T-spectra, or spectrum objects with generalized "suspension" parameters, first appeared in the construction of motivic stable categories. The motivic stable model structure lives within a localized model structure of simplicial presheaves in which the affine line is formally collapsed to a point, and the suspension object is the projective line. Because of these constraints and limitations of the tools then at hand, the original construction of the motivic stable category was technical, and made heavy use of the Nisnevich descent theorem.

This talk will begin with a general introduction to the concepts around T-spectra. I shall display a short list of axioms on the parameter object T and the ambient f-local model category which together lead to the construction of a well behaved f-local stable model structure of T-spectra. Examples include the motivic stable category, but the construction is much more general. The resulting stable category has many of the basic calculational features of the motivic stable category, including slice filtrations. The overall construction method is to suitably localize an easily defined strict model structure for T-spectra. The localization trick is an old idea of Jeff Smith, but assumptions (the axioms) are required for the recovery of normal features of stable homotopy theory from the localized structure.

A Lagrangian inclusion is a smooth map from a compact real surface into $C^2$ which is a local Lagrangian embedding except a finite set of singular points. The singular points can be taken to be either transverse double self-intersection points or the so-called open Whitney umbrellas. In the first talk I introduce relevant terminology and will formulate a recent result (joint with A. Sukhov) concerning rational convexity of a Lagrangian inclusion with one umbrella point. As an application I will explain how Lagrangian surgery can be used to obtain some approximation results on real surfaces.

In the present talk, I will review, from a mathematical viewpoint, the computations of instanton partition functions of supersymmetric gauge theories in four dimension by means of quiver varieties and their relations to representation theory of vertex algebras.

By extending the canonical trace of Kontsevich-Vishik to Connes' pseudodifferential operators on noncommutative tori, we study various spectral invariants associated to elliptic operators in this setting. We also consider a family of Cauchy-Riemann operators over noncommutative 2-torus and using the machinery of canonical trace, we compute the curvature form of the associated Quillen determinant line bundle.

We will present a theorem due to R.M.F. Moss which says, roughly, that 3-fold Massey products of permanent cycles in the Adams spectral sequence converge to the corresponding 3-fold Toda brackets in stable homotopy.

The Kepler conjecture asserts that no packing of congruent balls in space can have density greater than the familiar cannonball arrangement. If every logical inference of proof has been checked all the way to the fundamental axioms of mathematics, then we say that the proof has been formally verified. The Kepler conjecture has now been formally verified by computer, in a massive cloud computation. This talk will report on this and other massive formal verification projects.

The index of a bounded operator $T\in B(H)$ of a Hilbert space $H$ is defined as the difference between the dimensions of kernel and cokernel. That is, $${\rm Ind}(T):=\dim(\ker T)-\dim({\rm coker}T)$$ This index, if defined, is called the Fredholm index. The Fredholm index of an operator on a finite dimensional Hilbert space $H$ by the dimension theorem in linear algebra. However, the case of infinite dimensional Hilbert spaces requires more delicate analysis and an operator with nonzero index exists. The celebrated local index formula in noncommutative geometry (Connes and Moscovici 1995) relates the index of Dirac type operators and the residue cocycle in the cyclic cohomology. In the classical case, this formula equates topology and geometry. In my talk, I will prove two special cases of local index formula following closely the chapter 5 in Noncommutative geometry and particle physics by Walter Van Suijlekom. If the time is allotted, I will demonstrate the strength of the formula using simple classical spectral triples such as the circle $S^1$.

We present a chain lemma for tensor products of any number of quaternion algebras over fields of cohomological dimension 2. We discuss the connection to other objects, such as quadratic forms and the symplectic group.