ICTS

• Universal operations on the Hochschild complex of algebras over a prop
• String topology operations via the Hochschild complex
• Algebraic structures on Tate-Hochschild complex
   

• Universal operations on the Hochschild complex of algebras over a prop
• String topology operations via the Hochschild complex
• Algebraic structures on Tate-Hochschild complex
   

• quasisyntomic rings and ring spectra
• Motivic filtrations on THH, TC^{-}, TP, and TC
• Syntomic cohomology and prismatic cohomology of truncated Brown--Peterson spectra
   

For a \(C_2\)-commutative ring spectrum \(R\), a twisted \(R\)-algebra is an \(R\)-module equipped with a multiplication whose order is switched by the \(C_2\)-action; equivalently, such objects are algebras over the little \(\sigma\)-disks operad. In this talk, we present a construction of quotients of \(C_2\)-commutative ring spectra as twisted algebras using equivariant Thom spectra, under suitable cofree and evenness assumptions.

As a motivating example, we consider \(KR/2\) as a twisted \(KR\)-algebra and compute its real topological Hochschild homology over \(KR\). This involves a computation of the splitting of the spectrum of units \(gl_1(KR)\), together with an identification of the real topological Hochschild homology of such Thom spectra as Thom spectra themselves.

This talk is based on joint work with Samik Basu.

By Jones's theorem, the negative cyclic homology $\HC^-_\ast(C^\ast(Y))$ of a simply connected space $Y$ computes the $S^1$-equivariant cohomology of its free loop space $LY$. Rationally, the right-hand side is computable via Sullivan minimal models (Burghelea--Vigu\'e-Poirrier), but integrally HKR fails for exterior algebras over $\Z$ and explicit cyclic torsion enters.

I describe an integral computation of $\HC^-_\ast(\Lambda_\Z(e_1, \ldots, e_r))$, equivalently the integral $S^1$-equivariant cohomology of $LX$ for $X = \prod_i S^{2k_i - 1}$ a finite product of odd spheres. Eilenberg--Zilber reduces the negative cyclic complex to a tensor-product mixed complex $(W, B^\otimes) = \bigotimes_i (V_i, B_i)$ with single-factor input $H_j(V_i, B_i) = \Z/j$ for $j \ge 1$. Iterated K\"unneth produces gcd-indexed cyclic torsion with binomial multiplicities, and the kernel of $B^\otimes$ has rank generating function
\[
K(t) \;=\; \frac{(1 - t)^r + 2^r t}{(1 + t)(1 - t)^r}.
\]
The Borel spectral sequence for $LX$ collapses at $E_3$ integrally with no extensions. Time permitting, I will indicate a degree-$D_G$ naturality identity transporting this to compact Lie groups via the Hurewicz-scaled map $\varphi_G: X_G \to G$.

In this talk we will present some results on real and higher real K-theories of complex projective spaces, \(KO_*(\mathbb{CP}^n)\), and \(EO_{p-1}^*(\mathbb{CP}^n)\) for odd primes \(p\). We will give a brief review of the cohomology theories \(EO_{p-1}\) and discuss the methods involved, including the filtration spectral sequence and the homotopy fixed point spectral sequence. Finally, using these computations, we will discuss various applications concerning exotic smooth structures on complex projective spaces, the existence of free smooth \(S^1\) and \(S^3\)-actions on exotic spheres, and the existence of complex hyperbolic manifolds with certain curvature properties.

• Chas–Sullivan loop product (chain/homology level)
• Cohen–Godin operations via fatgraphs
• Moduli space operations
• Compactifications and higher structures