments have shown that when a sheet is repeatedly crumpled, the total crease length grows logarithmically [1]. This talk will offer insight into this surprising result by developing a correspondence between crumpling and fragmentation processes. We show how crumpling can be viewed as fragmenting the sheet into flat facets that are outlined by the creases, and we use this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon [2].
This study was made possible by large-scale data analysis of crease networks from crumpling experiments. We will describe recent work to use the same data with machine learning methods to probe the physical rules governing crumpling. We will look at how augmenting experimental data with synthetically generated data can improve predictive power and provide physical insight [3].
[1] O. Gottesman et al., Commun. Phys. 1, 70 (2018).
[2] J. Andrejevic et al., Nat. Commun. 12, 1470 (2021).
[3] J. Hoffmann et al., Sci. Advances 5, 6792 (2019).
Arithmetic hyperbolic manifolds are constructed as quotients of hyperbolic space by subgroups of isometries commensurable with integer points in algebraic groups. The rich connection between the geometry and arithmetic give these manifolds a special beauty. In this talk, I will introduce arithmetic methods to construct hyperbolic manifolds and describe how arithmeticity helps us understand the geometry and topology of these manifolds and their finite covers.
In 1957, Grothendieck defined the K-groups of vector bundles on an algebraic variety to formulate his Riemann-Roch theorem, bringing an algebraic perspective to a theorem in complex analysis. The higher K-groups of varieties were discovered in 1973 by Quillen and have remained a source of both mystery and inspiration in mathematics, and interacts with many vastly different subjects.
In this talk, I will provide a panoramic tour on K-theory and its cousins - cycles and cobordisms. In particular I will explain how the birational geometry of Hilbert schemes (joint with Bachmann) and p-adic Hodge theory (joint with Morrow) can be used to shed new light on these creatures.
Interacting agent-based systems are ubiquitous in science, from modeling of particles in Physics to prey-predator and colony models in Biology, to opinion dynamics in economics and social sciences. Oftentimes the laws of interactions between the agents are quite simple, for example they depend only on pairwise interactions, and only on pairwise distance in each interaction. We consider the following inference problem for a system of interacting particles or agents: given only observed trajectories of the agents in the system, can we learn what the laws of interactions are? We would like to do this without assuming any particular form for the interaction laws, i.e. they might be "any" function of pairwise distances. We consider this problem both the mean-field limit (i.e. the number of particles going to infinity) and in the case of a finite number of agents, with an increasing number of observations, albeit in this talk we will mostly focus on the latter case. We cast this as an inverse problem, and present a solution in the simplest yet interesting case where the interaction is governed by an (unknown) function of pairwise distances. We discuss when this problem is well-posed, we construct estimators for the interaction kernels with provably good statistically and computational properties, and discuss extensions to second-order systems, more general interaction kernels, and stochastic systems. We measure empirically the performance of our techniques on various examples, that include extensions to agent systems with different types of agents, second-order systems, and families of systems with parametric interaction kernels. We also conduct numerical experiments to test the large time behavior of these systems, especially in the cases where they exhibit emergent behavior. This is joint work with F. Lu, J. Miller, S. Tang and M. Zhong.
Recent experimental studies in the properties of atomically thin materials such as graphene and black phosphorus have offered insights into useful aspects of the collective motion of electrons in 2D. A wealth of intriguing optical phenomena can arise in these systems because of the coupling of the electron motion with incident electromagnetic fields.
In many applications of photonics at the nanoscale, 2D materials such as graphene may behave as conductors, and allow for the excitation and propagation of electromagnetic waves with surprisingly small length scales. These surface waves are tightly confined to the material. They can possibly beat the optical diffraction limit, in the sense that the wavelength of the excited surface waves can be much smaller than that of the incident wave in a frequency range of practical interest. A broad goal in mathematical modeling is to understand how distinct kinetic regimes of 2D electron transport can be probed, and even controlled, by electromagnetic signals.
I will discuss recent work in describing the dispersion of electromagnetic modes that may propagate along edges of flat, anisotropic conducting sheets. Some emphasis will be placed on an emergent concept for the existence of such modes. The starting point is a boundary value problem for Maxwell's equations coupled with the physics of the moving electrons in monolayer and bilayer structures.
The mapping class group of a surface is a remarkable group with connections to low-dimensional topology, algebraic geometry, dynamics, and many other areas. I will give an elementary survey of some aspects of this topic with a focus on finiteness properties.
After around 25 years as an algebraic number theorist, I switched fields in 2017 and started to teach number theory to a computer proof system called Lean. I now believe that these systems will be playing an inevitable role in the future of mathematical research. I'll tell the story of what I've been doing and why I think it's important. I will assume no prior knowledge of computer proof systems.
Hecke algebras are ubiquitous in number theory and geometric representation theory. In this talk we describe the appearance of various Hecke algebras such as the affine Hecke algebra and the double affine Hecke algebra (DAHA) in Floer theory, through the higher-dimensional analog of Heegaard Floer homology. This is joint work with Yin Tian and Tianyu Yuan.
The existence and stability of concentrated vorticity solutions (vortices in 2D and vortex filaments in 3D) to the Euler equation is a long-standing problem. In this talk I will discuss our new approach--the inner-outer-gluing-scheme--to this problem. I will first report the result on desingularization of vortices in 2D: Let $ \xi (t)$ be any solution of Kirtchhoff-Ruth dynamical system. Then one can construct solutions to 2D Euler equations with concentrating vorticity along the trajectory of $\xi (t)$. Then I will discuss recent work on the Leapfrogging phenomenon in 3D Euler: Let $ (r_j, z_j)$ be any solution of Leapfrogging Dynamics. Then one can construct an axially symmetric 3D Euler solution with vortex filaments following the Leapfrogging dynamics. Vortex Filament Conjecture and relation of 3D Euler with generalized Adler-Moser polynomials will be mentioned.
The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer in 1968, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. Our proof of this conjecture is based on a new arithmetic algebraization theorem, which has its root in the classical Borel--Dwork rationality criterion. In this talk, we will discuss some ingredients in the proof and a variant of our arithmetic algebraization theorem, which we will use to prove the irrationality of certain 2-adic zeta value. This is joint work with Frank Calegari and Vesselin Dimitrov.
Cellular networks are ubiquitous in nature. Most technologically useful materials arise as polycrystalline microstructures, composed of a myriad of small monocrystalline cells or grains, separated by interfaces, or grain boundaries of crystallites with different lattice orientations. A central problem in materials science is to develop technologies capable of producing an arrangement of grains that provides for a desired set of material properties. One method by which the grain structure can be engineered is through grain growth (also termed coarsening) of a starting structure.
The evolution of grain boundaries and associated grain growth is a very complex multiscale process. It involves, for example, dynamics of grain boundaries, triple junctions, and the dynamics of lattice misorientations/grains rotations. In this talk, we will discuss recent progress in mathematical modeling, simulation and analysis of the evolution of the grain boundary network in polycrystalline materials.
I will give an introduction to some recent progress on scalar curvature rigidity using the Dirac operator method. In particular, I will discuss how the Dirac operator method can be used to resolve several of Gromov's conjectures on scalar curvature. I will make this talk accessible to graduate students and non-experts.
The Andre-Oort conjecture concerns special points of a Shimura variety S --- points that are in a certain sense "maximally symmetric". It states that if a variety V in S contains a zariski-dense set of such points, then V must itself be a Shimura variety. It is an example of the field now known as "unlikely intersections theory" which seeks to explain "arithmetic coincidences" using geometry. In fact, there is a very natural sense in which the Andre-Oort conjecture can be seen as an analogue of Faltings theorem concerning rational points on curves. The proof of this conjecture involves a wide range of disparate mathematical ideas --- functional transcendence, mondromy, point counting in transcendental sets, upper bounds for arithmetic complexity (heights of special points), and p-adic hodge theory. We will survey these concepts and how they relate to each other in the proof, aiming to give an overview of the relevant ideas. We will also discuss the current status of the field, now spearheaded by the (still extremely open!) Zilber-Pink conjecture, and what is required to make further progress.
A very common theme in number theory (and many other areas) is the study of various "local-to-global principles", such as the Hasse-Minkowski theorem for rational points on a quadric. The three talks will be devoted to several such instances in the context of rational points/algebraic cycles on varieties over global fields and their relation to special values of L-functions. In the first talk (colloquium), we will discuss the Hasse principle for rational points on intersection of two quadrics in P^N over function fields (of algebraic curves over a finite field). Among other things, the new tools here include an on-going work with Zhiwei Yun to establish a function field analog of (a strengthened) Kolyvagin's theorem for elliptic curves and our previous work on a Higher Gross-Zagier formula relating intersection numbers of certain cycles on moduli space of Drinfeld Shtukas to L-functions of elliptic curves.
The theory of complex multiplication developped in the 19th century for imaginary quadatic fields via the moduli of elliptic curves with complex multiplication, and extended by Shimura, Taniyama and Weil to CM fields, leads to (a partial) explicit class field theory for these base fields. The case where the base field is not a CM field appears more mysterious. I will describe a largely conjectural approach, for the simplest case of real quadratic fields, which rests on replacing modular functions by "rigid meromorphic cocycles". This is an account of ongoing joint work with Alice Pozzi and Jan Vonk.
Pattern formation in ecological systems is driven by counteracting feedback mechanisms on widely different spatial scales. Moreover, ecosystem models typically have the nature of reaction-diffusion systems: the dynamics of ecological patterns can be studied by the methods (geometric) singular perturbation theory. In this talk we give an overview of the surprisingly rich cross-fertilization between ecology, the physics of pattern formation and the mathematics of singular perturbations. We show how a mathematical approach uncovers mechanisms by which real-life ecosystems may evade (catastrophic) tipping under slowly varying climatological circumstances. This insight is based on two crucial ingredients: the careful study of Busse balloons in (parameter, wavenumber)-space associated to spatially periodic patterns and the validation of the model predictions by field observations. Vice versa, ecosystem models motivate the study of classes of singularly perturbed reaction-diffusion equations that exhibit much more complex behavior than the models so far studied by mathematicians: we present several novel research directions initiated by ecology.
The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$, known to the ancient Greeks in dimensions two and three, states that among all sets ("bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the sphere $\mathbb{S}^n$ and on Gauss space $\mathbb{G}^n$. Furthermore, one may consider the "multi-bubble" partitioning problem, where one partitions the space into $q \geq 2$ (possibly disconnected) bubbles, so that their total common surface-area is minimal. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $q=2$; the case $q=3$ is called the double-bubble problem, and so on.
In 2000, Hutchings, Morgan, Ritoré and Ros resolved the Double-Bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently resolved in $\mathbb{R}^n$ as well) -- the optimal partition into two bubbles of prescribed finite volumes (and an exterior unbounded third bubble) which minimizes the total surface-area is given by three spherical caps, meeting at 120-degree angles. A more general conjecture of J. Sullivan from the 1990's asserts that when $q \leq n+2$, the optimal Multi-Bubble partition of $\mathbb{R}^n$ (as well as $\mathbb{S}^n$) is obtained by taking the Voronoi cells of $q$ equidistant points in $\mathbb{S}^{n}$ and applying appropriate stereographic projections to $\mathbb{R}^n$ (and backwards).
In 2018, together with Joe Neeman, we resolved the analogous Multi-Bubble conjecture on the optimal partition of Gauss space $\mathbb{G}^n$ into $q \leq n+1$ bubbles -- the unique optimal partition is given by the Voronoi cells of (appropriately translated) $q$ equidistant points. In this talk, we will describe our approach in that work, as well as recent progress on the Multi-Bubble problem on $\mathbb{R}^n$ and $\mathbb{S}^n$. In particular, we show that minimizing partitions are always spherical when $q \leq n+1$, and we resolve the latter conjectures when in addition $q \leq 6$ (e.g. the triple-bubble conjecture in $\mathbb{R}^3$ and $\mathbb{S}^3$, and the quadruple-bubble conjecture in $\mathbb{R}^4$ and $\mathbb{S}^4$).
Based on joint work (in progress) with Joe Neeman.
I will survey some recent results on unique continuation and homogenization. In particular, I will discuss the Landis conjecture, the embedded eigenvalue problem for periodic elliptic equations, and applications of homogenization to quantitative unique continuation.