Thursday, November 29, 2018
Time  Items 

All day 

11am 
11/29/2018  11:00am The classical Fano scheme $F_d(X)$ of a variety X parametrises $d$dimensional linear spaces contained in X. In this talk I am going to define a tropical analogue of the Fano scheme $F_d( trop X)$ and I will show its relation with the tropicalization $trop F_d(X)$ of the classical Fano scheme. In particular I will focus on the tropical version of Fano schemes of tropicalized linear spaces and tropicalized toric varieties. Location:
Hillhouse 17, Room 101 (TEAL)

2pm 
11/29/2018  2:00pm For a Hyperkähler variety which admits a Lagrangian fibration, an increasing filtration is defined on its rational cohomology using the perverse $t$structure. We will discuss the role played by this filtration in the study of the topology and geometry of Hyperkähler varieties. First, we will focus on the perverse filtration for the moduli of Higgs bundles with respect to the Hitchin fibration. We will discuss our recent proof of de Cataldo, Hausel, and Migliorini’s $P=W$ conjecture for parabolic Higgs bundles labelled by affine Dynkin diagrams. Then I will present a surprising connection between the perverse filtration for a projective Hyperkähler variety and the (pure) Hodge structure on itself. Based on joint work with Qizheng Yin and Zili Zhang. Location:
WTS A53

4pm 
11/29/2018  4:00pm Let $p$ be a fixed prime. A $k$cycle in $\mathbb{F}_p^n$ is an ordered $k$tuple of points that sum to zero; we also call a $3$cycle a triangle. Let $N=p^n$, (the size of $\mathbb{F}_p^n$). Green proved an arithmetic removal lemma which says that for every $k$, $\epsilon>0$ and prime $p$, there is a $\delta>0$ such that if we have a collection of $k$ sets in $\mathbb{F}_p^n$, and the number of $k$cycles in their cross product is at most a $\delta$ fraction of all possible $k$cycles in $\mathbb{F}_p^n$, then we can delete $\epsilon N$ elements from the sets and remove all $k$cycles. This is closely related to the graph removal lemma, which essentially says that if a graph $G$ has few copies of a fixed subgraph $H$, then we can remove a small number of edges from $G$ and get rid of all copies of $H$. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to our work, the best known bound for any $k$, due to Fox, showed that $1/\delta$ can be taken to be an exponential tower of twos of height logarithmic in $1/\epsilon$ (for a fixed $k$). In this talk, we will discuss recent work on Green’s problem. For triangles, we prove an essentially tight bound for Green’s arithmetic triangle removal lemma in $\mathbb{F}_p^n$, using the recent breakthroughs with the polynomial method. For $k$cycles, we also prove a polynomial bound. We also prove a lower bound on the exponent by proving a lower bound on the $k$multicolored sumfree problem. However, the question of the optimal exponent is still open. The triangle case is joint work with Jacob Fox, the $k$cycle case with Jacob Fox and Lisa Sauermann, and the lower bound for general $k$ with Lisa Sauermann. Location:
DL 431
11/29/2018  4:00pm Semialgebraic sets are those defined by polynomial inequalities over an algebraically closed field. In this talk I will discuss their tropicalization and analytification. We show that the real analytification is homeomorphic to the inverse limit of real tropicalizations, analogously to a result of Payne. We also show a real analogous of the fundamental theorem of tropical geometry. The talk will be accessible to graduate students with background on basic algebraic geometry. This is based on joint work with Philipp Jell and Claus Scheiderer. Location:
LOM 214
