The UChicago math department hosts colloquia to introduce its members, and anyone else who wishes to attend, to recent developments and important ideas in mathematics.

**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:

-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-assist ed 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 Erd*ő*s on Ramsey complete sequences, for which Erd*ő*s 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 Erd*ő*s 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 Erd*ő*s 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.

## Alex Lubotzky (Hebrew University): Stability, Non-Approximated Groups and High-Dimensional Expanders

**Stability, Non-Approximate d 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-

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

## Song Sun (UC Berkeley): Complex Structure Degeneration and Metric Collapsing of Calami-Yau Manifolds

**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