Department Colloquia

The Fundamental Group of Random 3-Manifolds and Other Random Objects

The fundamental group of a 3-manifold varies greatly from one manifold to the next. But what do they typically look like? Dunfield and Thurston formulated a precise version of this question by giving a definition of a random 3-manifold, and made partial progress on it. Building on their work, in joint work with Melanie Matchett Wood we answer this question, at least for the profinite completion of the fundamental group. I will explain how this relates to the classical problem of determining a probability distribution from its moments and discuss how this generalizes to the study of random objects in a wide class of categories.

Cohomology of Moduli Spaces of Curves

The moduli space M_g of genus g curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of M_g is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of M_g called the tautological ring. The definition of the tautological ring was later extended to the compactification M_g-bar and the moduli spaces with marked points M_{g,n}-bar. While the full cohomology ring of M_{g,n}-bar is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I'll ask the question: which cohomology groups H^k(M_{g,n}-bar) are tautological? And when they are not, how can we better understand them?

This is joint work with Samir Canning and Sam Payne.

The Weil Conjectures and A1-homotopy Theory

The celebrated Weil conjectures from 1948 propose a beautiful connection between algebraic topology and the number of solutions to equations over finite fields: the zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the Betti numbers of the associated complex manifold. This talk will describe the Weil conjectures, some A1-homotopy theory, and then enrich the zeta function to have coefficients in a group of bilinear forms. The enrichment provides a connection between the solutions over finite fields and the topology of the associated real manifold.

The new work in this talk is joint with Tom Bachmann, Margaret Bilu, Wei Ho, Padma Srinivasan and Isabel Vogt.

Feb. 21 2024

Hidden Structures on de Rham Cohomology of P-adic Analytic Varieties

I will survey what is known about extra structures (Hodge filtration, Frobenius, monodromy) appearing on de Rham cohomology of analytic varieties over local fields of mixed characteristic.

Eigenmatrix for Unstructured Sparse Recovery

This talk discusses the unstructured sparse recovery problems of a general form. The task is to recover the spike locations and weights of an unknown sparse signal from a collection of its unstructured observations. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are the noise in the sample values and the unstructured nature of the sample locations. We propose the eigenmatrix construction, a data-driven approach to this problem.  The eigenmatrix turns this non-linear inverse problem into an eigen-decomposition problem with desired eigenvalue and eigenvector pairs. This approach extends the classical Prony's method and offers a new way for these sparse, unstructured recovery problems.

Categorical Noncommutative Geometry

I will give a brief overview of the categorical approach to algebraic geometry. The idea is to study algebraic varieties (or schemes) via their derived categories of coherent sheaves, considered as differential graded (dg) categories. One can consider more general dg categories (with some finiteness conditions) as noncommutative algebraic varieties (or schemes). It turns out that the notions of smoothness and properness make sense in this context, as well as classical invariants such as de Rham cohomology and K-theory. I will give an overview of some results in this field.

Symmetries, Anomalies and Anomaly Inflow

Anosov Fllows on 3-Manifolds

Abstract:

Anosov flows are a fascinating class of dynamical systems, generalizing and including geodesic flows on manifolds of negative curvature.  These systems exhibit "local chaos but global stability" - individual orbits diverge wildly, but the systems as a whole are stable under perturbation.  This stability means there is some hope to classify them by discrete algebraic invariants.  Even on 3-dimensional spaces, this is an interesting and challenging problem.   In this talk, I will describe some of the history and motivation for classification (dating back to work of Anosov and Smale in the 60s), connections with low-dimensional geometric topology, and will describe recent joint work with Barthelmé, Bowden, Frankel and Fenley (in various combinations) giving answering one thread of the classification problem in dimension 3.  

Random Planar Geometry and the Directed Landscape

Abstract:

Consider the lattice Z^2, and assign length 1 or 2 to every edge by flipping a series of independent fair coins. This gives a random weighted graph, and looking at distances in this graph gives a random planar metric. This model, along with most natural models of random planar metrics and random interface growth (the so-called `KPZ universality class'), is expected to converge to a universal scaling limit: the directed landscape. The goal of this talk is to introduce this object, describe some of its properties, and describe at least one model where we can actually prove convergence.

Spectra and Definability

Abstract:

In this talk we will see some recent results regarding the spectra, i.e. the set of possible sizes of, of combinatorial sets of reals, sets traditionally associated to the combinatorial cardinal characteristics of the continuum  and the projective complexity to witnesses of different cardinalities.

Random surface, Planar Lattice Model, and Conformal Field Theory

Abstract:

Liouville quantum gravity (LQG) is a theory of random surfaces that originated from string theory. Schramm Loewner evolution (SLE) is a family of random planar curves describing scaling limits of many 2D lattice models at their criticality. Before the rigorous study via LQG and SLE in probability, random surfaces and scaling limits of lattice models have been studied via  another approach in theoretical physics called conformal field theory (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics on random surfaces and planar lattice models, including the law of the random modulus of the scaling limit of uniform triangulation of the annular topology, and the crossing formula for critical planar percolation on an annulus. I will then present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT. 

The Ramanujan Conjecture

I will explain the statement of the Ramanujan conjecture for modular forms, and something of its history, finishing with a gentle overview of joint work in progress with George Boxer, Frank Calegari and James Newton, in which we prove the Ramanujan conjecture for Bianchi modular forms. No prior knowledge of these topics will be assumed!

For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). In this talk, I will present recent progress on this topic.

Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.

A Tale of Two Theorems of Thurston

Abstract:

In the 20th century, Thurston proved two classification theorems, one for surface homeomorphisms and one for branched covers of surfaces.
While the theorems have long been understood to be analogous, we will present new work with Belk and Winarski showing that the two theorems
are in fact special cases of one Ubertheorem.  We will also discuss joint work with Belk, Lanier, Strenner, Taylor, Winarski, and Yurttas on further algorithmic and theoretical aspects of Thurston’s theorems.

This talk is meant to be accessible to a wide audience.

2/8/23

Applications of Model Theory to Functional Transcendence

Abstract:

Over the last decades there has been a surge in interest around functional transcendence results, in part due to their connection with special points conjectures in number theory. The approaches to proving those results employ various techniques from group theory, complex variables, and number theory, but each one also shares a common element: a tool called o-minimality originating in model theory. In this talk I will discuss how an entirely new approach, centered around the model theory of differential fields, has been successfully used to tackle several open problems in functional transcendence theory. The talk will focus on recent progress around the Ax-Lindemann-Weierstrass and Ax-Schanuel Theorems for uniformizers of geometric structures.

2/1/23

Stochastic Processes on Trees and their Applications

We survey research involving stochastic processing on trees with emphasis on the broadcasting on trees model. We review the mathematical questions that arise in the analysis of this process and its inference via Belief Propagation", an iterative application of Bayes Law,. We discuss the mathematical connections to statistical physics, the study of random graphs, evolutionary inference in the biological sciences and understanding depth in inference in theoretical computer science.

Random Packings and Liquid Crystals

Let T be a subset of R^d, such as a ball, a cube or a cylinder, and consider all possibilities for packing translates of T, perhaps with its rotations, in some bounded domain in R^d. What does a typical packing of this sort look like? One mathematical formalization of this question is to fix the density of the packing and sample uniformly among all possible packings with this density. Discrete versions of the question may be formulated on lattice graphs.

The question arises naturally in the sciences, where T may be thought of as a molecule and its packing is related to the spatial arrangement of molecules of a material under given conditions. In some cases, the material forms a liquid crystal - states of matter which are, in a sense, between liquids and crystals. I will review ideas from this topic, mentioning some of the predictions and the mathematical progress. Time permitting, I will elaborate on a recent result, joint with Daniel Hadas, on the structure of high-density packings of 2x2 squares with centers on the square lattice.

Higher Traces in Arithmetic Geometry

A central theme in arithmetic geometry is the passage from geometric invariants to arithmetic information by taking the trace of Frobenius. I will describe a higher categorical version of this procedure with a particular focus on applications to the Langlands correspondence over function fields. In the global unramified case, we obtain that the space of automorphic forms is the categorical trace (aka Hochschild homology) of Frobenius acting on (an appropriate version of) the automorphic category. This, in particular, leads to an enhancement of V. Lafforgue’s spectral decomposition of the space of automorphic forms.  In the local case, we obtain an enhancement of the local Langlands conjecture of Fargues-Scholze in the case of function fields. This is based on joint works with Arinkin, Gaitsgory, Kazhdan, Raskin, and Varshavsky.

What is the most natural way of choosing a random surface (2d Riemannian manifold), say with the topology of the sphere? This question does not have an obvious answer since the space of all Riemannian metric tensors on the sphere is infinite-dimensional. One possible approach is to consider random triangulations of a sphere, with $n$ triangles, and take some sort of limit as $n\rta\infty$. This gives rise to a one-parameter family of random metric measure spaces called Liouville quantum gravity (LQG) surfaces, which can be thought of as canonical random surfaces". These surfaces have the same topology as the sphere, but very different geometric properties. For example, their Hausdorff dimension is strictly larger than two and the geodesics started from a fixed point form a tree-like structure. LQG surfaces are also of interest in physics, for example in bosonic string theory and conformal field theory. I will discuss the definition, motivation, and basic properties of LQG surfaces, assuming no background beyond a typical first-year graduate curriculum.

The Arithmetic of Power Series

A holomorphic function in a neighborhood of \(z=0\) is algebraic if it satisfies a polynomial equation with coefficients in \(\mathbb{Q}[z]\). An example of such a function is the square root of \(1-4z\). The power series expansion of this function turns out to have coefficients which are all integers, and a theorem of Eisenstein says that (up to scaling and suitably interpreted!) this is true of all algebraic functions. But under what conditions can we deduce that a power series with integral coefficients is an algebraic function? We shall explain how this problem is related not only to complex analysis and number theory, but also to Klein’s famous observation that not all finite index subgroups of \(SL_2(\mathbb{Z})\) are determined by congruence conditions.

Equiangular lines and eigenvalue multiplicities

Abstract:

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle.

A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.

My talk will discuss these problems and their connections. Here is an open problem that I would like to understand better: what is the maximum possible second eigenvalue multiplicity of a connected bounded degree graph?

Joint work with Zilin Jiang, Jonathan Tidor, Yuan Yao, and Shengtong Zhang

Volumes of Hyperbolic Polytopes, Cluster Polylogarithms, and the Goncharov Depth Conjecture

Lobachevsky started to work on computing volumes of hyperbolic polytopes long before the first model of the hyperbolic space was found. He discovered an extraordinary formula for the volume of an orthoscheme via a special function called dilogarithm. 

We will discuss a generalization of the formula of Lobachevsky to higher dimensions. For reasons I do not fully understand, a mild modification of this formula leads to the proof of a conjecture of Goncharov about the depth of multiple polylogarithms. Moreover, the same construction leads to a functional equation for polylogarithms generalizing known equations of Abel, Kummer, and Goncharov. 

When is a Mathematical Object Well-Behaved?

In this talk we will come at this question from two
different angles: first, from the viewpoint of model theory, a subject
in which for nearly half a century the notion of stability has played
a central role in describing tame behaviour; secondly, from the
perspective of combinatorics, where so-called regularity
decompositions have enjoyed a similar level of prominence in a range
of finitary settings, with remarkable applications.

In recent years, these two fundamental notions have been shown to
interact in interesting ways. In particular, it has been shown that
mathematical objects that are stable in the model-theoretic sense
admit particularly well-behaved regularity decompositions. In this
talk we will explore this fruitful interplay in the context of both
finite graphs and subsets of abelian groups.

To the extent that time permits, I will go on to describe recent joint
work with Caroline Terry (The Ohio State University), in which we
develop a higher-arity generalisation of stability that implies (and
in some cases characterises) the existence of particularly pleasant
higher-order regularity decompositions.

How Round is a Jordan Curve?

Abstract:

The Loewner energy for Jordan curves first arises from the small-parameter large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle. This class of curves has intriguingly more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and string theory, and has been studied since the eighties. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from probability theory inspire new results on Weil-Petersson quasicircles and discuss further directions.

Knotted handlebodies

Abstract:

Often, interesting knotting vanishes when allowed one extra dimension, e.g. knotted circles in 3-space all become isotopic when included into 4-space. Hughes, Kim and I recently found a new counterexample to this principle: for \(g>1\), there exists a pair of 3-dimensional genus-\(g\) solids in the 4-sphere with the same boundary, and that are homeomorphic relative to their boundary, but do not become isotopic rel boundary even when their interiors are pushed into the 5-dimensional ball. This proves a conjecture of Budney and Gabai (who previously constructed 3-balls in the 4-sphere with the same boundary that are not isotopic rel boundary) for \(g>1\) in a very strong sense.

In this talk, I’ll describe some motivation from 3-dimensional topology and useful/weird facts about higher-dimensional knots (e.g. knotted surfaces in 4-manifolds), show how to construct interesting codimension-2 knotting in dimensions 4 and 5 (joint with Mark Hughes and Seungwon Kim), and talk about related open problems.

February 16, 2022

Phase Transitions in Hyperbolic Space

Abstract:

Many questions in probability theory concern the way the geometry of a space influences the behaviour of random processes on that space, and in particular how the geometry of a space is affected by random perturbations. One of the simplest models of such a random perturbation is percolation, in which the edges of a graph are either deleted or retained independently at random with retention probability p. We are particularly interested in phase transitions, in which the geometry of the percolated subgraph undergoes a qualitative change as p is varied through some special value. Although percolation has traditionally been studied primarily in the context of Euclidean lattices, several new and interesting phenomena can arise when one considers more exotic geometric settings. In this talk, I will discuss conjectures and results concerning percolation on nonamenable graphs and hyperbolic spaces and explain the main ideas behind our recent result that percolation in any hyperbolic graph has a non-trivial phase in which there are infinitely many infinite clusters.

Big Fiber Theorems and Ideal-Valued Measures

In various fields of mathematics one encounters statements of the following type: For any map in a suitable class there is a point in the target whose preimage is big". This includes the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-Displaceable Fiber Theorem in symplectic topology. I'll present a unified viewpoint at these results based on Gromov's notion of ideal-valued measures and its adaptation to symplectic topology. Joint work with Adi Dickstein, Yaniv Ganor, and Frol Zapolsky.

This talk will be followed by a short course over the next few weeks expanding on these ideas.  (Details will be announced in a future email and on our web page; Leonid's lectures will be followed by a course by Danny Calegari on the foundations of symplectic topology).

Some New Algebraic K(pi,1)s

Configuration spaces of unordered distinct points in C are extremely familiar and important examples of complex algebraic varieties that also happen to be K(pi,1)s (from which many interesting properties follow). Given how well-studied these examples are, it is rather astonishing (to me) that a very simple variation gives rise to a new infinite family of (algebraic surface) K(pi,1)s. We explain these examples and some interesting and tantalizing connections to braids, polynomials, and complex dynamics.

Period Integrals of Algebraic Varieties

Abstract:

A period integral of a complex algebraic variety is the integral of an algebraic differential form along a topological cycle.  While such integrals are transcendental in nature, they have had pervasive importance in algebraic and arithmetic geometry going back to the foundational work of Abel and Jacobi.  The functions one obtains from taking period integrals in families of algebraic varieties moreover recur in many areas of mathematics.  In this talk I will first survey how algebraic information can be extracted from period integrals, as well as their connection to various conjectural frameworks.  I will then discuss recent progress on understanding the algebraic properties of period integrals in families and recent applications to Hodge theory, algebraic geometry, arithmetic geometry, and logic. 

Helly Type Problems

Abstract:

Helly’s theorem from 1912 asserts that for a finite family of convex sets in a d-dimensional Euclidean space, if every d + 1 of the sets have a point in common then all of the sets have a point in common. This theorem found applications in many areas of mathematics and led to numerous generalizations. Helly’s theorem is closely related to two other fundamental theorems in convexity: Radon’s theorem asserts that a set of d + 2 points in d-dimensional real space can be divided into two disjoint sets whose convex hulls have non empty intersection. Caratheodory’s theorem asserts that if S is a set in d-dimensional real space and x belongs to its convex hull then x already belongs to the convex hull of at most d + 1 points in S. I will mention problems around Helly's theorem with combinatorial, topological and geometrical flavours.

Height Gaps, an Arithmetic Margulis Lemma and Almost Laws

Abstract:

We discuss some open problems and some new results about the topology of arithmetic locally symmetric spaces. Among the new results is a proof of a conjecture of Gelander stating that the topology of these manifolds can be bounded just in terms of the volume. Some of the proofs are based on a resolution of a non-abelian version of Lehmer's conjecture due to Breuillard, for which we also provide a new proof coming from the study of some curious word maps called almost laws.

All notions will be explained. Based on joint projects with M. Fraczyk, J. Raimbault, Lvzhou (Joe Chen) and Homin Lee.

Noncollapsed Gromov-Hausdorff Limit Spaces with Ricci Curvature Bounded Below

Abstract:

By a fundamental (relatively soft) theorem of Gromov, the collection of riemannian manifolds (Mn, g) with Ricci curvature bounded below, say RicMn ≥ −(n − 1), and diameter ≤ d, is precompact in the Gromov-Hausdorff topology. Thus, any sequence of such manifolds has a subsequence (Mni, gi) which converges in a weak geometric sense to some limiting metric space (X, d). Intuitively, this means that no matter how acute our vision, if i is sufficiently large, we will be unable to distinguish between (Mni, gi) and (X, d). 

The limits of such sequences can be thought of as counter parts of distributions or Sobolev functions. Thus, even for applications to the smooth case, it is important to understand their structure. In the noncollapsing case, where by definition, Xn has Hausdorff dimension n, the structure is much more constrained. From our work with Colding in the mid 1990’s, it is known that for all   > 0, there is a closed subset S   ⊂ Xn of Hausdorff dimension ≤ n−2, such that Xn\S  is θ( )-bi-Holder equivalent to a smooth ¨ n-dimensional riemannian manifold. Moreover, θ( ) → 1 as   → 0. Until relatively recently (2018) little else was known about the structure of S. 

We will discuss some structural results obtained in joint work with Aaron Naber an Wenshuai Jiang. They show that in several precise senses, S   strongly resembles a smooth submanifold of dimension ≤ n − 2. In particular, if S   has postive (n − 2)-dimensional Hausdorff measure, then it is (n − 2)-rectifiable, which is the measure theoretic version of being a manifold of dimension n − 2. Examples of Naber and Nan Li show that in actuality, S   need have no manifold points.

Supplemental materials can be found here:

-Slides

-Structure Theory and Convergence in Riemannian Geometry

-Degeneration of Riemannian Metrics Under Ricci Curvature Bounds

Symmetries in Algebraic Geometry

In this talk I will discuss different notions of symmetry of algebraic varieties. First we will examine automorphisms. The structure of the group of automorphisms of a projective variety often encodes some of its relevant geometric properties. However, the notion of automorphism is too rigid in the scope of birational geometry. We are then led to consider another class of symmetries of projective varieties, its birational self-maps. Birational self-maps of the projective space $\mathbb{P}^n$ are called Cremona transformations. Describing the structure of the group of Cremona transformations of the plane is a classical problem that goes back to the 19th century. In higher dimensions, not so much is known, and a natural problem is to construct interesting subgroups of the Cremona group. I will end the talk by discussing a recent work with Alessio Corti and Alex Massarenti, where we investigate subgroups of the Cremona group consisting of symmetries preserving special geometric structures.

Quantum Symmetry

Abstract:

In this talk, I will discuss “quantum symmetry” from an algebraic viewpoint, especially for symmetries of algebras. The term “quantum” is used as algebras here are usually noncommutative. I will mention some interesting results on when symmetries of algebras must factor or do not factor through symmetries of classical gadgets (such as groups or Lie algebras), that is, when we must enter the realm of quantum groups (or Hopf algebras) to understand symmetries of a given algebra. This all fits neatly into the framework of studying algebras in monoidal categories, and if time permits, I will give some recent results in this direction. I aim to keep the level of the talk down-to-earth by including many basic definitions and examples.

Stationary Measure for the Open KPZ Equation

Abstract:

The KPZ equation has attracted great attention since its inception in the 80s, first in physics, then in the fields of random matrix theory, integrable probability, and stochastic PDEs. In this talk we will focus on the "open KPZ equation" which models stochastic interface growth over a spatial interval [0,1] with mixed Neumann boundary conditions at 0 and 1. For given boundary parameters, when started from any initial height profile the interface should converge to a unique stationary measure in terms of height-function differences. We will describe how to construct such stationary measures. Along the way, we will encounter orthogonal polynomials in the Askey-Wilson scheme, the asymmetric simple exclusion process, and precise asymptotics of q Gamma functions.

This talk is based on my joint work with Alisa Knizel and Hao Shen, and also involves important contributions from Wlodzimierz Bryc, Jacek Wesolowski and Yizao Wang.

Macdonald and Schubert Polynomials from Markov Chains

Abstract:

Two of the most famous families of polynomials in combinatorics are Macdonald polynomials and Schubert polynomials. Macdonald polynomials are a family of orthogonal symmetric polynomials which generalize Schur and Hall-Littlewood polynomials and are connected to the Hilbert scheme.  Schubert polynomials also generalize Schur polynomials, and represent cohomology classes of Schubert varieties in the flag variety. Meanwhile, the asymmetric exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice, which was initially introduced by Macdonald-Gibbs-Pipkin to provide a model for translation in protein synthesis.  In my talk I will explain how two different variants of the ASEP have stationary distributions which are closely connected to Macdonald polynomials and Schubert polynomials, respectively.  This leads to new formulas and new conjectures.

This talk is based on joint work with Corteel-Mandelshtam, and joint work with Donghyun Kim.

The Extended Smale's 9th problem — On hardness of approximation in computation and non-computable problems in computer-assisted proofs

Abstract:

In the list of problems for the 21st century S. Smale calls for extended '[Computational] models which process approximate inputs'. In addition, Smale's 9th problem asks for an algorithm over the reals that decides feasibility and produces a minimiser of linear programs (LPs) in polynomial time (or strongly polynomial time in the Turing case). The extended model — in combination with Smale's 9th problem — yields an extended version of Smale's 9th problem for which we show several surprises:

(1) Given the extended model, for any eps_0 > 0 there is a class of LPs such that no algorithm can compute an epsilon-approximation to a minimiser (even in the randomised case) if epsilon < eps_0. However, computing an epsilon-approximation is in P (computable in polynomial time in the number of variables) if epsilon > eps_0. This result is independent of the P vs NP question unlike the typical hardness of approximation phenomenon in computer science.

(2) Hence, the extended Smale's 9th problem leads to new type of hardness of approximation results not previously known. Moreover, similar behaviour occurs in statistical estimation with the Lasso and in sparse regularisation with Basis Pursuit. Thus, one computes with non-computable functions on a daily basis.

(3) The extended Smale's 9th problem demonstrates how the proof of Kepler's conjecture by T. Hales was done based on successfully computing with non-computable problems. This phenomenon is not exclusive and happens also in the proof of the Dirac-Schwinger conjecture of C. Fefferman and L. Seco.

The above results demonstrate — paradoxically — the need for a complexity theory for non-computable functions. We will discuss this issue and the implications in mathematics, computations and computer-assisted proofs.
 

Liouville Quantum Gravity as a Metric Space and a Scaling Limit

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results. In this talk, we will discuss the relationship between these objects in addition to how it can be used to study statistical mechanics models on random planar maps. 

Parts of this talk will describe joint works with Ewain Gwynne and Scott Sheffield.

My Favorite Algebraic Cycle

Notes for this talk can be found here.

Abstract:

Hasse Weil L-functions L(H^n(X),s) are functions of a complex variable s, analytic in a half-plane, associated to an algebraic variety X defined over Q (or more generally any number field), together with a choice of cohomological degree n. They have an Euler product expansion with local factors at good reduction primes p of the form (f_p = geometric frobenius at p)

1/det(1-f_pp^{-s}|H^n(X_{\F_p})).

The simplest is the Riemann zeta function. Another example is L(H^1(E),s) with E an elliptic curve. Significant progress has been made in understanding these in recent years due to the introduction of Euler systems by Kolyvagin and Kato.

The basic conjecture (first formulated by Birch and Swinnerton-Dyer, and then greatly generalized by Beilinson) is that the behavior of L at integer values of s should be controlled by algebraic cycles.

I will focus on the case dim X = 2m-1 is odd, and n=2m-1. The order of vanishing of L(H^{2m-1}(X),s) at s = m is then conjectured to be the rank of the Chow group of codim m algebraic cycles homologous to 0 modulo rational equivalence on X, CH^m(X)^0. For example, when dim X=1, this becomes the Mordell-Weil group of 0-cycles on X of   degree 0 modulo divisors of functions. The order of 0 at s=1 should be the rank of the Mordell-Weil group, and the value of the first non-vanishing term in the Taylor series expansion of L(H^1(X),s) in s at s=1 should be given upto elementary factors by the discriminant of the height pairing on the Mordell Weil group.

Not much is known in the case dim X=2m-1>1. The talk will focus on a class of elementary examples of varieties X and cycles Z. The construction is geometric rather than arithmetic, and there are interesting links with hypergeometric motives and limiting mixed Hodge structures.

Green Function vs. Geometry

Abstract:

In this talk we will discuss connections between the geometric and PDE properties of sets. The emphasis is on quantifiable, global results which yield true equivalence between the geometric and PDE notions in very rough scenarios, including domains and equations with singularities and structural complexity. The main result establishes that in all dimensions $d<n$, a $d$-dimensional set in $\RR^n$ is regular (rectifiable) if and only if the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes). To the best of our knowledge, this is the first free boundary result of this type for lower dimensional sets and the first free boundary result in the classical case $d=n-1$ without restrictions on the coefficients of the equation. 

Exotic Calabi-Yau Metrics

Abstract:

I will survey the problem of constructing complete Calabi-Yau metrics on noncompact manifolds, and trace its history from the foundational work of Tian-Yau to some more recent developments. Some emphasis will be given to a concrete construction of a nontrivial Calabi-Yau metric on C^3 with maximal volume growth, which turns out to be also relevant in describing collapsing Calabi-Yau metrics.

Trigonometric Functions and Modular Symbols

In his fantastic book “Elliptic functions according to Eisenstein and Kronecker,” Weil writes: "As Eisenstein shows, his method for constructing elliptic functions applies beautifully to the simpler case of the trigonometric functions. Moreover, this case provides […] the simplest proofs for a series of results, originally discovered by Euler." 
 
The results Weil alludes to are relations between product of trigonometric functions. I will first explain how these relations are quite surprisingly governed by relations between modular symbols (whose elementary theory I will sketch). I will then show how this story fits into a wider picture that relates the topological world of group homology of some linear groups to the algebraic world of trigonometric and elliptic functions. To conclude I will briefly describe a recent number theoretical application. 
 
This is based on a work-in-progress with Pierre Charollois, Luis Garcia and Akshay Venkatesh.

Algebraic K-theory, Traces, and Arithmetic

Abstract:

The connections between l-adic algebraic K-theory and \'etale cohomology have played a central role in the development of the former. In the p-adic context, a key computational tool is the cyclotomic trace to topological cyclic homology, which has close connections to prismatic cohomology. I will describe some aspects of the p-adic picture.

Counting Problems: Open Questions in Number Theory, From the Perspective of Moments

Abstract:

Many questions in number theory can be phrased as counting problems. How many number fields are there? How many elliptic curves are there? How many integral solutions to this system of Diophantine equations are there? If the answer is “infinitely many,” we want to understand the order of growth for the number of objects we are counting in the “family." But in many settings we are also interested in finer-grained questions, like: how many number fields are there, with fixed degree and fixed discriminant? We know the answer is “finitely many,” but it would have important consequences if we could show the answer is always “very few indeed.” In this accessible talk, we will describe a way that these finer-grained questions can be related to the bigger infinite-family questions. Then we will use this perspective to survey interconnections between several big open conjectures in number theory, related in particular to class groups and number fields, and recent joint work with Caroline Turnage-Butterbaugh and Melanie Matchett Wood.

Subset Sums

Abstract:

In this talk, I will discuss novel techniques which allow us to prove a diverse range of results relating to representing integers as subset sums, including solutions to several long-standing open problems in the area. These include: solutions to the three problems of Burr and Erdős on Ramsey complete sequences, for which Erdős later offered a combined total of $350 for their solution; analogous results for the new notion of density complete sequences; the answer to a question of Alon and Erdős on the minimum number of colors needed to color the positive integers less than n so that n cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erdős and Graham and first studied by Alon on sets of integers avoiding a given subset sum; and, answering a question of Sárközy and of Tran, Vu and Wood, a common strengthening of seminal results of Szemerédi-Vu, Freiman, and Sárközy on long arithmetic progressions in subset sums. 

Based on joint work with David Conlon and Huy Tuan Pham.

Stability, Non-Approximated Groups and High-Dimensional Expanders

Abstract:

Several well-known open questions, such as: "are all groups sofic or hyperlinear?", have a common form: can all groups be approximated 
by asymptotic homomorphisms into the symmetricgroups Sym(n) (in the sofic case) or the  unitary groups U(n) (in the hyperlinear case)?

      In the case of U(n), the question can be asked with respect to different metrics and norms. 

We answer, for the first time, some of these versions, showing that there exist finitely presented groups which are  not approximated by U(n) with respect to the Frobenius (=L_2) norm and many other norms.

The strategy is via the notion of "stability": Some higher dimensional cohomology vanishing phenomena is proven to imply stability. Using Garland method  ( a.k.a. high dimensional expanders as quotients of Bruhat-Tits buildings)  , it is shown that  some non-residually-finite groups   are stable and hence cannot be approximated. These groups are  central extensions of some lattices in p-adic Lie groups (constructed via  a p-adic version of a result of Deligne).

All notions will be explained.  Based on joint works with M. De Chiffre, L. Glebsky and A. Thom and with I. Oppenheim.

Homology Growth in Towers and Aspherical Manifolds

Abstract:

Given a space and a tower of covering spaces, a natural game is to take a classical homological invariant and study its growth as one goes up the tower. If the covers are regular and residual (the intersection of the corresponding subgroups is the identity), one hopes that these limit towards an invariant of the universal cover with an analytic flavor. This turns out to be true for rational homology, and there has been a lot of recent work extending this to F_p-homology and torsion in integral homology, though the whole story remains conjectural. I'll survey various conjectures about rational/F_p homology growth and torsion growth in these covers. We'll then discuss constructions of closed aspherical manifolds that have F_p homology growth outside of the middle dimension, which disproves an F_p-version of a conjecture of Singer.

Algorithmic Solutions to Diophantine Problem

Abstract:

A basic problem in number theory is to find all integer (or rational) solutions to a system of polynomial equations.  This class of problem, known as Diophantine problems, includes many questions of traditional interest.  In 1900, Hilbert asked whether there is a general algorithm to solve all such problems; work of Matiyasevich and others shows that such an algorithm does not exist.  The prospects are much better when the system has only one degree of freedom -- the solution set is a curve.  In 1983, Faltings proved Mordell's conjecture, showing that a curve of genus at least 2 has only finitely many rational points.  Since then, significant progress has been made toward a general algorithm.  I will give an overview of the subject, including recent developments.

Combinatorics, Hodge Theory, and Beyond

Abstract:

I will survey applications of Hodge Theory to combinatorics, and, quite suprisingly, how Hodge-Riemann relations and Lefschetz type theorems can be proven using combinatorial methods, in settings that are beyond classical projective algebraic geometry.

I then go one step further, and discuss how many triangles a PL embedded simplicial complex in R^4 can have (the answer, generalizing a result of Euler and Descartes, is at most 4 times the number of edges). I discuss how to reduce this problem to a Lefschetz property beyond projective structure, and indicate the proof of the Lefschetz property in this case, replacing positivity with anisotropy of subspaces in the Hodge-Riemann pairing.
If time permits, I will then discuss relations to the Singer conjecture for aspherical manifolds.

The  Schrodinger Equations as Inspiration of Beautiful Mathematics

Abstract: 

In recent years great progress has been made in the study of dispersive and wave equations.  Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem the periodic 2D cubic nonlinear Schrodinger equation. I will start by giving the  physical  derivation of the equation starting from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to  dynamical systems, and I will end with some results, such as the non-squeezing theorem, that one can obtain once the equation is viewed in the frequency space as an  infinite dimension Hamiltonian system.

January 22, 2020

Growth, Distortion, and Isoperimetry in Topology

Abstract:

The contractibility of a loop in our universe becomes almost irrelevant if it takes longer than the age of the universe to contract it; in other words, the simple connectivity of our experience is ultimately a geometric, not a topological fact. In the 1990's, this and other considerations led Gromov to propose a program of quantitative topology: asking about the "size" or "complexity" of the objects (a homotopy between two maps; a filling of a nullcobordant manifold) whose existence is implied by the results of algebraic and geometric topology. I will discuss the questions and the motivations behind them, as well as some answers, most of them recent.

May 29, 2019

Complex Structure Degeneration and Metric Collapsing of Calami-Yau Manifolds

Abstract:

A Calabi-Yau manifold is a compact Kahler manifold with trivial canonical bundle. Yau’s solution to the Calabi conjecture yields canonical Ricci-flat Kahler metrics (Calabi-Yau metric) on such a manifold, and this has deep applications in many areas of mathematics. It is a longstanding question to understand how the Ricci-flat metrics develop singularities when the complex structure degenerates. An especially intriguing phenomenon is that these metrics can collapse to lower dimensions and exhibit very non-algebraic features, and it is challenging to describe the corresponding geometric behavior. In this talk I will review the status of this problem and explain some recent progress.

May 8, 2019

A New Approach to Higher Codimension Mean Curvature Flow

Abstract:

We will discuss very recent joint work with Bill Minicozzi about a new approach to higher codimension mean curvature flow. This is a subject that has been notoriously difficult and where much less is known than for hypersurfaces. Some of the inspiration for this new approach comes from function theory on manifolds.

May 1, 2019

Representation Stability

Abstract:

This talk will give an overview of the recent field of 'representation stability'. I will discuss how we can use representation theory to illuminate the structure of certain families of groups and topological spaces with actions of the symmetric groups SnSn, focusing on configuration spaces as an illustrative example.

April 26, 2019

Ancient Solutions to Geometric Flows

Abstract:

Some of the most important problems in geometric flows are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the partial differential equation involved. In the case of a parabolic equation the blow up analysis often leads to ancient solutions which are defined for all time −∞<t≤T−∞<t≤T for some T≤+∞T≤+∞. The classification of such solutions often sheds new insight to the singularity analysis. In some flows it is also important for performing surgery near a singularity. In this lecture we will give an overview of {\em uniqueness theorems} for ancient solutions to geometric partial differential equations such as the Mean curvature flow, the Ricci flow and the Yamabe flow.

April 17, 2019

D-Modules in Birational and Complex Geometry

Abstract:

I will give an overview of techniques based on the theory of mixed Hodge modules, which lead to a number of applications of a rather elementary nature in birational and complex geometry, as well as in the study of singularities. One of the main points I will emphasize is the existence (and usefulness) of a package of vanishing and positivity theorems in the context of filtered D-modules of Hodge theoretic origin.

March 15, 2019

Hodge Theory and Dynamical Systems

Abstract:

I will explain some interactions between dynamics and Hodge theory, with applications going in both directions. For certain low-dimensional systems, such as flows on surfaces, Hodge theory provides the necessary analytic tools. It also connects the subject to arithmetic questions such as real multiplication and torsion points on abelian varieties. In the reverse direction, I will describe dynamical considerations that provide tools for a global study of variations of Hodge structure. One consequence is a new uniformization result for special families of algebraic 3-manifolds.

March 14, 2019

Complex Dynamics and Arithmetic Equidistribution

Abstract:

I will explain a notion of arithmetic equidistribution that has recently found application in the study of complex dynamical systems. It was first introduced about 25 years ago, by Szpiro-Ullmo-Zhang, to analyze the geometry and arithmetic of abelian varieties. In 2011, Matt Baker and I used the theory to study periodic points of maps on P1P1. In this talk, I will explain some dynamical questions that were inspired by questions about elliptic curves, and then how the dynamical results allowed us to solve problems in the original setting of abelian varieties. The new results are joint with Holly Krieger and Hexi Ye.

March 13, 2019

Adding Numbers and Shuffling Cards

Abstract:

When numbers are added in the usual way 'carries' occur along the way. It turns out that the carries form a Markov chain with an 'amazing' transition matrix. This same matrix occurs in fractal geometry, the Veronese embedding AND in the analysis of shuffling cards. I will explain the connections 'in english'

March 13, 2019

The Cost of the Sphere Eversion and the 16\pi Conjecture

Abstract:

How much does it cost...to knot a closed simple curve ? To cover the sphere twice ? to realize such or such homotopy class ? ...etc. All these questions consisting of assigning a "canonical" number and possibly an optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together notions and techniques from topology, geometry and analysis.

In this talk we will concentrate on the operation consisting of turning inside out the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the existence of such an operation the search for effective realizations of such eversions has triggered a lot of fascination and works in the math community. The absence in nature of matter that can interpenetrate and the quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for the difficulty to develop an intuitive approach on the problem.

We will present the optimization of Sophie Germain conformally invariant elastic energy for the eversion. Our efforts will finally bring us to consider more closely an integer number together with a mysterious minimal surface.

February 6, 2019

Mean Field Games: What? Why? How?

Abstract:

Mean Field Games (MFG) are recent mathematical models that aim to describe the collective behavior of a large number of agents/players. We shall briefly present in this talk some motivation and a few applications to economics, crowd motion, mobile communication networks, and machine learning. Without going into any (technical) mathematical detail, the existing mathematical toolbox will be described together with some recent developments and perspectives.

February 1, 2019

The Complex and the Zilber Exponential

Abstract:

In 2005 Boris Zilber published a very influential paper, giving a sophisticated model-theoretic construction of an exponential field satisfying Schanuel's Conjecture and a kind of Nullstellensatz relating to extensions preserving a dimension arising from Schanuel's Conjecture. He proved some very remarkable properties of such exponential fields, and boldly conjectured that the classical complex exponential field is one of these fields. The conjecture of course assumes that Schanuel's Conjecture is true for the complex exponential, and subsequent research has generally made this assumption.

The conjecture survives, and one has been able to show that the two exponential fields share many properties, normally established by quite different proofs for the two fields. I will survey the situation, with special reference to a conjecture in complex analysis made sixty years ago by H. Shapiro.

October 29, 2018

Sparse Matrices in Sparse Analysis

Abstract:

In this talk, I will give two vignettes on the theme of sparse matrices in sparse analysis. The first vignette covers work from compressive sensing in which we want to design sparse matrices (i.e., matrices with few non-zero entries) that we use to (linearly) sense or measure compressible signals. We also design algorithms such that, from these measurements and these matrices, we can efficiently recover a compressed, or sparse, representation of the sensed data. I will discuss the role of expander graphs and error correcting codes in these designs and applications to high throughput biological screens. The second vignette flips the theme; suppose we are given a distance or similarity matrix for a data set that is corrupted in some fashion, find a sparse correction or repair to the distance matrix so as to ensure the corrected distances come from a metric; i.e., repair as few entries as possible in the matrix so that we have a metric. I will discuss generalizations to graph metrics, applications to (and from) metric embeddings, and algorithms for variations of this problem. I will also touch upon applications in machine learning and bio-informatics.

October 22, 2018

Systems of Points with Coulomb Interactions

Abstract:

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability. We will first review these motivations, then present the ''mean-field'' derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

October 17, 2018

Beyond the Age of Independence by Forcing?

Abstract:

Gödel’s consistency proof for the Axiom of Choice and the Continuum Hypothesis involves his discovery of the Constructible Universe of Sets. The axiom "V=LV=L" is the axiom which asserts that every set is constructible. This axiom settles the Continuum Hypothesis and more importantly, Cohen’s method of forcing cannot be used in the context of the axiom "V=LV=L".

However the axiom V=LV=L is false since it limits the fundamental nature of infinity. In particular the axiom refutes (most) strong axioms of infinity.

A key question emerges. Is there an "ultimate" version of Gödel’s constructible universe L yielding an axiom "V=ultimate LV=ultimate L" which retains the power of the axiom "V=LV=L" for resolving questions like that of the Continuum Hypothesis, which is also immune against Cohen’s method of forcing, and yet which does not refute strong axioms of infinity?

Until recently there seemed to be a number of convincing arguments as to why no such ultimate LL can possibly exist. But the situation is now changed.

May 31, 2017

Planar Graphs, Legendrian Surfaces, and Contact Homology

Abstract:

To any cubic planar graph, we can associate a surface inside R5R5, which is nicely compatible with a certain canonical geometry, called a contact structure. From here we can apply techniques from contact geometry to study graph theory; for example it was shown by Treumann-Zaslow that the space of constructable sheaves with singular support on the surface recovers the chromatic polynomial of the graph. Another strategy is to look at pseudo-holomorphic curves with boundary conditions on the surface, which defines an algebraic package known as Legendrian contact homology.

We'll explain what this gadget looks like in purely graph theoretical terms. In particular, the augmentation variety of this contact homology recovers the chromatic polynomial of the graph, though it does this in a highly non-trivial way and therefore yields a novel combinatorial definition of a graph coloring. But it also contains more subtle information about the graph, concerning how trajectories in the plane are required to interact with the graph. From here we can draw connections with mathematical physics, such as the spectral networks of Gaiotto-Moore-Neitzke and mirror symmetry in the style of Aganagic-Vafa.

May 10, 2017

The Triangulation Conjecture

Abstract:

Random triangulations have been studied as a discrete model for 2D quantum gravity since the 1980s. While there are many conjectures about the emergent geometry of such models, mathematical progress has been somewhat slower. Toward this end, Benjamini and Schramm (2001) defined the distributional limit of a sequence of finite graphs; the limit object is a unimodular random graph in the sense of Aldous and Lyons. Benjamini and Schramm showed that every distributional limit of finite planar graphs with uniformly bounded degrees is almost surely recurrent (the random walk returns to its starting point infinitely often almost surely).

Their approach uses the Koebe-Andreev-Thurston circle packing theorem to uniformize the geometry of the limit graph. In contrast, we consider intrinsic deformations of the path metric by a (random) weighting of the vertices. This leads to the notion of the conformal growth exponent of a unimodular random graph, which is the best degree of volume growth of balls that can be achieved by such a weighting of "unit area." The conformal growth exponent carries information about the underlying geometry; in particular, it bounds the almost sure spectral dimension.

We show that distributional limits of finite graphs that can be sphere-packed in RdRd have conformal growth exponent at most dd, and thus the connection to spectral dimension yields dd-dimensional lower bounds on the heat kernel. When the conformal growth exponent is bounded by 2, one obtains more precise information, including a conjectured generalization of the Benjamini-Schramm Recurrence Theorem to larger families of graphs.

These methods extend to models with unbounded degrees, giving new proofs of almost sure recurrence for the extensively studied uniform infinite planar triangulation (UIPT) and quadrangulation (UIPQ). The latter results were established only recently by Gurel-Gurevich and Nachmias (2013). Our approach yields quantitative lower bounds on the heat kernel, spectral measure, and speed of the random walk.

March 29, 2017

Random Groups from Generators and Relations

Abstract:

We consider a model of random groups that starts with a free group on nn generators and takes the quotient by nn random relations. We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the Cohen-Lenstra heuristics, which predict the distribution of class groups of number fields. We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen. Finally, we discuss joint work with Yuan Liu on the non-abelian random groups built in this way, including the existence of a limit of the random groups as nn goes to infinity.

January 11, 2017

Shifted Symplectic Structures, Quantization, and Applications

Abstract:

Many moduli problems of interest, such as moduli spaces of local systems, come equipped with a natural symplectic structure. Moreover, quantization of these symplectic structures is closely related to counting problems in geometry and topology, such as the Casson invariant and its generalizations, as well as Feynman integration in physics. The theory of shifted symplectic structures, a vast generalization of algebraic symplectic geometry, provides a natural framework for constructing and studying such symplectic structures and their quantizations. I will give a brief overview of this theory and describe applications in geometry and physics.

October 26, 2016

What is a G-spectrum?

Spectra in the sense of stable homotopy theory have been a major object of study in algebraic topology for half a century. During that time the basic definition has undergone some major revisions, including a major breakthrough in 1993 due to Peter May and three coauthors. Remarkably, these shifting foundations have not affected any of the computations made using earlier definitions. In the talk I will describe how the use of category theory has led to major simplifications.

May 25, 2016