Upcoming Seminars & Events

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April 5, 2017
4:30pm - 5:30pm
The Waring problem and group theory

The Waring problem asks whether for each positive integer n, every positive integer is the sum of a bounded number of nth powers. Although Hilbert solved it more than 100 years ago, it remains the focus of much current research. I will discuss some questions in group theory, especially algebraic group theory, inspired by it. In particular, I will present work with Aner Shalev and Pham Tiep on Waring-type word maps on finite simple groups and work with Dong Quan Ngoc Nguyen on the unipotent Waring problem. The emphasis throughout will be on unsolved problems.

Speaker: Michael Larsen , Indiana University
Location:
Fine Hall 314
April 6, 2017
10:45am - 11:45am
TBA - Sebastian Hurtado
Speaker: Sebastian Hurtado , University of Chicago
Location:
IAS Room S-101
April 6, 2017
12:30pm - 1:30pm
TBA - Pan Lue
Speaker: Pan Lue , Princeton University
Location:
Fine Hall 110
April 6, 2017
2:00pm - 3:30pm
TBA - Dmitry Dolgopyat
Speaker: Dmitry Dolgopyat , University of Maryland
Location:
Jadwin Hall 111
April 6, 2017
3:00pm - 4:00pm
Symmetrized topological complexity of the circle

We prove that it is impossible to have two continuous rules telling how to move between any two points on the circle in such a way that the path from Q to P is the reverse of the path from P to Q. It is easy to see that this can be done with three such rules.

Speaker: Don Davis , Lehigh University
Location:
Fine Hall 322
April 6, 2017
4:30pm - 5:30pm
Embedding problems in affine algebraic geometry and slice knots

We first discuss classical questions about polynomial embeddings of the complex line C into complex spaces such as C^m and affine algebraic groups. Next, we consider knots and different notions of sliceness for knots.
Finally, we use a knot theory perspective to indicate proofs for the embedding questions discussed first. 

Speaker: Peter Feller , Max Planck Institute for Mathematics
Location:
Fine Hall 314
April 6, 2017
4:30pm - 5:30pm
TBA - Andrej Zlatos
Speaker: Andrej Zlatos , UC San Diego
Location:
Fine Hall 322
April 6, 2017
4:30pm - 5:30pm
Basic loci of Shimura varieties

In mod-p reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-p reductions of more general Shimura varieties, there is a ``Newton stratification'' decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves. In certain cases, the basic locus admits a simple description as a union of classical Deligne-Lusztig varieties. The precise description in these case has proved to be useful for several purposes: to compute intersection numbers of special cycles and to prove the Tate conjecture for certain Shimura varieties. We will describe a group-theoretic approach to understand this phenomenon. We will show that this phenomenon is closely related to the Hodge-Newton decomposition, and many other nice properties on the Shimura varieties. This talk is based on the joint work with Ulrich Gortz and Sian Nie.

Speaker: Xuhua He , IAS
Location:
Fine Hall 214
April 7, 2017
5:00pm - 6:00pm
Solving packing problems by linear programming

Part 4:  The solution of the sphere packing problem in dimensions 8 and 24.

The sphere packing problem asks which biggest portion of the euclidean d-dimensional space can be covered by non-overlapping unit balls. In most dimensions d this question is believed be an extremely difficult combinatorial geometric problem. However, in dimensions 8 and 24 the sphere the sphere packing problem has a surprisingly simple solution based on linear programming bounds.The goal of this series of talks is to explain the ideas behind this solution. 

Speaker: Maryna Viazovska , Humboldt University
Location:
Fine Hall 314
April 10, 2017
3:00pm - 4:00pm
Square functions for directional operators on the plane

We show that the truncated Hilbert transform along a Lipschitz field of monomial curves x+(t,u(x)t^\alpha) is almost diagonalized by Littlewood-Paley projections in the direction transversal to the field. As a consequence we obtain L^p estimates in the curved case \alpha\neq 1. In the flat case \alpha=1 our method removes the auxiliary C^{1+\epsilon} regularity hypothesis from the conditional (on a certain maximal inequality involving u(x)) L^2 result of Lacey and Li. Joint work with F. di Plinio, S. Guo, and C. Thiele.

Speaker: Pavel Zorin-Kranich , Bonn University
Location:
Fine Hall 314
April 10, 2017
4:00pm - 5:00pm
Physics in the complex plane

The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian $H=p^2+ix^3$, which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory.  Evidently, the axiom of Dirac Hermiticity is too restrictive. While $H=p^2+ix^3$ is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past few years, some of these properties have been observed and verified in laboratory experiments. A particularly interesting PT-symmetric Hamiltonian is $H=p^2-x^4$, which contains an upside-down potential. We explain in intuitive and in rigorous terms why the energy levels of this potential are real, positive, and discrete.

Speaker: Carl M. Bender, Washington University in St. Louis
Location:
Fine Hall 214
April 11, 2017
4:30pm - 5:30pm
TBA - Linquan Ma
Speaker: Linquan Ma , University of Utah
Location:
Fine Hall 322
April 12, 2017
2:30pm - 3:30pm
A random walk on the upper triangular matrices

We study the following random walk on the group of n n upper triangular matrices with coecients in Z=pZ and ones along the diagonal. Starting at the identity, at each step we choose a row at random and either add it to or subtract it from the row immediately above. The mixing time of this walk is conjectured to be asymptotically n2p2. While much has been proven in this direction by a number of authors, the full conjecture remains open. We sharpen the techniques introduced by Arias-Castro, Diaconis, and Stanley to show that the dependence on p of the mixing time is p2. To prove this result, we use super-character theory and comparison theory to bound the eigenvalues of this random walk.

Speaker: Evita Nestoridi, Princeton University
Location:
Fine Hall 224
April 12, 2017
3:00pm - 4:00pm
Min-max minimal hypersurfaces with free boundary

I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. Our result allows the min-max free boundary minimal hypersurface to be improper; nonetheless the hypersurface is still regular.

Speaker: Xin Zhou , UC Santa Barbara
Location:
Fine Hall 314
April 12, 2017
3:00pm - 4:00pm
TBA - Sergey Bobkov
Speaker: Sergey Bobkov , university of Minnesota
Location:
Fine Hall 214
April 12, 2017
4:30pm - 5:30pm
Statistics for random linear combinations of Laplace eigenfunctions

There are several questions about the behavior of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of their number of critical points, the study of the size of their zero set, the study of the number of connected components of their zero set, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for random linear combinations of eigenfunctions. In this talk I will present several results in this direction. This talk is based on joint works with Boris Hanin and Peter Sarnak.

Speaker: Yaiza Canzani, University of North Carolina at Chapel Hill
Location:
Fine Hall 314
April 13, 2017
9:30am - 10:30am
TBA - Sheng-Fu Chiu

Please note special time (9:30 a.m.)

Speaker: Sheng-Fu Chiu, Northwestern University
Location:
IAS Room S-101
April 13, 2017
10:45am - 11:45am
TBA - Maia Fraser
Speaker: Maia Fraser , University of Ottawa
Location:
IAS Room S-101
April 13, 2017
12:30pm - 1:30pm
TBA - Charlie Stibitz
Speaker: Charlie Stibitz, Princeton University
Location:
Fine Hall 110
April 13, 2017
2:00pm - 3:30pm
TBA - Subhro Ghosh
Speaker: Subhro Ghosh , Princeton University
Location:
Jadwin Hall 111

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