Colloquia 2012-2013: Difference between revisions

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It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain.
It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain.
===Emanuele Macri (University of Bonn)===
''Stability conditions and Bogomolov-type inequalities in higher dimension''
Stability conditions on a derived category were originally introduced by Bridgeland to give a mathematical foundation for the notion of \Pi-stability in string theory, in particular in Douglas’ work. Recently, the theory has been further developed by Kontsevich and Soibelman, in relation to their theory of motivic Donaldson-Thomas invariants for Calabi-Yau categories. However, no example of stability condition on a projective Calabi-Yau threefold has yet been constructed.
In this talk, we will present an approach to the construction of stability conditions on the derived category of any smooth projective threefold. The main ingredient is a generalization to complexes of the classical Bogomolov-Gieseker inequality for stable sheaves. We will also discuss geometric applications of this result.
This is based on joint work with A. Bayer, A. Bertram, and Y. Toda.
===Marcus Roper (Berkeley)===
''Modeling microbial cooperation''
Although the common conception of microbes is of isolated and individualistic cells, cells can also cooperate to grow, feed or disperse in challenging physical environments. I'll show how new mathematical models reveal the benefits and developmental and stability barriers to inter-cellular cooperation, by highlighting three paradigmatic examples: 1. The primitive multicellularity of our closest-related protozoan cousins, Salpingoeca rosetta. 2. The cooperative hydrodynamics that enhance spore dispersal in the devastating crop pathogen Sclerotinia sclerotiorum. 3. The architectural adaptations that allow filamentous fungi to thrive in many different ecological niches by harboring mixed communities of genotypically different nuclei.
=== Ana-Maria Castravet (Arizona) ===
''Hypertrees and moduli spaces of stable rational curves''
The Grothendieck-Knudsen moduli space \bar M_{0,n} of stable
rational curves with n marked points is a building block towards many
moduli spaces (stable curves, Kontsevich stable maps). Although there are
several explicit descriptions of \bar M_{0,n}, its geometry is far from
being understood. In this talk, I will introduce new combinatorial
structures called hypertrees and use them to approach the string of open
problems about effective cycles on \bar M_{0,n}. In particular, we will
construct new exceptional divisors - which lead to new birational models
of \bar M_{0,n} - and rigid curves - which are important for the
Faber-Fulton conjecture on the structure of the Mori cone of \bar M_{0,n}.
This is based on joint work with Jenia Tevelev.
=== Xinyi Yuan (Columbia) ===
''Equidistribution in algebraic dynamics''
Abstract: Algebraic dynamics studies iterations of algebraic
endomorphisms of algebraic varieties. One problem is to study
distribution of points with special dynamical properties. In this
talk, I will describe some an equidistribution result of small points
in algebraic dynamics. Its proof lies in arakelov geometry.
=== Christian Schnell (UIC) ===
''On the locus of Hodge classes and its generalizations''
The Hodge conjecture is one of several conjectures that attempt to
relate algebraic cycles on a complex algebraic variety to the cohomology
groups of the variety. Among other things, the conjectures predict that
certain loci, a priori defined by holomorphic equations, can actually be
defined by polynomials. Although we do not know how to prove the
conjectures, those predictions have now all been verified. In the talk,
I will survey this story, starting from the famous 1995 paper by
Cattani-Deligne-Kaplan, and ending with recent work by
Brosnan-Pearlstein, Kato-Nakayama-Usui, and myself. I promise to make
the talk accessible to everyone.
=== Omri Sarig (Penn State and Weizmann Institute) ===
''Measure rigidity for dynamical systems on (very) non-compact spaces''
The long term statistical behavior of the orbits of a dynamical system T:X --> X can be studied by analyzing the invariant measures of T. Different measures correspond to different statistics of meandering in space. Usually, there are many different invariant measures, and listing all possible types of statistical behavior is intractable. But sometimes there are few invariant measures and this is possible. In such a case we speak of "measure rigidity".
There are many classical examples of measure rigidity where X is compact.The phenomenology is by now understood. But the situation in the non-compact case is not clear. I will survey some examples (mostly horocycle flows on infinite genus hyperbolic surfaces) which indicate the exotic phenomena which may appear in seriously non-compact situations.
=== Jeff Weiss (Colorado) ===
''Nonequilibrium Statistical Mechanics and Climate Variability''
The natural variability of climate phenomena has significant human impacts but is difficult to model and predict. Natural climate variability self-organizes into well-defined patterns that are poorly understood. Recent theoretical developments in nonequilibrium statistical mechanics cover a class of simple stochastic models that are often used to model climate phenomena: linear Gaussian models which have linear deterministic dynamics and additive Gaussian white noise. The theory for entropy production is developed for linear Gaussian models and applied to observed tropical sea surface temperatures (SST). The results show that tropical SST variability is approximately consistent with fluctuations about a nonequilibrium steady-state. The presence of fluctuations with negative entropy production indicates that tropical SST dynamics can, on a seasonal timescale, be considered as small and fast in a thermodynamic sense. This work demonstrates that nonequilibrium statistical mechanics can address climate-scale phenomena and suggests that other climate phenomena could be similarly addressed by nonequilibrium statistical mechanics.
=== Roger Howe (Yale) ===
''Hibi Rings in Invariant Theory''
Since its beginnings in the early 19th century, invariant theory has provided impetus for advances in algebra,
especially commutative algebra. This continues today, especially with the use of toric deformation
to understand the structure of rings arising in invariant theory and representation theory.
This talk will review the general notion of toric deformation and some of its applications in invariant theory.
In this context, the class of Hibi rings has a particularly elegant theory, and encompasses many of the rings
=== Sylvain Cappell (Courant) ===
''Compact aspherical manifolds whose fundamental groups have center.''
Classical work of Borel had shown that an action of the
circle on a manifold with contractible universal cover yields non-trivial
center in the manifold's fundamental group. In the early 70's, Conner and
Raymond made further deep investigations which led them to conjecture
a converse to Borel's result. We construct counter-examples to this
conjecture, i.e., we exhibit aspherical manifolds (in all dimensions greater
than or equal to 6) which have non-trivial center in their fundamental
groups but no circle actions (and hence no compact Lie group actions). The
constructions involve synthesizing disparate ideas of geometric topology,
geometric group theory and hyperbolic geometry. (This is joint work with
Shmuel Weinberger and Min Yan.)
=== Pham Huu Tiep (Arizona) ===
''Representations of finite simple groups and applications''
In the first part of the talk we will
survey recent results on representations of
finite simple groups. In the second part
we will describe some applications of these results in
group theory and algebraic geometry. In particular,
we will discuss recent proofs by the speaker and
his collaborators of some conjectures of
Katz, Kollar, Larsen, and Ore.
=== Amy Ellis (UW-Madison) ===
''Do algebra students need a reality check? How quantitative reasoning can support function understanding.''
Understanding functions is a critical aspect of algebraic reasoning, and yet
research investigating students? abilities in algebra suggests that students experience
severe difficulty in correctly creating, representing, and analyzing functions. I will
share data from two studies describing how reasoning directly with real-world quantities
can support middle-school students? developing understanding of linear and quadratic
functions. These studies shed new light on the role of concrete problem contexts for
abstract ideas, and suggest an alternate approach to helping students understand
functional relationships in beginning algebra.
=== Max Gunzburger (Florida State) ===
''A nonlocal vector calculus and finite element methods for nonlocal diffusion and mechanics''
We develop a vector calculus for nonlocal operators that mimics the classical differential vector calculus. Included are the definition of nonlocal divergence, gradient, and curl operators and derivations of nonlocal Gauss and Stokes theorems and Green's identities. Through appropriate limiting processes, relations between the nonlocal operators and their differential counterparts are established. The nonlocal calculus is applied to nonlocal diffusion and mechanics problems; in particular, strong and weak formulations of these problems are considered and analyzed, showing, for example, that unlike elliptic partial differential equations, these problems do not necessary result in the smoothing of data. Finally, we briefly consider finite element methods for nonlocal problems, in particular focusing on solutions containing jump discontinuities; in this setting, nonlocal problems can lead to optimally accurate approximations.
=== Alan Weinstein (UC Berkeley) ===
''Symplectic and Quantum Categories''
A useful approach to quantization is to represent classical and quantum mechanics by
categories and then to try to construct functors between them.  The most widely used "classical"
category has symplectic manifolds as its objects and
includes relations as well as mappings among its morphisms.
One possible "quantum" category has as objects rigged Hilbert spaces (also known as Gelfand triples), with
the morphisms being certain partially defined continuous linear mappings.  Another has as objects spaces
of smooth functions or distributions parametrized by a small parameter such as Planck's constant.
In each situation, the main problem is that the composition of certain nice morphisms is no longer
nice.  I will present a general approach to this problem and show how it applies in parallel ways on the classical
and quantum sides.
=== Jane Hawkins (U. North Carolina) ===
''Dynamical properties and parameter space of elliptic functions ''
We iterate the Weierstrass elliptic function, a doubly periodic meromorphic map, in order to understand the dependence of the dynamics on the underlying period lattice.
We use the holomorphic dependence on the classical invariants (g2,g3)  to parametrize the dynamics as well as the lattices themselves; a wide variety of behavior is shown to occur even for a fixed shape of lattice (like square).    In parameter space we see both quadratic-like attracting orbit behavior and pre-pole dynamics, attracting behavior and ergodic or chaotic behavior.
For some lattices  the Julia set of the maps is the entire sphere, while the entire spectrum  of  Mandelbrot-like dynamics that appears in quadratic polynomials also occurs.  The interplay between
the elliptic function properties and the dynamics provides insight into the iteration of meromorphic function and gives rise to some interesting algorithms for viewing Julia sets.
=== Jaroslaw Wlodarczyk (Purdue) ===
''Algebraic Morse Theory and factorization of birational maps''
We develop a Morse-like theory for complex algebraic varieties.
In this theory a Morse function is replaced by a multiplicative group
<i>C<sup>*</sup></i>-action. The critical points
of the Morse function correspond to connected fixed point components.
"Passing through the fixed points" induces some simple birational
transformationscalled blow-ups, blow-downs and flips which are analogous to spherical
modifications.
In classical Morse theory by means of a Morse function we can decompose
the manifold into elementary pieces -- "handles". In  algebraic Morse
theory we decompose a birational map between two smooth complex algebraic
varieties  into a sequence of blow-ups and blow-downs with smooth centers.
=== Olga Holtz (UC Berkeley) ===
''On complexity of linear problems''
This talk will offer a survey of the algebraic complexity theory
for essentially all linear problems of interest (matrix multiplication,
matrix inversion, solving linear systems, QR, LU, Cholesky, SVD
decompositions etc.), along with several glimpses into recent
developments in algebra, algebraic geometry, combinatorics,
numerical analysis, and theoretical computer science that shed
additional light on the true meaning of complexity.
=== Avraham Aizenbud (MIT) ===
''Gelfand pairs and Invariant distributions: Representation Theory and Harmonic Analysis''
The lecture is intended for the general audience and most important to graduate students.
First we will introduce the notion of Gelfand pair. This is an important notion in representation theory. It has
applications to classical representation theory and harmonic analysis. More recently it was also applied to automorphic forms and number theory.
Next, we will discuss the connection of this notion to invariant distributions. We will list some recent results on Gelfand
pairs and demonstrate the tools used to achieve those results on a simple example $(GL_2, GL_1)$. 
If we have time in the end we will discuss the question of when a symmetric pair is a Gelfand pair.

Revision as of 19:00, 26 June 2011

Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Fall 2011

date speaker title host(s)
Sep 30 Alex Kontorovich (Yale) On Zaremba's Conjecture Shamgar
oct 19, Wed Bernd Sturmfels (UC Berkeley) TBA distinguished lecturer
oct 20, Thu Bernd Sturmfels (UC Berkeley) TBA distinguished lecturer
oct 21 Bernd Sturmfels (UC Berkeley) TBA distinguished lecturer
oct 28 Peter Constantin (University of Chicago) TBA distinguished lecturer
oct 31, Mon Peter Constantin (University of Chicago) TBA distinguished lecturer
nov 18 Robert Dudley (University of California, Berkeley) From Gliding Ants to Andean Hummingbirds: The Evolution of Animal Flight Performance Jean-Luc
dec 9 Xinwen Zhu (Harvard University) TBA Tonghai

Spring 2011

date speaker title host(s)
jan 21 Emanuele Macri (University of Bonn) Stability conditions and Bogomolov-type inequalities in higher dimension Andrei Caldararu
jan 28 Marcus Roper (Berkeley) Modeling microbial cooperation Paul Milewski
jan 31, 2:30pm, room 901 Ana-Maria Castravet (Arizona) Hypertrees and moduli spaces of stable rational curves Andrei
feb 4 Xinyi Yuan (Columbia University) Equidistribution in algebraic dynamics Tonghai
feb 11 Christian Schnell (U. Illinois at Chicago) On the locus of Hodge classes and its generalizations Andrei
feb 25 Omri Sarig (Penn State and Weizmann Institute) Measure rigidity for dynamical systems on (very) non-compact spaces Shamgar
mar 4 Jeff Weiss (Colorado) Nonequilibrium Statistical Mechanics and Climate Variability Jean-Luc
mar 11 Roger Howe (Yale) Hibi Rings in Invariant Theory Shamgar
mar 22, Tue Sylvain Cappell (Courant, NYU) Compact aspherical manifolds whose fundamental groups have center Laurentiu
mar 25 Pham Huu Tiep (Arizona) Representations of finite simple groups and applications Martin Isaacs
apr 1 Amy Ellis (Madison) Do algebra students need a reality check? How quantitative reasoning can support function understanding. Steffen
apr 8 Alan Weinstein (Berkeley) Symplectic and Quantum Categories Yong-Geun
apr 15 Max Gunzburger (Florida State) A nonlocal vector calculus and finite element methods for nonlocal diffusion and mechanics James Rossmanith
apr 21, Thu Jane Hawkins (U. North Carolina) Dynamical properties and parameter space of elliptic functions WIMAW (Diane Holcomb)
apr 29 Jaroslaw Wlodarczyk (Purdue) Algebraic Morse Theory and factorization of birational maps Laurentiu
may 2, Mon Olga Holtz (Berkeley) On complexity of linear problems LAA Lecture (Shamgar)
may 6 Rami Aizenbud (MIT) Gelfand pairs and Invariant distributions Shamgar

Abstracts

Alex Kontorovich (Yale)

On Zaremba's Conjecture

It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain.

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