Spring 2026

This seminar usually meets on Wednesdays 2:00-3:00 PM in Blocker 302.

The organizers are Simone Cecchini, Runije Hu, Qiaochu Ma, Jesus Sanchez Jr, Zhizhang Xie and Guoliang Yu.

Date	Speaker	Affiliation	Title	Abstract
January 14, 2026	Jintao Deng	-	The Coarse Baum–Connes Conjecture and Group Extensions	View Abstract
January 14, 2026	Liyuan Chen	Harvard University	Toward useful quantum computation: from near term to fault tolerance	View Abstract
January 20, 2026	Arthur Jaffe	Harvard University	Some Fantastic Connections between Mathematics and Physics	View Abstract
January 30, 2026	Luca Di Cerbo	University of Florida	Curvature, Macroscopic Dimensions, and Symmetric Product of Curves	View Abstract
February 25, 2026	[Jie Xu]	Texas A&M University	A Series of Rosenberg-Stolz Conjectures	View Abstract
March 23, 2026	Assaf Naor	Princeton University	Sparsest Cut	View Abstract
March 24, 2026	Assaf Naor	Princeton University	The Ribe program and metric smoothness	View Abstract
March 25, 2026	Assaf Naor	Princeton University	Distortion growth and the gap possibility	View Abstract
March 25, 2026	Rufus Willett	University of Hawai’i at Mānoa	A secondary pairing for K-theory and K-homology	View Abstract
March 27, 2026	Eric Chen	University of Illinois Urbana-Champaign	Ricci flow and integral curvature pinching	View Abstract
April 06, 2026	Dennis Sullivan	Stony Brook University, City University of New York	Symmetry underlying “abstract 3D incompressible fluid motion” created by the discrete group of integer 3x3 matrices of unit determinant I	View Abstract
April 07, 2026	Dennis Sullivan	Stony Brook University, City University of New York	Symmetry underlying “abstract 3D incompressible fluid motion” created by the discrete group of integer 3x3 matrices of unit determinant II	View Abstract
April 08, 2026	Dennis Sullivan	Stony Brook University, City University of New York	Symmetry underlying “abstract 3D incompressible fluid motion” created by the discrete group of integer 3x3 matrices of unit determinant III	View Abstract
April 15, 2026	Junrong Yan	Northwestern University	Graph integral on Kahler manifolds	View Abstract
April 22, 2026	Jesus Sanchez Jr	Texas A&M University	TBD	TBD
April 29, 2026
May 06, 2026

Past Seminars

• Fall 2025

Related Links

• Departmental seminar website (not working as of Aug 21, 2025)
• Departmental website of the geometry and topology group (very out of date of Aug 21, 2025)

The coarse Baum–Connes conjecture is a central problem in noncommutative geometry, capturing deep connections between operator algebras and the large-scale geometry of metric spaces. While the conjecture is inherently coarse-geometric in nature, understanding its behavior under group extensions presents significant challenges, as extensions often obscure large-scale geometric properties. In this talk, I will discuss recent progress on the coarse Baum–Connes conjecture for groups arising from extensions, highlighting the techniques used to analyze their coarse geometry.

I give a personal overview of several different areas of research that span these two subjects, with past results and some projects for the future.

Two central goals in quantum science are (i) near-term simulation of many-body physics and (ii) scalable, fault-tolerant quantum computation in the long term. I will present two recent advances toward both aims. For (i), we present theoretical and experimental studies of globally controlled analog simulators, positioning them as principled platforms for quantum simulation. For (ii), we develop a non-Abelian topological framework in two dimensions that provides a route to fault-tolerant universality and remains compatible with current devices. Together, these results outline a practical trajectory from near-term simulation to scalable quantum computation.

In 1994 and 2006 survey articles, Rosenberg and Stolz stated a series of conjectures by "persisting" non-positive scalar curvature metrics from a closed manifold to the product space between and a real line, the 2-plane, or a circle, respectively. There are two classical methods to study this type of problem: one is the method of index-theoretic obstructions, and the other is through geometric measure theory initiated by minimal hypersurfaces, then developed by e.g. Gromov's -bubble method. In this talk, we introduce a different, conformal geometry and PDE approach to partially answer a series of Rosenberg-Stolz conjectures, and Gromov-Lawson type scalar and mean curvature comparison results for all dimensions from "transposing the positivity" point of view. We also show its application in complex geometry. Some of my work involves collaboration with S. Rosenberg.

The primary pairing between K-theory and K-homology of a C*-algebra is an important Z-valued invariant. It can be computed using index theory, and several other techniques from noncommutative geometry. However, it cannot see torsion information. In this talk, I will introduce a pairing between the torsion subgroups of K-theory and K-homology that takes values in Q/Z, and in good cases (e.g. under the UCT, and if K-homology is finitely generated) captures all the information that the primary pairing misses. I will then relate this to the relative eta invariants of Atiyah-Patodi-Singer, and also to the zeta map of Gong-Lin-Niu, and Carrión-Gabe-Schafhauser-Tikuisis-White.

In this talk, I will present a detailed study of the curvature and symplectic asphericity properties of symmetric products of curves. I show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, symmetric products of curves sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and Dranishnikov. As a natural generalization of this circle of ideas, I will also address the Gromov–Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions. This is joint work with Alexander Dranishnikov and Ekansh Jauhari.

The Sparsest Cut problem is a major algorithmic task which asks for an efficient procedure to partition a data sets into two parts whose interface is (approximately) as small as possible. In the mid-1990s Goemans and Linial devised an algorithm for Sparsest Cut that to date remains the best-known algorithm. Understanding how well this algorithm performs was a central project that over the past 30 years occupied many researchers and exhibited fascinating mathematical twists and turns. Now, we at last understand the worst-case performance of the Goemans--Linial algorithm up to universal constant factors, as the size of the inputted data increases (the final step here is recent joint work with Chang and Ren). This talk, which is intended to a general audience and assumes no previous knowledge of this area, will explain the problem and its challenges, and will describe ideas that contributed to its solution from a range of fields including geometric group theory, harmonic analysis, combinatorics, Gaussian processes, and insights in metric geometry that were inspired in part by Banach space theory.

The Ribe program is an intricate web of analogies and conjectures that aim to translate deep phenomena and concepts from the geometry of Banach spaces to the realm of general metric spaces. After many decades of research, this longstanding endeavor has by now become a massive and deep edifice with multifaceted applications that is impossible to describe in a single lecture . Nevertheless, this talk, which is intended for mathematicians who do not specialize in related areas, will illustrate the underlying principles that guide the Ribe program. It will conclude with a very recent completion (joint with Eskenazis) of a foremost open step in the Ribe program that explains how one could define a concept that applies to any metric spaces, but when the metric space in question is a normed space, it coincides with classical measurements of the degree of smoothness of spheres.

Infinite metric spaces of interest typically do not admit any bi-Lipschitz embedding into a Hilbert space. The pertinent question thus becomes understanding the Euclidean distortion growth of a given infinite metric space X, which is defined to be the rate at which the most non-Euclidean n-point subset of X differs from a subset of R^n. Even though this is natural and useful, determining the Euclidean distortion growth rate of classically studied spaces (e.g. Banach spaces, groups) has proved to be very difficult. One inspiration here is the search for nonlinear versions of a classical theorem of John, as initiated by visionary work of Johnson and Lindenstrauss in the early 1980s, which inspired a lot of subsequent work. This talk will describe what is known here, including very recent progress (joint with Ren) that resolves an old question which arose from a seminal and highly original 1985 contribution of Bourgain, will state some stubbornly open longstanding open questions, and will end with a tantalizing new conjecture that I call the "gap possibility."

The right coset space S of 3 x 3 real matrices of unit determinant SL(3,R) by the maximal compact subgroup of rotations of space (SO(3,R)) is a concrete construction of the manifold of positive definite inner products (with unit volume for the unit ball) on any abstract real vector space of dimension three. This manifold has a canonical metric because a tangent vector at a particular inner product is a symmetric two tensor which has a definite size using that same definite inner product to measure the tangent vector. The manifold S becomes a 5D complete Riemannian manifold with this “tautologous metric” and has only non-positive gaussian curvature at each 2D tangent subspace. These sectional curvatures fill up a finite interval say [-1,0]. This manifold S of “ellipsoids of unit volume” has (for component of the identity) the entire Lie group of unit determinant linear automorphisms of the three dimensional vector space. This implies for example there is for each point of S a global isometry fixing the chosen point which induces the antipodal map on the tangents to that point. This means even more than that “the manifold S is a homogeneous riemannian manifold”. It means that it is a symmetric space in the sense of E.Cartan (which he introduced as part of the classification of Lie groups).

The discrete subgroup SL(3,Z) of this Lie group SL(3,R) of isometries acts properly discontinuously and “virtually freely” on the space S of unit vol ellipsoids. Namely there is a proper fundamental domain for a subgroup of finite index in SL(3,Z). This fundamental domain has finite 5D volume and is non compact but complete for the induced metric. It has finitely many ends whose volume outside a ball of radius R decreases exponentially fast as R tends to infinity.The double coset space obtained by taking cosets of the left action of SL(3,Z) on S, the space of unit vol ellipsoids, describes geometrically the “space of shapes of lattices” in R^3 (unit vol and “up to rotation”) which is of “critical interest” for 3D fluid motion on the three torus T^3. By three torus one means “triply periodic three space”, that is, R^3 / standard integer lattice. This “critical interest” arises because given any volume preserving smooth motion, for any pair of times s less than t, the motion from time s to time t, for each point of the three torus, linearly distorts any lattice shape to a new lattice shape. Thus the motion determines for each point in T^3 a parametrized curve, starting from time zero with the background lattice shape, which then moves through the double coset space of shapes of lattices.

Fix the standard parallel translation on (R^3 modulo any unit volume lattice). Then any Riemannian metric on such a quotient Torus of R^3: with this volume and whose canonical connection induces this fixed notion of parallel translation: by the standard discussions (using this metric as a background metric) determine the PDE(s) of initial value problems whose solutions are the volume preserving 3D fluid motions (both ideal and viscous). This will be recalled in the lecture. Our discrete group SL(3,Z) acts on such (background) metrics and “transports” one initial value problem isomorphically to the other. This entire structure is tantamount to the “abstract fluid motion” referred to in the title. The extra information in the metric ( beyond the volume form and the flat connection which are fixed) can be quotiented by SL(3,Z) without loss of essential information. This follows from the important “transport” statement just made. The group parameter space of such metrics is SL(3,R) acting transitively on the set of metrics, with isotropy group of a given metric being a conjugate of the rotation subgroup.Thus essential information about abstract fluid motion should then be found in the structure of the geometry of the moduli space S of such metrics by the isometric action of SL(3,Z) described above as the “shape space of unit vol lattices” in R^3. Finally we will discuss the fact that SL(3,Z) is a finitely generated discrete group with a remarkable property for its orthogonal representations on any Hilbert space without any fixed unit vectors. Namely, independently of the chosen orthogonal representation, given any finite set of generators there is a positive number epsilon (depending only on the set of generators) so that “some generator moves some unit vector by at least epsilon”. This property of SL(3,Z), called property T, helps to understand “abstract incompressible 3D fluid motion” a little better.

This is joint work with Minghao Wang. In this talk, I will explain the convergence of Feynman graph integrals on closed real-analytic Kähler manifolds: Using Getzler’s rescaling technique, the graph integrands extend naturally to the Fulton–MacPherson compactification as forms with divisorial singularities, allowing a rigorous definition as Cauchy principal value integrals. As an application, this yields a mathematical construction of the higher-genus B-model invariants on Calabi–Yau threefolds in the sense of Bershadsky–Cecotti–Ooguri–Vafa (BCOV).

Early applications of the Ricci flow by Hamilton and others characterized Riemannian manifolds with certain pointwise curvature pinching conditions as spherical space forms. In some cases, curvature pinching in averaged, integral senses can extend such results on topological restrictions. I will describe some works on critical, scale-invariant integral curvature pinching and smoothing obtained using the Ricci flow and consequences of Perelman's W-entropy, joint with Guofang Wei and Rugang Ye.

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