Fall 2025

This seminar 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
August 27, 2025	Qiaochu Ma	Texas A&M University	Small Scale Index Theory, Scalar Curvature, and Gromov’s Simplicial Norm	View Abstract
September 3, 2025	Ryo Toyota	Texas A&M University	Twisted coarse Baum-Connes conjecture and relatively hyperbolic groups	View Abstract
September 10, 2025	Simone Cecchini	Texas A&M University	Positive scalar curvature with point singularities	View Abstract
September 17, 2025	Jesus Sanchez Jr	Texas A&M University	Hypoelliptic Operators on Contact Manifolds	View Abstract
September 24, 2025	Hongyi Liu	Princeton University	Poincaré-Einstein 4-manifolds with conformally Kähler geometry	View Abstract
October 1, 2025
October 8, 2025	Tao Mei	Texas A&M University	On Connes’ quantized derivative on Free groups	View Abstract
October 15, 2025
October 22, 2025	Runjie Hu	Texas A&M University	Application of Finiteness Obstruction to Varieties	View Abstract
October 29, 2025	Jianchao Wu	Fudan University	Quasi-representations and a K-theoretic invariant	View Abstract
November 5, 2025	Thomas Tony	University of Muenster	Higher Index Theory and Scalar Curvature Rigidity	View Abstract
November 12, 2025
November 19, 2025
November 26, 2025
December 3, 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• Departmental website of the geometry and topology group (very out of date of Aug 21, 2025)

In this talk, I will present a recent generalization of Goette and Semmelmann’s rigidity theorem, where the original topological condition on the Â-degree is replaced by a less restrictive condition involving the so-called higher mapping degree. A key challenge in the proof is that a non-vanishing higher index does not necessarily give rise to a non-trivial kernel of the corresponding Dirac operator. I will present a new method that extracts geometrically useful information even in this more general setting. Moreover, I will explain a cut-and-paste principle for the higher index, which leads to a new index theorem relating the higher index of a certain Dirac operator to the higher mapping degree and the Euler characteristic.

Scalar curvature encodes the volume information of small geodesic balls within a Riemannian manifold, making it, to some extent, the weakest curvature invariant. This raises a natural question: what topological constraints does scalar curvature impose on manifolds? In this talk, we shall show that for a manifold with a scalar curvature lower bound, the simplicial norm of certain characteristic classes can be controlled by its volume and isoperimetric constant. This is joint work with Guoliang Yu.

We introduce twisted coarse Baum–Connes conjecture with stable coarse algebras, a geometric analogue of the Baum–Connes conjecture with coefficients. We show that this twisted version has stronger permanence properties than the classical coarse Baum–Connes conjecture, particularly with respect to unions and subspaces. We apply this framework to relatively hyperbolic groups. For a finitely generated group $G$ that is hyperbolic relative to  $\{H_1,\cdots,H_n\}$, it is known that $G$ satisfies coarse Baum-Connes conjecture if each $H_i$ does and $H_i$ admits finite-dimensional simplicial model of the universal space proper actions. Using our permanence results, we can show that $G$ satisfies twisted coarse Baum-Connes conjecture with stable coefficients, if and only if each $H_i$ does. This is a joint work with Jintao Deng.

I will discuss obstructions to metrics of positive scalar curvature with uniformly Euclidean point singularities. This provides counterexamples to a conjecture by Schoen. I will also discuss the existence of metrics with uniformly Euclidean point singularities which cannot be smoothed by a geometric flow while preserving nonnegativity of the scalar curvature. This is based on joint work with Georg Frenck and Rudi Zeidler.

Historically the elliptic differential operators have held a special place for their importance in mathematical physics as well as their ability to connect various subjects within pure mathematics. In recent years, the Fourier theoretic approach to elliptic operators has seen much progress in its extension to a wider class of operators, those which are hypoelliptic. In this talk we will use the extended Fourier theoretic approach to provide a construction of a new hypoelliptic operator on the Heisenberg group and discuss its properties and generalizations to contact manifolds. This is joint work with Andres Franco Valiente.

Poincaré–Einstein metrics play an important role in geometric analysis and mathematical physics, yet constructing new examples beyond the perturbative regime is difficult. In this talk, I will describe a class of four-dimensional Poincaré–Einstein manifolds that are conformal to Kähler metrics. These metrics admit a natural symmetry generated by a Killing field, which reduces the Einstein equations to a Toda-type system. This approach leads to existence and uniqueness results in the case of complex line bundles over surfaces of genus at least one. The construction produces large-scale, infinite-dimensional families of new Poincaré–Einstein metrics with conformal infinities of non-positive Yamabe type. This is joint work with Mingyang Li.

The commutator operator [x,H], where x is a bounded function on the torus and H is the Hilbert transform, is a central object in classical analysis, lying at the intersection of Nehari’s theorem, Hankel operators, and BMO/VMO theory. In the terminology of A. Connes, [x,H] may be viewed as the quantized derivative of x. Connes further suggested a natural analogue of this operator on free groups and raised the question of how the properties of x are reflected in [x,H]. This was followed by G. Duchamp and C. Reutenauer, who proved that the finite-rank property of [x,H] characterizes the rationality of x. In this talk, I will review some history and share some of my recent thoughts on the boundedness and compactness of [x,H] in the setting of free groups and Voiculescu’s free semicircular von Neumann algebras.

Some characteristic p>0 varieties fail to lift to characteristic zero. A topological obstruction is whether the l-adic etale homotopy type is homotopy equivalent to a finite CW complex. The integral version of this obstruction is the so-called Wall's finiteness obstruction. In this talk, I will explain this finiteness obstruction and the idea of generalizing this topological discussion to varieties. These are joint works with Ruida Di and Siqing Zhang.

Generalizing the intriguing phenomenon of Voiculescu's almost commuting unitary matrices is the notion of a quasi-representation of a (discrete) group. As demonstrated by the work of Exel and Loring on Voiculescu's example, there may be topological obstructions to perturbing quasi-representations into genuine representations---this is where (topological or operator) K-theory enters the picture. Previous studies have mostly focused on fundamental groups of finite CW complexes. In this talk, which is based on joint work with Shmuel Weinberger and Guoliang Yu, we introduce the notion of a character as a more general and refined invariant for quasi-representations.

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Abstract