Talk

The average behaviour of class numbers of real quadratic fields is still poorly understood. Let $S(X)$ denote the sum of the class numbers of real quadratic fields with discriminant at most $X$. The class number formula gives the elementary bound $S(X)\ll X^{3/2}$, while a classical result of Siegel gives the best known upper bound $S(X)\ll X^{3/2}/\log X$. On the other hand, Hooley conjectured that $S(X)$ should be of order $X(\log X)^2$.

In this talk, we will discuss bounds on the largest prime factor of Fourier coefficients of normalized Hecke eigenforms. Further, we will study the positivity of Hecke eigenvalues of Ikeda lifts and bounds on the large prime divisors of these Hecke eigenvalues. These are joint works with Yuri Bilu, and Sanoli Gun.

In this talk we will focus on one arithmetic setting: the action of a finite group G on a curve X. The idea is to pass to its Jacobian variety and let the representation theory of G do the work. This splits the Jacobian, up to isogeny, into smaller factors. We will explain this method and give some examples.

For half a century, chromatic homotopy theory has provided us one framework for organizing computations and the search of large-scale phenomena. We can reasonably ask why. As one answer, I'll give a highly anecdotal and perhaps idiosyncratic survey of the field, past and present and future, with the explicit aim of highlighting the essential contributions of Neil Strickland in making it all go. 

 

Witten constructed a modular form-valued genus for any manifold with a $U<6>$-structure. Motivated by this,  Ando, Hopkins and Strickland characterized all ring maps from $MU<6>$ to even-periodic spectra in terms of cubical structures and characterized the Witten genus in these terms. 
In this talk, I will explore how one can combine their insights with equivariant topological modular forms to try to construct an equivariant string orientation, further refining the Witten genus.



 

In joint work with Ryan Quinn we developed a technique to produce structured equivariant orientations by lifting their underlying structured non-equivariant orientations. This allows us to lift many maps of interest, such as the Hirzebruch level-n genera or the Hahn-Shi orientations. In fact, our result is robust enough to lift structured equivalences resulting in a structured Real Snaith theorem, a structured Real Bökstedt periodicity, and finally the first multiplicative version of the Real Brown-Peterson spectrum.

The Thom isomorphism fails for equivariant vector bundles in $RO(G)$-graded cohomology, even for $G=\mathbb{Z}/2$. Costenoble--Waner developed a parametrized equivariant cohomology theory incorporating Thom isomorphisms for all vector bundles. This cohomology theory extends the usual representation grading $RO(G)$ to $RO(\Pi_G B)$, representations of the equivariant fundamental groupoid. We compute the parametrized cohomology of the classifying space for real $\mathbb{Z}/2$-line bundles, $B_{C_2}O(1)$.

For a compact Lie group G, we may study G-equivariant cohomology theories, and it is conjectured that there is an abelian category A(G) so that there is an equivalence between rational G-spectra  =  dg A(G).
The category A(G) is (in a sense to be made precise) a category of sheaves over the space X(G) of conjugacy classes of subgroups of G. In fact X(G) is a finite disjoint union of blocks V(G,H) (where H is a subgroup with finite Weyl group) so A(G)=A(V(G,H1)) x  …… x A(V(G,Hn)).