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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 quasiMonte Carlo methods for multidimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40year 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 "lowlying" 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 quasiMonte Carlo methods for multidimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40year 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 "lowlying" 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 Bogomolovtype inequalities in higher dimension''
 
− 
 
−  Stability conditions on a derived category were originally introduced by Bridgeland to give a mathematical foundation for the notion of \Pistability 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 DonaldsonThomas invariants for CalabiYau categories. However, no example of stability condition on a projective CalabiYau 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 BogomolovGieseker 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 intercellular cooperation, by highlighting three paradigmatic examples: 1. The primitive multicellularity of our closestrelated 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.
 
− 
 
−  === AnaMaria Castravet (Arizona) ===
 
−  ''Hypertrees and moduli spaces of stable rational curves''
 
− 
 
−  The GrothendieckKnudsen 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
 
−  FaberFulton 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
 
−  CattaniDeligneKaplan, and ending with recent work by
 
−  BrosnanPearlstein, KatoNakayamaUsui, 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) noncompact 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 noncompact 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 noncompact 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 selforganizes into welldefined 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 steadystate. 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 climatescale 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 nontrivial
 
−  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 counterexamples to this
 
−  conjecture, i.e., we exhibit aspherical manifolds (in all dimensions greater
 
−  than or equal to 6) which have nontrivial 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 (UWMadison) ===
 
−  ''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 realworld quantities
 
−  can support middleschool 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 quadraticlike attracting orbit behavior and prepole 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 Mandelbrotlike 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 Morselike 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 blowups, blowdowns 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 blowups and blowdowns 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.
 