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Feb. 27 colloquium: John Pardon (SCGP)

Special time, location: Tue. Feb. 27, 4:10-5:25pm, 407 Math


Universally counting curves in Calabi–Yau threefolds


Statements such as “there is a unique line between any pair of distinct points in the plane” and “there are 27 lines on any cubic surface” have given rise to the modern theory of enumerative geometry.
To define such “curve counts” in a general setting usually involves choosing a particularly nice compactification of the space of smooth embedded curves (one which admits a natural “virtual fundamental class”).  I will propose a new perspective on enumerative invariants which is based instead on a certain “Grothendieck group of 1-cycles” and the “universal” curve enumeration invariant taking values in this group.  It turns out that if we restrict to complex threefolds with nef anticanonical bundle, this group has a very simple structure: it is generated by “local curves”.  This generation result implies some new cases of the MNOP conjecture relating Gromov–Witten and Donaldson–Pandharipande–Thomas invariants of complex threefolds.

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Feb. 28. colloquium: Hans Ringström (KTH)
Title: Curvature blow up at big bang singularities
Abstract: Singularities have been accepted as a natural feature in general relativity since the appearance of the singularity theorems of Hawking and Penrose. But these theorems do not say much concerning the nature of singularities. Do the gravitational fields become unbounded? Can the spacetime be extended through the singularity? Recently, many results demonstrating the stability of spatially homogeneous solutions with big bang singularities have appeared. As a consequence, there is an open set of initial data yielding big bang singularities with curvature blow up. However, the purpose of the talk is to illustrate that it is possible to go beyond stability results. In fact, I will present a new result (joint work with Hans Oude Groeniger and Oliver Petersen) in which we identify a general condition on initial data ensuring big bang formation. The solutions need, in this case, not be close to symmetric background solutions. Moreover, the result reproduces previous results in the Einstein-scalar field and Einstein-vacuum settings. Finally, the result is in the Einstein-non-linear scalar field setting, and therefore yields future and past global non-linear stability of large classes of spatially locally homogeneous solutions.
Time and location: Wed. Feb. 28, 4:30-5:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
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January 17: Mehtaab Sawhney (MIT)
Title: On High Girth Steiner-Triple Systems and Subspace Designs
Abstract: We discuss the recent resolutions of the 1973 conjecture of Erdős on the existence of high girth Steiner triple systems and the existence of subspace designs. The talk will focus on placing these results within the context of classical design theory and within recent advances in the absorption method in combinatorics.
Based on joint works w. Peter Keevash, Matthew Kwan, Ashwin Sah, and Michael Simkin
Time and location: Wed. Jan. 17, 4:30-5:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
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March 4: Horng-Tzer Yau (Harvard)

Title: A dynamical approach to universality in probability theory with applications in random matrices read more »

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Oct. 16: Amol Aggarwal (Harvard)

Title: The Local Behavior of Random Lozenge Tilings read more »

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Sept. 25: Henry Cohn (Microsoft)

Title: Sphere packing, Fourier interpolation, and ground states in 8 and 24 dimensions read more »

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Nov. 7: Robin Pemantle (U Penn)

Title: Combinatorial Applications of Computational Topology and Algebraic Geometry read more »

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Oct. 24: Maciej Zworski (Berkeley)

Title: Microlocal methods in chaotic dynamics   read more »

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Sept. 5: Jonathan Luk (Stanford)

Title: The strong cosmic censorship conjecture in general relativity
read more »

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Sept. 6: Sabin Cautis (UBC)

Title: [N choose k] equals [N choose N-k]
read more »

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