Title: Positive scalar curvature and exotic structures on simply connected four manifolds.Abstract: We address Gromov’s band width inequality and Rosenberg’s S1-stability conjecture for smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless, we show that both of these results hold upon considering simply […]
Title: An intrinsic approach to moduli theoryAbstract: A central problem in algebraic geometry is to construct and study moduli spaces of objects of interesting geometric objects. The classical tool for this is geometric invariant theory, which requires you to approximate a moduli problem by an orbit space X/G for some reductive group G acting on […]
Title: Euler Systems for certain non-unique models Abstract: Pushforwards of Beilinson's Eisenstein symbols into the motives of Shimura varieties provide fertile grounds for exploring instances of the Beilinson–Bloch–Kato conjectures, particularly when these constructions are accompanied by period integral representations. About a decade ago, Pollack and Shah introduced two novel Rankin–Selberg integral representations: one for the […]
Title: Circle bundles with PSC over large manifolds Abstract: In his seminal work on metric inequalities for scalar curvature, Gromov asked whether total spaces of circles bundles over enlargeable manifolds can admit metrics with positive scalar curvature. We answer this question in all dimensions and construct infinitely many such examples over manifolds dimension 4 and above, […]
Title: Period sheaves on Bun_G. Reference: Feng and Wang's duality periods .
Title: Arnold-Thom conjecture for the arrival time of surfacesAbstract: Following Łojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. In this talk, I will explain the proof of Łojasiewicz's theorem and Arnold's conjecture in the context of arrival time […]
Title: Near-center derivatives and arithmetic 1-cycles Abstract: Degrees of arithmetic special cycles on Shimura varieties are expected to appear in first derivatives of automorphic forms and L-functions, such as in the Gross--Zagier formula, Kudla's program, and the Arithmetic Gan--Gross--Prasad program. I will explain some “near-central” instances of an arithmetic Siegel--Weil formula from Kudla’s program, which […]
Title: Transfer operators in the language of quantization. Reference: https://arxiv.org/abs/2111.03004
Title: Some recent developments on the fully nonlinear Yamabe problems Abstract: In recent joint work with YanYan Li and Zongyuan Li, we broaden the scope of fully nonlinear Yamabe problems by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions […]
Title: Stable Reduction via the Log Canonical Model.Abstract: We will discuss a natural perspective on stable reduction that extends Deligne--Mumford's stable reduction for curves to higher dimensions. From this perspective, we will outline a proof of stable reduction for surfaces in large characteristic.
Title: Beyond Endoscopy: Poisson Summation Formula and Kuznetsov Trace Formula on GL_2. Abstract: In the first part of the talk, we will give a direct proof of a Poisson summation formula on the Kuznetsov quotient of GL_2 that is responsible for the local Hankel transform calculated by H. Jacquet. In the second part of the […]
Title: The Sato–Tate ConjectureAbstract: We will motivate and state the Sato–Tate conjecture for elliptic curves. If we have time remaining, we will indicate how potential automorphy results are used in its proof.
Title: Covering Lemmas in Analysis and GeometryAbstract: Covering lemmas are fundamental tools in mathematical analysis, with deep connections to the geometry of Euclidean spaces. In this talk, we will explore key classical results, such as the Vitali and Besicovitch covering theorems, and discuss their significance in measure theory, differentiation, and geometric analysis.
Title: Finite time explosion for SPDEs Abstract: The classical existence and uniqueness theorems for SPDEs prove that, under appropriate assumptions, SPDEs with globally Lipschitz continuous forcing terms have unique global solutions. This talk outlines recent results about SPDEs exposed to superlinearly growing deterministic and stochastic forcing terms. I describe sufficient conditions that guarantee that, despite […]
Title: Developing a library of formalized mathematics from a univalent point of view Abstract: The agda-unimath project started in November 2021, at a meeting called Univalent Foundations for Daily Applications. At this meeting, I proposed to formalize the Symmetry book project, in which group theory is studied from the perspective that symmetries are identifications in […]
Title: Cantor Dynamical Systems, Bratteli Diagrams, and Their Associated Full GroupsAbstract: If a countable group G acts on a Cantor set X, one can enlarge G to a (still countable) group F(G) by adding homeomorphisms of X that act locally as elements of G. This group, known as the full group of the dynamical system […]
Title: Du Bois invariants for isolated singularities.Abstract: The Du Bois invariants are natural invariants for singularities. In this talk, I will explain what they are, why they are interesting, and report on recent and ongoing work studying their properties.
Title: A survey of enriched categoriesAbstract: We'll recall some rudiments of the notion of a category, including some examples such as the category Set of sets and functions, the category Vect of vector spaces and linear maps, the category Top of topological spaces and continuous maps, etc. We'll then delve into the notion of an enriched category, where […]
Title: Affine and asymptotic Hecke algebras Abstract: Affine Hecke algebras play a prominent role the representation theory of p-adic groups, where a large subcategory of representations are equivalent to modules over the affine Hecke algebra. The important subcategory of tempered representations is equivalent to modules over a larger ring, the Harish-Chandra Schwartz algebra. However, this […]
Title : Scalar curvature comparison and rigidity of 3-dimensional weakly convex domains Abstract: I will discuss a comparison and rigidity result of scalar curvature and scaled mean curvature on the boundary for weakly convex domain in Euclidean space, which is a joint work with Xuan Yao. This result is a smooth analog of Gromov's dihedral […]
Title: "On applications of Topology" Abstract: I will explain how knowledge of Topology helped with winning the Nobel Prize in Physics 2016. No familiarity with advanced math beyond Calculus will be assumed from the audience.
Title: A characterization of uniruled Kähler manifolds.Abstract: We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kähler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Paun and of Druel to foliations on compact Kähler manifolds. As an application, we prove that a compact Kähler manifold is uniruled if and […]
Title: A Comparison of Two Supercharacter TheoriesAbstract: In 2008, P. Diaconis and I.M. Isaacs introduced a generalization of classical character theory, which they called supercharacter theory. Let X be a partitioning of the irreducible characters of a finite group G such that the trivial character is in its own block of this partition. Let Y […]
Title: Non-uniqueness of mean curvature flow Abstract: The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes […]
Title: Wait! That Cannot Be RightAbstract: In this talk proofs will be presented that contain an incorrect step or include an incorrect assumption. This will lead us to prove concepts that are clearly false. This incorrect step or assumption will not be obvious, in fact the proofs will be presented in a way to try […]