Talk Archive

Abstract: We will introduce the notion of \(C^\infty\)-rings and study Poisson structures in this setting. We will then discuss their relationship with singular foliations and, more generally, with Lie-Rinehart algebras.

Abstract: We will introduce the notions of regular and singular foliations, discuss their main features, and present some motivating examples.

Abstract: If one takes the view of Witt vectors as the \(t\)-adic completion of \(\operatorname{End}_0(R)\), then one can essentially circumvent the use of the trace property and the Witt vectors with coefficients used in DKNP, and recover every property of the Witt vectors as a consequence of the K-theoretic properties of \(\operatorname{End}_0(R)\). This second talk will focus on an alternate definition of Witt vectors and the characteristic map, and additionally give an external pairing of Witt vectors and \(\operatorname{Nil}_0(R)\).

Abstract: For a commutative ring \(A\), the ring of Witt vectors \(W(A)\) has been useful in number theory, algebraic topology, and algebraic K-theory. Classical results of Almkvist, Stienstra, Grayson, and Weibel demonstrate that the characteristic map bridges the ring \(\operatorname{End}_0(R)\) to its \(t\)-adic completion \(W(A)\), and the nil groups \(\operatorname{Nil}_0(A)\) are modules over \(W(A)\). Recently Dotto, Krause, Nikolaus, and Patchkoria (DKNP) gave an algebraic description of Witt vectors for a non-commutative ring \(R\). While attempting to generalize this framework to non-commutative rings \(R\) strips away significant algebraic structure, \(\operatorname{End}_0(R)\) is reduced to an abelian group with only an external product. Furthermore, the foundational arguments about the interaction of Frobenius, Verschiebung, and the pairings rely on the Cayley-Hamilton theorem, which breaks down in the non-commutative setting. This first talk will give an introduction to these new definitions of Witt vectors and the DKNP definition of the characteristic map.

Abstract: In this talk, we will discuss the existence of a 1-parameter family of arbitrary genus maxfaces with an arbitrary number of spacelike ends and infinitely many swallowtails. Our primary tool of construction will be node-opening technique. All maxfces of these families are embedded in a wider sense. We will also see families of Lorentzian Costa and Costa-Hoffman-Meeks surfaces, as well as periodic surfaces of infinite genus, as examples.

Abstract: In this talk, we will discuss minimal and maximal surfaces in three-dimensional Euclidean and Lorentz-Minkowski space respectively. We will explore how they are related and how to construct them. In particular, we will cover Björling problem and Weierstrass-Ennerper representations for them. I will also define the term maxface on which my next two lectures will be.

Abstract: Having completed the introductions, we will summarize some recent progress in using diffeology to understand geometry. Among these will include: a generalization of the Serre-Swan theorem using diffeological vector bundles; an integration of Lie algebroids to diffeological groupoids, even when no integrating Lie groupoid exists; and a diffeological approach to the question of geometric quantization.

Abstract: We will introduce two other approaches to singular and infinite-dimensional spaces, namely Lie groupoids (a step towards differentiable stacks), and convenient infinite-dimensional manifold structures. We will then explore how relevant diffeological spaces appear in both contexts.

Abstract: We will introduce the category of diffeological spaces, and give several examples: the cusp \(x^2 = y^3\), as a subset of \(\mathbb{R}^2\); the irrational tori \(\mathbb{R}/(\mathbb{Z} + \alpha\mathbb{Z})\), for irrational \(\alpha\); and the diffeomorphism group of the real line, \(\operatorname{Diff}(\mathbb{R})\). These will illustrate how the category of diffeological spaces is closed under taking, respectively, subsets, quotients, and mapping spaces, and help give a sense of the data contained within a diffeological structure.

Abstract: The goal of this lecture is to explain how closed 3-manifolds can be constructed via surgery along framed links. We briefly review the notion of handle decompositions of manifolds and then describe how framed knots and links give rise to 3-manifolds.

Abstract: We introduce the Kauffman bracket and the Jones polynomial, which arise naturally in quantum topology and statistical mechanics. As an extension of the Jones polynomial, we also introduce the colored Jones polynomial.

Abstract: We introduce basic definitions and fundamental results in knot theory. One of the simplest knot invariants is the coloring invariant, which can detect certain nontrivial knots. As a generalization of the coloring invariant, we define an algebraic structure called a quandle.

Abstract: Here we will briefly see the ingredients of rational homotopy theory, like rationalization of spaces, Sullivan (minimal) models, LS category, etc, and compute some basic examples. Finally, we will relate these to our case of self-homotopy equivalences, like how $\mathcal{E}(X)$ and $\mathcal{E}(X_\mathcal{P})$ are related, some other results and possible open directions.

Abstract: We survey explicit computations of the group of self-homotopy equivalences $\mathcal{E}(X)$ for key spaces in homotopy theory. This includes results for pseudo-projective planes, with emphasis on lens spaces, as well as for Moore spaces $M(G, n)$ and twisted Eilenberg-MacLane spaces. We also discuss the structure of $\mathcal{E}(X)$ for real, complex, and quaternionic projective spaces. Additionally, we present computations of $\mathcal{E}_H(X)$ for rank one $H$-spaces, specifically $S^1$, $S^3$, $S^7$, $\mathbb{R}P^3$, $\mathbb{R}P^7$, and for certain higher-dimensional examples such as $S^3\times S^7$, $S^3\times S^3$, $S^7\times S^7$, $SU(3)$, and $Sp(2)$. These cases illustrate the diversity and depth of self-equivalence groups across notable spaces in topology.

Abstract: We provide an overview of the group of self-homotopy equivalences, $\mathcal{E}(X) = \{[f] \mid f: X \simeq X\}$, focusing on key subgroups such as those acting trivially on homotopy or cohomology groups. We summarize the known structures of $\mathcal{E}(X)$ for spaces like spheres and low-dimensional complexes. The discussion includes exact sequences relating $\mathcal{E}(X)$ for products and wedge sums. Finally, we explore various generalizations obtained by considering free-, fiber-, equivariant-, and $H$-homotopy equivalences, highlighting their algebraic and geometric significance through classical results.