The r-th cactus variety of a subvariety X in a projective space generalises secant variety of X and it is defined using linear spans of finite schemes of degree r. It's original purpose was to study the vanishing sets of catalecticant minors. We propose adding a scheme structure to the cactus variety and we define it via relative linear spans of families of finite schemes over a potentially non-reduced base. In this way we are able to study the vanishing scheme of the catalecticant minors. For X which is a sufficiently large Veronese reembedding of projective variety, we show that r-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on an open and dense subset which is the complement of the (r-1)-st cactus variety/scheme. As an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties. Based on a joint work with Hanieh Keneshlou.

One of the classical and most fascinating problems at the intersection between combinatorics and number theory is the study of the parity of the partition function. Even though p(n) in widely believed to be equidistributed modulo 2, progress in the area has proven exceptionally hard. The best results available today, obtained incrementally over several decades by Serre, Soundarajan, Ono and many otehrs, do not even guarantee that, asymptotically, p(n) is odd for /sqrt{x} values of n/neq x, In this talk, we present a new, general conjectural framework that naturally places the parity of p(n) into the much broader, number-theoretic context of eta-eqotients. We discuss the history of this problem as well as recent progress on our "master conjecture," which includes novel results on multi-and regular partitions. We then show how seemingly unrelated classes of eta-equotients carry surprising (and surprisingly deep) connections modulo 2 to the partition function. One instance is the following striking result: If any t-multiparition function, with t/neq 0(mod 3), is odd with positive density, then so is p(n). (Note that proving either fact unconditionally seems entirely out of reach with current methods.) Throughout this talk, we will give a sense of the many interesting mathematical techniques that come into play in this area. They will include a variety of algebraic and combinatorial ideas, as well as tools from modular forms and number theory.

In this talk, we consider some polynomials which define Gaussian Graphical models in algebraic statistics. First, we briefly introduce background materials and some preliminary on this topic. Next, we regard a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle graph and explain how to prove it. We also report a result on linear syzygies of any model coming from block graphs. The former work was done jointly with A. Conner and M. Michalek and the latter with J. Choe.