Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 2, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 2, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, October 2, 2018

#### Generalized square knots and the 4-dimensional Poincare Conjecture

###### Alex Zupan (University of Nebraska-Lincoln)

Abstract: The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. In this talk, we consider homotopy 4-spheres X built with two 2-handles and two 3-handles, which are uniquely determined by the attaching link L for the 2-handles in the 3-sphere. We prove that if one of the components of L is the connected sum of a torus knot T(p,2) and its mirror (a generalized square knot), then X is diffeomorphic to the standard 4-sphere. This is joint work with Jeffrey Meier.

1:00 pm in 347 Altgeld Hall,Tuesday, October 2, 2018

#### From Neumann to Steklov via Robin: the Weinberger way

###### Richard Laugesen   [email] (Illinois Math)

Abstract: Lord Rayleigh asserted in 1877, and Faber and Krahn proved fifty years later, that “If the area of a membrane be given, there must evidently be some form of boundary for which the pitch (of the principal tone) is the gravest possible, and this form can be no other than the circle.” In modern terminology, Rayleigh was claiming that among all planar domains of given area, the one that minimizes the first eigenvalue of the Laplacian (under Dirichlet boundary conditions) is the disk. In terms of heat flow in 3 dimensions, that means a room of given volume whose boundary is maintained at temperature zero will cool off slowest when it is a ball. What about a room whose boundary is perfectly insulated (Neumann boundary conditions)? Szego and Weinberger discovered in the 1950s that the room’s temperature will equilibrate fastest when it is spherical - which is admittedly an unlikely shape for a room, except for fans of Futuro Flying Saucer homes. Insulation is never perfect, as every homeowner knows, and so we are led to the Robin boundary condition for partial insulation, and to recent progress and open problems on “isoperimetric type” eigenvalue inequalities that extend the work of Rayleigh-Faber-Krahn and Szego-Weinberger.

2:00 pm in 243 Altgeld Hall,Tuesday, October 2, 2018

#### Long paths and large matchings in ordered and convex geometric hypergraphs

###### Alexandr Kostochka (Illinois Math)

Abstract: An ordered $r$-graph is an $r$-uniform hypergraph whose vertex set is linearly ordered, and a convex geometric $r$-graph (cg $r$-graph, for short) is an $r$-uniform hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and cg graphs have rich history.

We consider extremal problems for two types of paths and matchings in ordered $r$-graphs and cg $r$-graphs: zigzag and crossing paths and matchings. We prove bounds on Turán numbers for these configurations; some of them are exact. Our theorem on zigzag paths in cg $r$-graphs is a common generalization of early results of Hopf and Pannwitz, Sutherland, Kupitz and Perles for cg graphs. It also yields the current best bound for the extremal problem for tight paths in uniform hypergraphs. There are interesting similarities and differences between the ordered setting and the convex geometric setting.

This is joint work with Zoltán Füredi, Tao Jiang, Dhruv Mubayi and Jacques Verstraëte.

3:00 pm in 243 Altgeld Hall,Tuesday, October 2, 2018

#### S(ymplectic) duality

###### Justin Hilburn

Abstract: In this talk I would like to briefly sketch how one can use the tools of derived symplectic geometry and holomorphically twisted gauge theories to derive a relationship between symplectic duality and local Langlands. Our starting point will be an observation due to Gaiotto-Witten that a 3d N=4 theory with a G-flavor symmetry is a boundary condition for 4d N=4 SYM with gauge group G. By examining the relationship between boundary observables and bulk lines we will be able to derive constructions originally due to Braverman, Finkelberg, Nakajima. By examine the relationship between boundary lines and bulk surface operators one can derive new connections to local geometric Langlands. This is based on joint work with PhilsangYoo, Tudor Dimofte, and Davide Gaiotto.