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4:00pm to 5:00pm - RH 340N - Geometry and Topology Dragos Oprea - (UCSD) On the genera of Quot schemes of zero dimensional quotients on curves We study Quot schemes of rank 0 quotients on smooth projective curves. These Quot schemes exhibit a rich and highly structured geometry, with formal parallels to the Hilbert schemes of points on surfaces. In this talk, we first note formulas for the twisted \chi_y-genera with values in tautological line bundles pulled back from the symmetric product via the Quot-to-Chow morphism, and for the associated twisted Hodge numbers. Going further, we give formulas for the level 2 (twisted) elliptic genus for quotients of a vector bundle of even rank. We also discuss the case of level \ell elliptic genera, for higher values of \ell. |
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3:00pm to 3:50pm - RH 440R - Logic Set Theory Garrett Ervin - (Caltech) An arithmetic characterization of additive commutativity for order types For which pairs of linear orders $A$ and $B$ are the sums $A + B$ and $B + A$ isomorphic? In the 1930s, Tarski conjectured that $A + B = B + A$ if and only if (i.) There is an order $C$ and natural numbers $n$ and $m$ such that $A = nC$ and $B = mC$, or (ii.) There is an order $M$ such that $B = \omega A + M + \omega^* A$, or (iii.) There is an order $N$ such that $A = \omega B + N + \omega^* B$. Notably, these conditions on $A$ and $B$ are “arithmetic” in the sense that they are expressed in terms of finitary and $\omega$-ary sums of linear orders. Tarski proved his conjecture over the class of scattered linear orders, but Lindenbaum was able to produce a non-scattered counterexample. Building on Lindenbaum’s work, Aronszajn found a structural characterization of all additively commuting pairs of linear orders. Aronszajn’s characterization is somewhat complicated: in modern language it can be described in terms of orbit equivalence relations of groups of translations on $\mathbb{R}$. Tarski lamented that Aronszajn’s result could not be formulated arithmetically — that is, purely in terms of sums — and the line of work was abandoned. Building on our recent work on sums of linear orders, Eric Paul and I showed that there is an arithmetic condition equivalent to commutativity for linear orders. And in fact, the condition is a natural extension of the one appearing in Tarski’s original conjecture. In this talk, I will state our result, outline the proof, and discuss some related problems. |
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9:00am to 9:50am - Zoom - Inverse Problems Ali Feizmohammadi - (University of Toronto) Inverse spectral problems with sparse data and applications to photoacoustic tomography |
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3:00pm to 4:00pm - RH 306 - Number Theory Noah Lebowitz-Lockard - (UCI) The Mondrian Puzzle The Mondrian Puzzle asks whether it is ever possible to partition a square into at least two disjoint rectangles of integer side lengths and the same area where no two rectangles have the same dimension. In this talk, we show that for a positive proportion of squares, this task is impossible. Along the way, we discuss several classic results in analytic number theory. |