Spring 2011
This semester the talks and preparatory lectures were organized by Orit Davidovich, Brandy Guntel, Karin Knudson, Alice Mark, Allison Moore, Verónica Quítalo, and Sarah Rich.
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Alice Chang, Princeton University
Topic: Geometry and PDE
Title: Fully non-linear PDE in conformal geometry
Abstract: I will discuss a class of integral conformal invariants and the role they have
played in a special case of a uniformization theorem for 4-spheres. The main tool is
a study of fully non-linear elliptic PDE of Monge-Amphere type. I will also discuss
the connection of these conformal invariants to geometric invariants on conformal
compact Einstein manifolds in the CFT/ADS setting.
This presentation was also the Horton-Jacobs/WINS Lecture for the 2010 - 2011 year.
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Winnie Li, Pennsylvania State University
Topic: Number Theory
Title: Zeta Functions in Combinatorics and Number Theory
Abstract: Roughly speaking, a zeta function is a counting function.
Well-known zeta functions in number theory include the Riemann zeta function
and the zeta function attached to an algebraic variety defined over a finite field.
The former counts integral ideals of a given norm, while the latter counts solutions
over a finite field. A combinatorial zeta function counts tailless geodesic
cycles of a given length in a finite simplicial complex. One-dimensional complexes
are graphs; attached to graphs are the well-studied Ihara zeta functions. Zeta functions
attached to 2-dimensional complexes are recently obtained by myself and students Ming-Hsuan
Kang and Yang Fang by considering finite quotients of the Bruhat-Tits buildings associated
to SL(3) and Sp(4) over a p-adic field. The purpose of this talk is to show connections
between combinatorics and number theory, using zeta functions as a theme. We shall give
closed form expressions of the combinatorial zeta functions mentioned above, and compare
their features, in particular, the role of the Riemann Hypothesis, with those of the zeta
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Spring 2010
This semester the talks and preparatory lectures were organized by Orit Davidovich, Brandy Guntel, Kim Hopkins and Heather
Van Ligten.
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Antonella Grassi, University of Pennsylvania
Topic: Algebraic Geometry
Title: A, D, E (B and C etc.)
Abstract: I will give an overview of certain
singularities arising in algebraic geometry (like in elliptic
fibrations) and in other contexts including the physics of string
theory. |
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Kristin Lauter, Microsoft Research
Topic: Cryptography
Title: Expander graphs and their applications to cryptography
Abstract: Hash functions are ubiquitous in cryptography: they are used
in encryption, key exchange,
signatures and more. We will review
these functions and discuss the requirement that they be resistant
to collision. We will then recall the notion of expander graphs and
explain how to construct collision-resistant hash functions from
graphs in which it is hard to find cycles. Finally, we will discuss a
family of graphs that were constructed by A. Pizer: the
vertices of Pizer's graphs are supersingular elliptic curves in
characteristic p, while the edges are n-isogenies between
supersingular elliptic curves. (Here, n is a fixed integer prime to p.)
For
Pizer's graphs, cycles are hard to find because it is difficult to
compute isogenies between supersingular elliptic curves.
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Fall 2009
This semester the talks and preparatory lectures were organized by
Aynur Bulut, Orit Davidovich, Brandy Guntel, Kim Hopkins and Heather
Van Ligten, and the invited speakers were Susan Friedlander and Dusa
McDuff.
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Dusa McDuff, Columbia University
Topic: Symplectic Geometry
Title: Symplectic embedding of ellipsoids and continued fractions
This presentation was also the Horton-Jacobs/WINS Lecture for the 2009 - 2010 year.
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Susan Friedlander, University of Southern California
Topic: Mathematical Fluid Dynamics
Title: Instabilities in fluid motion
Abstract: Instabilities in fluid motion are ubiquitous and yet
instabilities come in various "flavors". The partial differential
equations of fluid dynamics are very challenging nonlinear systems. A
classical approach to detecting instabilities is to study the spectral
problem associated with the linearized equations. We will discuss how
in some situations it is possible to prove that linear instability
implies instability for the full nonlinear equations. Examples where
this can be proved include the cases of the 2-dimensional Euler
equations, the 3-dimensional Navier-Stokes equation and an interesting
equation arising in oceanography called the surface quasi-geostrophic
equation.
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Spring 2009
This semester the talks and preparatory lectures were organized by Orit
Davidovich, Brandy Guntel, Adriana Salerno and Elizabeth Thoren, and
the invited speakers were Abigail Thompson, Ruth Charney, Gigliola
Staffilani.
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Abigail Thompson, University of
California, Davis
Topic Knot Theory; 3-Manifolds
Title: The stabilization problem for 3-manifolds
Abstract: Surprisingly, any closed orientable 3-manifold can be split
into two simple pieces, called handlebodies. The simplicity stops
there, sadly, and understanding the relationships among different
splittings of the same manifold is an ongoing task. I'll describe the
stabilization problem for such splittings of 3-manifolds, and some
recent examples which underscore the difficulty of the problem. This is
joint work with Joel Hass and William Thurston.
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Ruth Charney, Brandeis University
Topic: Geometric Group Theory
Title: Groups and their automorphisms, from free to free abelian.
Abstract: Automorphism groups of free groups and free abelian groups
play an important role in mathematics. Surprisingly, they share much in
common. Between free groups and free abelian groups lies a large class
of groups known as right-angled Artin groups. We investigate which of
these properties hold for automorphisms of all such groups.
This presentation was the first in the Horton-Jacobs/WINS Lecture Series.
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Gigliola Staffilani, Massachusetts
Institute of Technology
Topic: Partial Differential Equations and Harmonic Analysis
Title: On Dispersive Waves
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Fall 2008
This semester the talks and preparatory lectures were organized by Orit
Davidovich, Brandy Guntel, Adriana Salerno, and Elizabeth Thoren, and
the invited speakers were Eleny Ionel and Catharina Stroppel.
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Eleny Ionel, Universität Bonn
Topic : Symplectic Geometry
Title: Gromov-Witten invariants and symplectic degenerations
Abstract: The moduli spaces of holomorphic curves have an intriguing
structure, which is reflected in the structure of the Gromov-Witten
invariants. One way to get a glimpse into it is to follow what happens
to the moduli spaces during (symplectic) degenerations, like those
coming from a (generalized) symplectic sum. This in particular involves
extending the notion of Gromov-Witten invariants to (mildly) singular
settings. I will survey what is known so far about this problem,
mention some of its applications and discuss some of the current open
problems.
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Catharina Stroppel,
Universität Bonn
Topic: Representation Theory
Title: Crystal bases, Hecke algebras and equivalences of categories
Abstract: The classical Schur-Weyl duality relates modules for the
general linear Lie algebra with modules over the symmetric group. I
first explain a higher level version of this where cyclotomic versions
of degenerate Hecke algebras occur. Afterwords I will indicate how this
picture can be categorified. The combinatorics of crystal graphs plays
an important role here. Finally I want to illustrate in two examples
how this setup can be used to derive equivalences of categories where
the aforementioned Hecke algebras play the key role.
This was a joint presentation with the GRASP Colloquium and they have posted a video of the lecture.
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Spring 2008
This semester the talks and preparatory lectures were organized by Orit
Davidovich, Adriana Salerno, and Andrea Young, and the invited speakers
were Vyjayanthi Chari, Panagiota Dakalopoulos, and Paula Tretkoff.
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Vyjayanthi Chari, University of
California, Riverside
Topic: Representation Theory
Title: Affine Algebras, Quivers and Koszulity
Abstract: The representation theory of the affine Lie algebras and
their quantum analogs have been intensively studied in recent years.
The subject has connections with number theory, topology and
mathematical physics. The study of finite dimensional representations
of these algebras is surprisingly complex, and is related to the
mathematical structures which arise from solvable models in statistical
mechanics. There are a number of different approaches to this study: a
geometric approach via quiver varieties, a combinatorial approach using
crystal bases and an algebraic approach using the classical methods of
representation theory. In this talk, I will discuss some of these ideas
and formulate some recent results which establish a connection between
these representations and those of finite dimensional associative
algebras.
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Panagiota Dakalopoulos, Columbia
University
Topic: Geometric Flow
Title: Surface Evolution under Curvature Flows: Existence and Optimal
Regularity
Abstract: We will discuss the evolution of a hyper-surface in
R^{n+1} by functions of its principal curvatures. Typical examples
include the Mean Curvature flow, the Gauss Curvature flow, the Inverse
Mean Curvature flow and the Harmonic Mean Curvature flow. These flows
are described by non-linear parabolic equations for the local embedding
map. We will discuss the existence and optimal regularity for such
equations as well as the formation of singularities in certain cases.
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Paula Tretkoff, Texas A&M
University
Topic: Number Theory
Title: Aspects of transcendental number theory
Abstract: We discuss an assortment of results on the transcendence
properties of special values of modular and hypergeometric functions.
The emphasis will be on open problems and connections with other
branches of mathematics.
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