It is a classical topic dating back to Maclaurin (1698–1746) to study certain special points on smooth cubic plane curves, such as the 9 inflection points (Maclaurin and Hesse), the 27 sextatic points (Cayley), and the 72 points "of type 9" (Gattazzo). Motivated by these algebro-geometric constructions, we ask the following topological question: is it possible to choose n distinct points on a smooth cubic plane curve as the curve varies continuously in family, for any integer n other than 9, 27 and 72? We will present both constructions and obstructions to such continuous choices of points, state a classification theorem for them, and discuss conjectures and open questions.

Understanding joint behaviour of $(f(n),g(n+1))$ where f and g are given multiplicative functions play key role in analytic number theory with potentially profound consequences such as Riemann hypothesis, twin prime conjecture, Chowla's conjecture and many others.

In the the first part of this talk, I will discuss joint work with A. Mangerel, answering an old question of Katai about distribution of points $\{(f(n),g(n+1))\}_{n\ge 1}\in \mathbb{T}^2,$ where f and g are unimodular multiplicative functions.  

In the second part of the talk, which is based on a joint work with P. Kurlberg, answering a question of M. Lemanczyk, we construct deterministic example of multiplicative function $f:{\mathbb{N}\to \{+1,-1\}$ with various ergodic properties with respect to the Mirsky measure and discuss its relation to the interplay between Chowla conjecture and Riemann hypothesis. 

Short generating functions were first introduced by Barvinok to enumerate integer points in polyhedra. Adding in Boolean operations and projection, they form a whole complexity hierarchy with interesting structure. We study them in the computational complexity point of view. Assuming standard complexity assumption, we show that these functions cannot effectively represent certain truncated theta functions. Along the way, we will draw connection to ordinary number theoretic objects, such as the set of prime or square numbers. This talk assumes no prior knowledge of the subject. Some open questions will be offered at the end. Joint work with Igor Pak.

I will introduce the mod p derived spherical Hecke algebra of a p-adic group, and discuss its structure via a derived version of the Satake homomorphism. Then, I will survey some speculations about its action on the cohomology of arithmetic manifolds.

Suppose f is an n-variate polynomial with integer coefficients and degree d. Many natural computational problems involving the real zero set Z of f are algorithmically difficult. For instance, no known algorithm for computing the number of connected components of Z has complexity polynomial in n+d. Furthermore, no known general algorithm for deciding whether f has root over the p-adic has sub-exponential complexity. So it is worthwhile to seek families of polynomials where these questions are tractable.

Assuming f has n+2 monomial terms, and its exponent vectors do not all lie on an affine hyperplane, we prove that the isotopy type of Z can be determined in time polynomial in log d, for any fixed n. (This family of polynomials --- polynomials supported on circuits ---is highly non-trivial, since it already includes Weierstrass normal forms and several important examples from semi-definite programming.) We also show that a 1998 refinement of the abc-Conjecture (by Baker) implies that our algorithm is polynomial in n as well. Furthermore, the original abc-Conjecture implies that p-adic rational roots for f can be detected in the complexity class NP.

     These results were obtained in collaboration with Kaitlyn Phillipson and Daqing Wan.