# Mathematics Colloquium

**2013-14 Department of Mathematics Colloquia**### This colloquium program is supported in part by a generous donation from Neil Zimmerman.

### Next Department Colloquium

Information for Colloquium speakers is available

**here**.

*The colloquium runs Thursday afternoons at 4 PM in Phillips Hall 332 with a tea preceding at 3:30 PM in Phillips Hall 330. In Fall 2014, talks can be scheduled from August 28th to December 4th avoiding the Fall break October 16th and Thanksgiving break November 27th. In Spring 2015, talks can be scheduled from February 19th to April 23rd, excepting March 12th due to the Spring break. Colloquium speakers for the 2014-2015 series will include Ken Ono (September 25th), Ingrid Daubechies (October 2nd), Luis Vega (November 20th), Carlos Simpson (March 19th), Carolyn Gordon (March 26th), David Lannes (April 23rd) and more to come.*

*Other special lectures this year have not yet been scheduled but will be posted here when available. The Brauer Lectures for this year will be TBD and the Ergodic theory workshop will run from TBD.*

### *** SPRING 2015 Colloquiua in Order: ***

*March 19th, 2015*

Carlos Simpson

`Bio to come. `

`Colloquium Committee`` `

**ABSTRACT:***The Riemann-Hilbert correspondence between the moduli space of vector bundles with connections, and the character variety or moduli space of representations, has exponential behavior at infinity. The WKB problem of estimating the asymptotics of this map has highest order term a Higgs field. In turn, the main aspect of the Higgs field is its spectral curve. We discuss recent work with Katzarkov, Noll and Pandit about constructing a universal harmonic map to a building with given spectral curve. The singularities of the building are related to the spectral networks recently introduced by Gaiotto, Moore and Neitzke. Our conjectural construction should determine the WKB exponents for the case of $SL_3$.*` `

*March 26th, 2015*

Carolyn Gordon

`Bio to come. `

`Colloquium Committee`` `

**ABSTRACT:***Inverse spectral problems ask how much geometric information is encoded in spectral data. For example, Mark Kac´s question ``Can you hear the shape of a drum?´´ asks whether a plane domain, viewed as a vibrating membrane, is determined by the Dirichlet eigenvalue spectrum of the associated Laplacian, equivalently, by the characteristic frequencies of vibration. The lecture will focus on Kac´s question and its generalization to Riemannian manifolds. We will give a partial survey of both positive and negative results and techniques.*` `

*April 23rd, 2015*

David Lannes

`Bio to come. `

`Colloquium Committee`` `

**ABSTRACT:***We will present in this talk several mathematical issues related to the water waves problem, many of them suggested by phenomena arising in coastal oceanography. After a brief historical survey, we will present several mathematical approaches to the water waves problem (based on complex analysis, harmonic analysis, geometry, etc). We will in particular show how this free boundary problem (the domain in which the solution lives is itself the main unknown of the problem) can be reduced to a more standard evolution equation. We will then comment on some of the mathematical issues arising in the local well posedness theory for these equations. Finally, we will also explain how the solutions thus constructed can be described using simpler equations that are used in oceanography for the numerical simulation of coastal flows and comment on the mathematical challenges raised by specific physical problems, such as wave breaking for instance.*` `

### *** FALL 2014 Colloquiua in Order: ***

*September 25th, 2014*

Ken Ono

`Ken Ono is a professor of mathematics at Emory University. Upon completing his Ph.D. at UCLA under the direction of Professor Basil Gordon in 1993, Prof. Ono spent time at Penn State and Wisconsin before coming to Emory in 2010. He has been awarded an NSF Career/PECASE Award (1998/1999), a Sloan Fellowship (1999), a Packard Fellowship (1999), a Guggenheim Fellowhip (2003) and given numerous distinguished lectures throughout his career. He is regarded as an expert in the areas of number theory and combinatorics. See for instance his works "The web of modularity: arithmetic of the coefficients of modular forms and q-series" from 2004, "Distribution of the partition function modulo m" from 2000 and "Unearthing the visions of a master: harmonic Maass forms and number theory" from 2010. `

`Colloquium Committee`` `

**ABSTRACT:***The speaker will discuss recent work on Moonshine and the Rogers-Ramanujan identities. The Rogers-Ramanujan identities are two peculiar identities which express two infinite product modular forms as number theoretic q-series. These identities give rise to the Rogers-Ramanujan continued fraction, whose values at CM points are algebraic integral units. In recent work with Griffin and Warnaar, the speaker has obtained a comprehensive framework of identities for infinite product modular forms in terms of Hall-Littlewood q-series. This work characterizes those integral units that arise from this theory. In a related direction, the speaker revisits the classical Moonshine Theorem which asserts that the coefficients of the modular j-functions are “dimensions” of virtual characters for the Monster, the largest of the simple sporadic groups. There are 194 irreducible representations of the Monster, and it has been a longstanding open problem to determine the distribution of these representations in Moonshine. In joint work with Griffin and Duncan, the speaker has obtained exact formulas for these distributions.*` `

*October 2nd, 2014*

Ingrid Daubechies

`Ingrid Daubechies is a professor of mathematics at Duke University. Upon completing her Ph.D. in Physics at Vrije Universiteit Brussel under the direction of Professors Jean Reignier and Alex Grossman in 1980, Prof. Daubechies spent time at Bell Labs and Princeton before coming to Duke in 2011. She has been awarded the Steele Prize (1994), a MacArthur Fellowship (1992), the Satter Prize (1997), the Nemmers Prize (2012) and the Benjamin Franklin Medal in Electrical Engineering (2011). She was elected to the National Academy of Sciences and as a Fellow of the IEEE (1993), and delivered the Emmy Noether AMS Lectures (2006) and a plenary ICM talk (1994). She is regarded as an expert in the areas of applied harmonic analysis and analysis. See for instance her works "Ten lectures on wavelets" from 1994, "Orthonormal bases of compactly supported wavelets" from 1990 and "Iteratively solving linear inverse problems under general convex constraints" with G. Teschke and L. Vese from 2008. `

`Colloquium Committee`` `

**ABSTRACT:***The talk describes new distances between pairs of two-dimensional surfaces (embedded in three-dimensional space) that use both local structures and global information in the surfaces. This is work done in collaboration with Yaron Lipman. These are motivated by the need of biological morphologists to compare different phenotypical structures. At present, scientists using physical traits to study evolutionary relationships among living and extinct animals analyze data extracted from carefully defined anatomical correspondence points (landmarks). Identifying and recording these landmarks is time consuming and can be done accurately only by trained morphologists. This necessity renders these studies inaccessible to non-morphologists and causes phenomics to lag behind genomics in elucidating evolutionary patterns. Unlike other algorithms presented for morphological correspondences, our approach does not require any preliminary marking of special features or landmarks by the user. It also differs from other seminal work in computational geometry in that our algorithms are polynomial in nature and thus faster, making pairwise comparisons feasible for significantly larger numbers of digitized surfaces. The approach is illustrated using three datasets representing teeth and different bones of primates and humans; it is shown that it leads to highly accurate results.*` `

*November 20th, 2014*

Luis Vega

`Luis Vega is a professor of mathematics at Universidad Pais Vasco and Scientific Director at the Basque Center for Applied Mathematics. Upon completing his Ph.D. at the Universidad Autonoma de Madrid under the direction of Professor Antonio Cordoba Barba in 1988, Prof. Vega spent time at Chicago and Madrid before coming to Bilbao in 1993. He was an ICM speaker (2006), is a Fellow of the AMS (2012) and the Euskadi Research Prize from the Basque Government (2012). He is regarded as an expert in the areas of harmonic analysis and PDE. See for instance his works "Oscillatory integrals and regularity of dispersive equations" with C. Kenig and G. Ponce from 1992, "Compactness at blow-up time for L^2 solutions of the critical nonlinear Schrödinger equation in 2D" with F. Merle from 1999 and "Bilinear virial identities and applications" with F. Planchon from 2010. `

`Colloquium Committee`` `

**ABSTRACT:***Vortex filaments are mathematical idealizations of some coherent structures of high concentrated vorticity that naturally appear in turbulent fluids. Examples are the classical smoke rings and bathtub vortices. Helicoidal shapes or even logarithmic spirals are also common examples that appear for example in a Francis turbine. We shall present a simplified model, the so called binormal flow, that describes the evolution in 3d of vortex filaments, with special emphasis in filaments with singularities as corners. In the first half of the talk I shall look at the particular case where just one corner is present (joint work with V. Banica). In the second half I shall describe a recent work with F. De la Hoz for regular polygons and its connection with the multi-fractal conjecture proposed by Frisch and Parisi for fully developed turbulence.*` `

### *** SPRING 2014 Colloquiua in Order: ***

*February 27th, 2014*

Dusa McDuff

`Dusa McDuff is a professor of mathematics at Barnard College and Columbia University. Upon completing her Ph.D. at Cambridge under the direction of Professor George A. Reid in 1971, Prof. McDuff spent time at MIT, IAS and the bulk of her career at Stony Brook, before coming to Barnard/Columbia in 2007. She has been awarded the Satter Prize (1991), has been appointed to the National Academy of Sciences (1999), spoken at the ICM (1998), and received the Senior Berwick prize from the London Mathematical Society in 2010. She is regarded as an expert in the areas of symplectic geometry and topology. See for instance her works "J-holomorphic curves and symplectic topology" with D. Salamon from 2004, "The structure of rational and ruled symplectic 4-manifolds" from 1991 and "The geometry of symplectic energy" from 1995. `

`Colloquium Committee`` `

**ABSTRACT:***A survey of current problems and advances, for nonspecialists.*` `

*March 6th, 2014 - Cancelled*

Nicolai Reshetikhin

`Nicolai Reshetikhin is a professor of mathematics at the University of California, Berkeley and the University of Amsterdam. Upon completing his Ph.D. at the Steklov Institute of Mathematics under the direction of Professor Ludvig Fadeev in 1984, Prof. Reshetikhin spent time at the Steklov Institute and Harvard before coming to Berkeley in the early 1990's. He has been awarded the Prize for Young Mathematicians by the St. Petersburg Mathematical Society (1988), given a plenary lecture at the European Mathematical Congress (2008), and spoken at the ICM (1990,2010). He is regarded as an expert in the area of quantum field theory in mathematical physics, with an emphasis on statistical mechanics, geometry and representation theory. See for instance his works "Invariants of 3-manifolds via link polynomials and quantum groups" with V. Turaev from 1991, "Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram" with A. Okounkov from 2004 and "Quantum Calabi-Yau and classical crystals" with A. Okounkov and C. Vafa from 2006. `

`Colloquium Committee`` `

**ABSTRACT:***TBD.*` `

*April 3rd, 2014*

Benjamin Weiss

` Benjamin Weiss is the Miriam and Julius Vinik Professor of Mathematics at the Hebrew University of Jerusalem. He completed his PhD at Princeton University in 1965 under the direction of William Feller. He is considered as a world expert in Dynamical Systems, Probability Theory, Ergodic Theory and Topological Dynamics. His breadth, knowledge and impact on these fields has been very significant. His list of collaborators include A. Connes, H. Furstenberg, and D. Ornstein. His most recent research projects include work with Eli Glasner on large groups in dynamics and Matt Foreman on dynamical systems and descriptive set theory. He was an invited lecturer at the ICM, was elected as a foreign honorary member of the American Academy of Sciences in 2000, gave a noted Plenary Lecture at the IEEE conference on Information Theory in 2001, was a recipient of the Rothschild Prize in Mathematics in 2006 and became an AMS fellow in 2012. One of his doctoral students, E. Lindenstrauss was awarded the Fields medal in 2010. Prof. Weiss' lecture will occur in a special location as he is visiting us as the plenary speaker in the Ergodic theory workshop organized and arranged by Prof. Idriss Assani. Please see the conference web-site for further details. `

`Colloquium Committee`` `

**ABSTRACT:***In the last few years there has been great progress in extending the classical aspects of ergodic theory to actions of non-amenable groups. I will survey a part of this activity and in particular present a new proof of Kolmogorov’s theorem that isomorphic Bernoulli shifts have the same base entropy. This new proof applies almost verbatim to Bernoulli shifts over any sofic group.*` `

*April 10th, 2014*

Alessio Figalli

`Alessio Figalli is a professor of mathematics at the University of Texas at Austin. Upon completing his Ph.D. in Pisa under the direction of Professors Luigi Ambrosio and Cedric Villani in 2007, Prof. Figalli spent time at the Ecole Polytechnique before moving to Austin in 2009, where he now holds the R.L. Moore Chair. He has been awarded the European Mathematical Society Prize (2012), the Peccot-Vimont Prize (2011) and will be an ICM speaker in 2014. He is regarded as an expert in partial differential equations, especially in the areas of optimal transportation, probability and geometry. See for instance his works "Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients" from 2008, "Nearly round spheres look convex" with C. Villani from 2012 and "Regularity of optimal transport maps on multiple products of spheres" with Y.-H. Kim and R. McCann from 2013. `

`Colloquium Committee`` `

**ABSTRACT:***Given a Borel A in R^n of positive measure, one can consider its semisum S=(A+A)/2. It is clear that S contains A, and it is not difficult to prove that they have the same measure if and only if A is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of S is close to the one of A, is A close to his convex hull? More generally, one may consider the semisum of two different sets A and B, in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When n=1, one can approximate a set with finite unions of intervals to translate the problem to the integers Z. In this discrete setting the question becomes a well-studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk, which is intended for a general audience, I will review some results in the one-dimensional discrete setting and show how to answer to the problem in arbitrary dimension.*` `

*April 24th, 2014*

Sylvia Serfaty

`Sylvia Serfaty is a professor of mathematics at the Courant Institute and Paris 6. Upon completing her Ph.D. in Orsay under the direction of Professor Fabrice Bethuel in 1999, Prof. Serfaty spent time in Ecole Normal Cachan before coming to Courant in 2001, where she is now a global distinguished professor. She joined the faculty in Paris-6 in 2009. She has been awarded a Sloan Fellowship (2003), the European Mathematical Society Prize (2004), spoken at the ICM (2006), recently received the Henri Poincare Prize (2012) and last year received the French Academy of Sciences Prize. She is regarded as an expert in the area of partial differential equations and mathematical physics, and has made significant contributions to the theory of Ginzburg-Landau vortex theory. See for instance her works "Vortices in the magnetic Ginzburg-Landau model" with E. Sandier from 2008, "Néel and cross-tie wall energies for planar micromagnetic configurations. A tribute to J. L. Lions" with F. Alouges and T. Riviere from 2004 and "A deterministic-control-based approach to motion by curvature" with R. Kohn from 2006. `

`Colloquium Committee`` `

**ABSTRACT:***We are interested in systems of points with Coulomb interaction. An instance is the classical Coulomb gas, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns, named Abrikosov lattices in physics. In joint works with Etienne Sandier and with Nicolas Rougerie, we studied both systems and derived a "Coulombian renormalized energy". I will present it, examine the question of its minimization and its link with the Abrikosov lattice and weighted Fekete points. I will describe its relation with the statistical mechanics models mentioned above and show how it leads to expecting crystallisation in the low temperature limit.*` `

MIT The new homological algebra: p-complexes and categorification at roots of unity.

Toronto Consider the Icicle.

MIT Instanton Invariants of Knots.

### *** FALL 2013 Colloquiua in Order: ***

*September 19th, 2013*

Benjamin Elias

`Benjamin Elias is a Postdoctoral Fellow in the mathematics department at the Massachusetts Institute of Technology. Upon completing his Ph.D. at Columbia under the direction of Professor Mikhail Khovanov in 2011, Dr. Elias was named an NSF postdoctoral fellow at MIT. He is regarded as an expert in the area of representation theory, specifically with respect to categorification of finite quantum groups. See for instance his pre-print "An approach to categorification of some small quantum groups II" with Y. Qi from 2013 and his recent Masters Class entitled "Soergel bimodules and Kazhdan-Lusztig conjectures" held at Aarhus University with Geordie Williams from the Max Planck Institute. `

`Colloquium Committee`` `

**ABSTRACT:***Homological algebra has been at the foundation of the modern study of topology, representation theory, and geometry. Complexes, homotopies, derived categories, and dg-algebras are powerful tools and are, for many, a way of life. However, in a 2005 paper, M. Khovanov observed that homological algebra as we know it is but a theory attached to the finite (super) Hopf algebra $k[x]/x^2$, and that this whole framework can be generalized to any finite dimensional Hopf algebra! He called the resulting theory "Hopfological algebra," and it was developed further by Y. Qi. One special case holds particular interest: the (non-super) Hopf algebra $k[x]/x^p$ over a field $k$ of characteristic $p$. For this Hopf algebra, one should consider p-complexes, which are like ordinary complexes except that $d^p=0$ instead of $d^2=0$, and the tensor product rules have no signs. It turns out that many interesting algebras appearing in geometric representation theory can be equipped with p-differentials, so that they become p-dg-algebras. Interestingly, the homological shift acts on the Grothendieck group of a p-dg-algebra by multiplication by a $p$-th root of unity (instead of multiplication by -1). In this fashion, we can transform many of the recent categorifications of quantum groups and their representations into categorifications at roots of unity.*` `

*October 10th, 2013*

Stephen Morris

`Stephen Morris is a professor of physics at the University of Toronto. Upon completing his Ph.D. at Toronto in 1991, Prof. Morris spent time as an NSERC postdoctoral Fellow in Santa Barbara before eventually returning back to Toronto where he currently runs the Experimental Nonlinear Phsyics Group. He has been named an a Fellow of the American Physical Society (2012) and a Visiting Fellow at the Isaac Newton Institute in Cambridge (2005). He is regarded as an expert in the areas of nonlinear physics and nonequilibrium pattern formation, in particular with respect to the area of geophysical patterns. See for instance his works "Buoyant plumes and vortex rings in an autocatalytic chemical reaction" with Michael C. Rogers from 2005 and "On the origin and evolution of icicle ripples" with Antony Szu-Han Chen from 2013. `

`Colloquium Committee`` `

**ABSTRACT:***Icicles are harmless and picturesque winter phenomena, familiar to anyone who lives in a cold climate. The shape of an icicle emerges from a subtle feedback between ice formation, which is controlled by the release of latent heat, and the flow of water over the evolving shape. The water flow, in turn, determines how the heat flows. The air around the icicle is also flowing, and all forms of heat transfer are active in the air. Ideal icicles are predicted to have a universal "platonic" shape, independent of growing conditions. In addition, many natural icicles exhibit a ripply shape, which is the result of a morphological instability. The wavelength of the ripples is also remarkably independent of the growing conditions. Similar shape and ripple phenomena are also observed on stalactites, although certain details of their formation differ. We built a laboratory icicle growing machine to explore icicle physics. We learned what it takes to make a platonic icicle and the surprising origin of the ripples. Work done with Antony Szu-Han Chen.*

*papers here.*

*pictures and movies here.*` `

*November 14th, 2013*

Tomasz Mrowka

`Tomasz Mrowka is a professor of mathematics at the Massachusetts Institute of Technology. Upon completing his Ph.D. at Berkeley under the direction of Professors Clifford Taubes and Robion Kirby in 1988, Prof. Mrowka spent time at Stanford and Cal Tech, before coming to MIT in 1996. He has been awarded a Sloan Fellowship (1993), a National Young Investigator Award (1993), a Clay Professorship (1995), the Oswald Veblen prize with P. Kronheimer (2007), the Doob Prize with P. Kronheimer (2011), as well as been elected a member of the American Academy of Arts & Sciences (2007). He is regarded as an expert in the areas of lower dimensional topology and mathematical physics, in particular with respect to applications in geometry. See for instance his works "The genus of embedded surfaces in the projective plane" with P. Kronheimer from 1994, "Monopoles and contact structures" with P. Kronheimer from 1997 and "Monopoles and lens space surgeries" from 2007 with P. Kronheimer, P. Osvath and Z. Szabo. `

`Colloquium Committee`` `

**ABSTRACT:***Instantons, solutions to the Yang-Mills equations, have been used since the earlier 1980's to prove surprising results in low dimensional topology. In the past few years Kronheimer and I have been studying some interesting knot invariants that arise in this context. I will survey some background in gauge theory and low dimensional topology and some of the results that have been obtained using these tools and sketch some interesting new directions.*** **

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### *** SPRING 2013 Colloquiua in Order: ***

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*February 14th, 2013*

Ezra Miller

*February 14th, 2013*

Ezra Miller

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**Duke University**

**Biological applications of geometric statistics on stratified spaces.**

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`Ezra Miller is a professor of mathematics at Duke University as well as Associate Director of SAMSI in the Research Triangle. Upon completing his Ph.D. at Berkeley under the direction of Professor Bernd Sturmfels in 2000, Prof. Miller spent time as an NSF postdoc at MIT and MSRI, then moved to the University of Minnesota where he would become Associate Professor before moving to Duke in 2009. He has been named an NSF CAREER Award winner (2005-2010), as well as winning the University of Minnesota McKnight Land-Grant Professorship/Presidential Fellowship (2005/2007) and a Sloan Dissertation Fellowship (1999). He is regarded as an expert in the areas of combinatorics and algebraic geometry, in particular with applications in statistics, computer science and mathematical biology. See for instance his book "Combinatorial commutative algebra" with B. Sturmfels from 2005 and "Lattice point methods for combinatorial games" with Alan Guo from 2011.`

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`Colloquium Committee`** **

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Stanford University Spectral theory and Weil-Petersson geometry near the ends of moduli space.

Stanford University Minimal submanifolds in differential geometry.

University of Chicago The "General Case".

Massachusetts Institute of Technology Rigidity of generic singularities for mean curvature flow.

Cornell University Schubert Calculus and Puzzles

North Carolina State University Phylogenetic algebraic geometry

University of North Carolina A glimpse at the field of mathematics education research:Implications for mathematicians

Universität Bonn/University of Chicago Fusion algebras and quantum cohomology of Grassmannians.

University of Toronto Rotation of invariant tensors and geometric representation theory.

Williams College Measurable sensitivity for nonsingular and measure-preserving systems.

University of Chicago Invariant Manifolds and Nonlinear Evolution Equations

Duke University Branching Process Models of Cancer

Pomona College Hidden Symmetries in Everyday Operators - Special Colloquium in Honor of Warren Wogen's Retirement

College de France From Circle Diffeomorphisms to Interval Exchange Maps

University of Iowa Energy critical NLS on the exterior domain of a convex obstacle in three dimensions.

University of Illinois, Urbana-Champaign Stability, synchrony and Phase Transition in the Kuramoto Model.

University of California, Irvine

University Paris 7 Analytic Continuation of Polytopes and Wall Crossing

Université de Genève Y-systems and dilogarithm identities

École normale supérieure AdS/CFT correspondence and quantum integrability in four dimensions

Brown University The past, present and future of the Satake isomorphism

Princeton University Moebius function and statistical mechanics

University of Wisconsin, Madison The Pfaffian-Grassmannian derived equivalence

Columbia University Dynamics of nonlinear dispersive systems:

Analysis and Applications

Emory University A Hasse principle over function fields of p-adic curves

Université Paris 11 (Orsay) Random data for wave equations: From Paley and Zygmund to dispersive PDE's

MIT Geometry behind the fundamental lemma - an introduction to Ngo's work

Tata Institute Frobenius splitting: General theory

Columbia University Categorification of quantum groups

**ABSTRACT:***Processing samples of metric branched structures -- such as phylogenetic trees, blood vessels, and fruit fly wing veins -- in structural and evolutionary biology raises fundamental pure mathematical questions in geometric probability and metric geometry of stratified spaces.*` `

*February 28th, 2013*

Rafe Mazzeo

`Rafe Mazzeo is a professor of mathematics at Stanford University. Upon completing his Ph.D. at MIT under the direction of Professor Richard Melrose in 1986, Prof. Mazzeo spent time as an NSF postdoc and Assistant Professor at Stanford, then briefly visited the University of Washington before returning to Stanford as an Associate Professor in 1993. He has been named an NSF Young Investigator Award winner (1992-1997) and a Sloan Fellow (1991-1995), as well as given many invited lectures such as most recently the Yamabe Lectures at Northwestern in 2012. He is regarded as an expert in the areas of geometric analysis and partial differential equations, in particular with respect to the areas of microlocal analysis and differential geometry. See for instance his works "Elliptic theory of differential edge operators. I" from 1991, "Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature" with Richard Melrose from 1987 and "Hodge cohomology of gravitational instantons" from 2004 with T. Hausel and E. Hunsicker.`

`Colloquium Committee`` `

**ABSTRACT:***Moduli spaces of solutions to geometric structures typically carry natural L^2 metrics, the prototype being the Riemann moduli space of conformal structures on a Riemann surface and its Weil-Petersson metric. The associated compactification of this classical moduli space is stratified and the metric structure near the frontier is quite singular. I will report on recent work with Ji, Müller and Vasy concerning analytic properties of the scalar Laplacian associated to this incomplete metric, as well as work with Swoboda concerning the refined asymptotics of the metric itself. This latter result is necessary for an eventual study of index theory on moduli space. The talk will be largely expository, both concerning Weil-Petersson geometry as well as the more purely analytic aspects, which are extensions of classical PDE theorems about elliptic operators with irregular singularities.*` `

*March 7th, 2013*

Rick Schoen

`Richard Schoen is a professor of mathematics at Stanford University. Upon completing his Ph.D. at Stanford under the direction of Professors S.T. Yau and Leon Simon in 1977, Prof. Schoen spent time at UC Berkeley, NYU, the Institute for Advanced Study and UCSD before returning to Stanford as a Professor in 1987. He has received many prizes throughout his career, just some of which include a MacArthur Fellowship (1983), a Sloan Fellowship (1979), the Guggenheim Fellowship (1996), the Bocher Prize (1989), election to the Natioinal Academy of Sciences (1991) and multiple invitations to speak at the International Congress of Mathematicians (1983,1986). He is known for resolving both the Yamabe problem on compact manifolds and the positive mass theorem in General Relativity and is regarded as an expert in the area of differential geometry. See for instance his book "Lectures on differential geometry" with S.T. Yau from 1994, "Conformal deformation of a Riemannian metric to constant scalar curvature" from 1984, "On the proof of the positive mass conjecture in general relativity" with S.T. Yau from 1979 and more recently "Manifolds with 1/4-pinched curvature are space forms" with Simon Brendle from 2009.`

`Colloquium Committee`` `

**ABSTRACT:***The theory of minimal surfaces arose historically from work of J. L. Lagrange and physical observations of J. Plateau almost 200 years ago. Rigorous mathematical theory was developed in the 20th century. In more recent times the theory has found important applications to diverse areas of geometry and relativity. In this talk, which is aimed at a general mathematical audience, we will introduce the subject and describe a few recent applications of the theory.*` `

*March 21st, 2013*

Amie Wilkinson

`Amie Wilkinson is a professor of mathematics at the University of Chicago. Upon completing her Ph.D. at UC Berkeley under the direction of Professor Charles Pugh in 1995, Prof. Wilkinson spent several years at Northwestern University before coming to Chicago in 2011. She is a recipient of the Satter Prize from the AMS, an invited speaker at the ICM in 2010 and has given American Mathematical Society invited addresses in Salt Lake City (2002) and Rio de Janeiro (2007). She is regarded as an expert in the areas of smooth dynamical systems and ergodic theory, in particular with respect to foliation theory. She has recently published papers, jointly with K. Burns, that provided what has been described as a clean and applicable solution to a long-standing problem in stability of partially hyperbolic dynamical systems. She has also played a central role in recent major developments in related areas, including making some fundamental advances in understanding generic behavior of C1 diffeomorphisms. Prof. Wilkinson's lecture will occur in a special location as she is visiting us as the plenary speaker in the Ergodic theory workshop. Please see the conference web-site for further details. `

`Colloquium Committee`` `

**ABSTRACT:***In the early 1930's, the Ergodic theorems of von Neumann and Birkhoff put Boltzmann's Ergodic Hypothesis in mathematical terms, and the natural question was born: is ergodicity the "general case" among conservative dynamical systems? Oxtoby and Ulam tackled this question early on and showed that the answer to this question is "yes" for continuous dynamical systems. The work of Kolmogorov Arnol'd and Moser beginning in the 1950's showed that the answer to this question is "no" for C^infty dynamical systems. I will discuss recent work with Artur Avila and Sylvain Crovisier that addresses what happens for C^1 dynamical systems.*` `

*April 11th, 2013*

Bill Minicozzi

`William P. Minicozzi II is a professor of mathematics at the Massachusetts Institute of Technology. After finishing his Ph.D. at Stanford under Rick Schoen, he spent time the Courant Institute before landing at Johns Hopkins University, where he spent many years before coming to MIT in 2012. He won the Sloan Fellowship in 1998, spoke at the ICM in Madrid in 2006, and most recently won the Oswald Veblen Prize in Geometry with his long-time collaborator Tobias Colding. He is recognized as an expert in Differential Geometry, and in particular for his work on minimal surfaces. See for instance his works "Harmonic functions on manifolds" with T.H. Colding from 1997 and the sequence of works "The space of embedded minimal surfaces of fixed genus in a 3-manifold. I-IV" with T.H. Colding all from 2004. `

`Colloquium Committee`` `

**ABSTRACT:***Mean curvature flow (MCF) is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. Thus, in some sense, the topology is encoded in the singularities. I will discuss work with Toby Colding where we classify generic singularities of MCF, focusing on recent results showing rigidity and applications of this.*` `

### ***FALL 2012 Colloquia in Order:***

*September 20th, 2012 *

Allen Knutson

`Allen Knutson is a professor at Cornell University and an excellent juggler. Upon completing his Ph.D. at MIT (after starting at UC Santa Cruz and first transferring to Princeton) in symplectic geometry under the direction of Professors Victor Guillemin and Lisa Jeffrey in 1996, Prof. Knutson spent time as an NSF Postdoc at Brandeis, from where he moved to Berkeley then San Diego before coming to Cornell in 2008. He has been named a Sloan Fellow (2001-04) and with Terence Tao was awarded the Levi L. Conant Prize by the AMS in 2005. He is regarded as an expert in the areas of combinatorics and algebraic geometry. See for instance his works "The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture." with T. Tao from 1999 and "Gröbner geometry of Schubert polynomials." with E. Miller from 2005. `

`Colloquium Committee`` `

**ABSTRACT:**Let 0 < n_1 < n_2 < ... < n_k < n be a list of numbers, and M the "flag manifold" of chains of subspaces of complex n-space, where the subspaces have those dimensions. M's cohomology has a natural basis of "Schubert classes", and Schubert calculus is concerned with the structure constants of the cohomology product in this basis.

If k=1, then M is a Grassmannian, whose Schubert calculus is well-understood. I'll present a way of computing it using "puzzles" invented by me and Terry Tao, with new results on the geometric meaning of these puzzles.

If k>1, Schubert calculus is much less tightly related to other mathematics, and seems to be less important than a degeneration of it due to Belkale and Kumar. I'll explain how to compute the Belkale-Kumar product using a different set of puzzles. This is joint with Kevin Purbhoo.

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*October 11th, 2012 *

Seth Sullivant

`Seth Sullivant is a professor at North Carolina State University. Upon completing his Ph.D. at Berkeley under the direction of Professor Bernd Sturmfels in 2005, Prof. Sullivant spent time as a Junior Fellow at Harvard, from where he moved to North Carolina State. He has been named a Packard Foundation Fellow (2009-14) and an NSF CAREER Award winner (2010-2015). He is regarded as an expert in the area of algebraic statistics, which has appeared recently to advance the statistical studies of genetics mathematical biology through techniques from areas such as combinatorics and algebraic geometry. See for instance his works "Combinatorial secant varieties" with B. Sturmfels from 2006 and "Phylogenetic algebraic geometry" with N. Eriksson, K. Ranestad and B. Sturmfels from 2005. `

`Colloquium Committee`` `

**ABSTRACT:***The main problem in phylogenetics is to reconstruct evolutionary relationships between collections of species, typically represented by a phylogenetic tree. In the statistical approach to phylogenetics, a probabilistic model of mutation is used to reconstruct the tree that best explains the data (the data consisting of DNA sequences from homologous genes of the extant species). In algebraic statistics, we interpret these statistical models of evolution as geometric objects in a high-dimensional probability simplex. This connection arises because the functions that parametrize these models are polynomials, and hence we can consider statistical models as algebraic varieties. The goal of the talk is to introduce this connection and explain how the algebraic perspective leads to new theoretical advances in phylogenetics, and also provides new research directions in algebraic geometry.*` `

*October 25th, 2012 *

Marta Civil

`Marta Civil is a professor of Education at the University of North Carolina, Chapel Hill. Upon completing her Ph.D. at the University of Illinois under the direction of Professor Peter Braunfeld in 1990, Prof. Civil spent many years as a professor in the mathematics department at the University of Arizona, from where she moved to UNC only in the last year. She has written many articles on mathematics education and is recognized as an international expert in the subject, in particular in the areas of equity, teacher training and parental involvement. Recently, she led a project to establish the NSF funded Center for the Mathematics Education of Latinos/as. `

`Colloquium Committee`` `

**ABSTRACT:***First I will give a broad overview of the field of mathematics education research, including the most recent developments and focusing on some areas that I think are likely to be of interest for mathematicians. Then I will describe some of my research, which addresses equity in mathematics teaching and learning. I will illustrate the importance of context, beliefs about mathematics, and language(s) in understanding and improving the mathematics education of all students.*` `

*November 1st, 2012 *

Catharina Stroppel

` Catharina Stroppel is a professor of mathematics at Universität Bonn, though she is currently visiting the University of Chicago. Upon completing his Ph.D. in Freiburg under the direction of Professor Wolfgang Soergel in 2001, Prof. Stroppel spent time at the University of Leicester, then moved to Aarhus University, University of Glasgow before coming to Bonn in the late 2000's. She is regarded as an expert in the areas of representation theory and combinatorics. See for instance her results "Category O: gradings and translation functors" from 2003 and "Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors" from 2005. `

`Colloquium Committee`` `

**ABSTRACT:***The representation theory of a quantum group at a root of unity provides naturally interesting semisimple tensor categories. Their Grothendieck rings are called Verlinde or fusion algebras. It is well-known that the representation theory of the general linear Lie algebra as well as the cohomology of Grassmannians can be described using the combinatorics of symmetric functions. In the talk I will extend these connections to give a relationship between fusion algebras and quantum cohomology that was first observed and proved by Witten with mathematical physics methods.*` `

*November 15th, 2012 *

Joel Kamnitzer

`Joel Kamnitzer is an Assistant Professor of mathematics at the University of Toronto. Upon completing his Ph.D. at Berkeley under the direction of Professor Alan Knutson in 2005, Prof. Kamnitzer was named a 5 year fellow of the American Institute of Mathematics and spent time at MIT, Berkeley and MSRI before landing in Toronto in 2008. He has been named a Sloan Fellow (2012-2014) and won the Andre Aisenstadt Prize for outstanding research done by a young Canadian mathematician. He is regarded as an expert in the area of geometric representation theory, in particular with regard to complex reductive groups. See for instance his work "Crystals and coboundary categories" with Andre Henriques in 2006, "Mirković-Vilonen cycles and polytopes" from 2010 and "The crystal structure on the set of Mirković-Vilonen polytopes" from 2007. `

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**ABSTRACT:***We study the action of rotation on invariant spaces in tensor products of minuscule representations. We define a combinatorial notion of rotation of minuscule Littelmann paths. Using affine Grassmannians, we show that this rotation action is realized geometrically as rotation of components of Satake fibres. As a consequence we have a basis for invariant spaces which is permuted by rotation. Finally, we diagonalize the rotation operator by showing that its eigenspaces are given by intersection homology of quiver varieties. As a consequence, we generalize Rhoades’ work on the cyclic sieving phenomenon.*` `

*November 29th, 2012 *

Cesar Silva

`Cesar Silva is the Hagey Family Professor of Mathematics at Williams College, where he has held positions since 1984. He received his Ph.D. from the University of Rochester under the supervision of Dorothy Maharam and was hired by Williams immediately afterwards. He received the distinguished Hagey Family Professorship in 2007. He has held visiting positions at a number of universities including Maryland, Toronto, and Paris Sud. His B.S. is from Pontifica Universisdad Catolica in his native country of Peru. Silva is a co-founder of and regularly supervises undergraduate research in Williams’ successful SMALL Undergraduate Research Project funded by the NSF, in which he has participated for many years. These collaborations have resulted in more than 15 refereed journal articles in ergodic theory and p-adic dynamics, coauthored by Silva and his REU undergraduates. He has published a book in ergodic theory in addition to his many research articles, and is regarded as one of the leading experts in nonsingular dynamical systems, systems with natural measures that fail to be preserved over time. `

`Colloquium Committee`` `

**ABSTRACT:***The notion of sensitive dependence on initial conditions has been extensively studied in the context of topological dynamics. Recently, sensitivity has been defined for measurable dynamical systems. We present various notions of measurable sensitivity and explore their connections with other notions in ergodic theory. We discuss a classification theorem stating that a nonsingular conservative and ergodic dynamical systems on a standard space must be either W-measurably sensitive, or isomorphic mod 0 to a minimal uniformly rigid isometry, an analogue to similar theorems in topological dynamics. Also, a conservative and ergodic infinite measure-preserving transformation cannot be strong measurably sensitive. We also discuss restricted measurable sensitivity and its connection with entropy, and Li-Yorke measurable sensitivity.*` `

### *** PREVIOUS COLLOQUIA: ***

### *** SPRING 2012: ***

*February 23, 2012 *

Wilhelm Schlag

`Wilhelm Schlag is a professor at the University of Chicago. Upon completing his Ph.D. at Cal Tech under the direction of Professor Thomas Wolff in 1996, Prof. Schlag spent time at the Institute for Advanced Study and Princeton before returning to Cal Tech from 2001-2005, from where he moved to Chicago. He has been named a Sloan Fellow (2001-03), a Guggenheim Fellow (2009-10) and twice been a plenary speaker at the International Congress of Mathematical Physics (2004,2012). He is regarded as an expert in the areas of harmonic analysis, spectral theory and nonlinear partial differential equations, though he has also done some work in probability theory. See for instance his works "Time decay for solutions of Schrödinger equations with rough and time-dependent potentials" with I. Rodnianski from 2004, "Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions" with Yuval Peres from 2000 and his groundbreaking work "Stable manifolds for an orbitally unstable nonlinear Schrödinger equation" that appeared in 2009. `

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**ABSTRACT:***We will review recent work on the role that center-stable manifolds play in the study of dispersive unstable evolution equations. More precisely, by means of the radial cubic nonlinear Klein-Gordon equation we shall exhibit a mechanism in which the ground state soliton generates a center-stable manifold which separates a region of data leading to finite time blowup from another where solutions scatter to a free wave in forward time. This is joint work with Kenji Nakanishi from Kyoto University, Japan.*` `

*March 1, 2012 *

Rick Durrett

`Richard T. Durrett is a professor at Duke University. Upon completing his Ph.D. at Stanford under the direction of Professor Donald Iglehart in 1976, Prof. Durrett ended up at Cornell University from 1985-2010, where he founded and still runs the well-respected Cornell Probability Summer School. He is most known for his many contributions to the fields of probability and statistics, in particular those with applications in ecology and genetics. Prof. Durrett has both produced an incredible amount of new scientific work and written many books about his field, for instance his heavily cited book "Probability: theory and examples" most recently published in 1996 and his highly influential paper on mathematical biology "The importance of being discrete (and spatial)" from 1994 among many others. His work in these areas and others led to him being elected as a member of the National Academy of Sciences in 2007, where the citation particularly references the importance of his work to develop "mathematical models to study the evolution of microsatellites, impacts of selective sweeps on genetic variation, genome rearrangement, gene duplication, and gene regulation."`

`Colloquium Committee`` `

**ABSTRACT:***It is common to use a multitype branching process to model the accumulation of mutations that leads to cancer progression, metastasis, and resistance to treatment. In this talk I will describe results about the time until the first type k (cell with k mutations) and the growth of the type k population obtained in joint work with Stephen Moseley, and their use in evaluating possible screening strategies for ovarian cancer, work in progress with Duke undergraduate Kaveh Danesh, and with Evan Myers and Laura Havriletsky in Obstetrics and Gynecology at the Duke Medical Center. The point process representation of the limit, which is a one-sided stable law, together with results on that topic and on the Poisson-Dirichlet distribution lead to remarkable explicit formulas for Simpson's index and the size of the largest clone. These results are important in understanding tumor diversity which can present serious obstacles to treatment. The last topic is joint work with Jasmine Foo, Kevin Leder, John Mayberry, and Franziska Michor.*` `

*March 15, 2012 *

Stephan Garcia

`Stephan Garcia is an Associate Professor of Mathematics at Pomona College. After completing his Ph. D at Berkeley in 2003 under the direction of Don Sarason, he took a visiting position at Santa Barbara prior to accepting his current position in 2006. Most of Stephan’s research has been in Operator Theory and Linear Algebra. He has over 30 published papers and has had NSF funding since 2006. He has made important contributions in Operator Theory, both on function spaces and on abstract Hilbert space. Stephan also has an exceptional record as a teacher and expositor. He has earned campus-wide teaching awards at Berkeley, Santa Barbara, and Pomona.`

`Colloquium Committee`` `

**ABSTRACT:***What do a 2 by 2 matrix, a Jordan block, a complex Hankel matrix, the adjacency matrix of a graph, integration, and the Fourier transform have in common? They each enjoy hidden symmetries which are part of a general theory, only recently developed by the speaker and his collaborators (in particular, we highlight some of the key contributions of Warren Wogen). This talk should be accessible to graduate students and advanced undergraduates.*` `

*March 22, 2012 *

Jean-Christophe Yoccoz

`Jean-Christophe Yoccoz is a professor at the prestigious College de France in Paris. Upon completing his Doctorat d'Etat under the direction of Professor Michael Herman in 1985, Prof. Yoccoz quickly established himself as a leading expert in dynamical systems, which he continues to be to this day. He has produced groundbreaking results in the theory of small divisors and the structural and dynamic stability of dynamical systems, let us mention "Siegel theorem, Bryuno numbers and quadratic polynomials" from 1995 and "Stable intersections of regular Cantor sets with large Hausdorff dimensions" from 2002 with C. de A. Moreira among others. His work in dynamical systems led to the Salem Prize in 1988 and subsequently the Fields medal in 1994. He has been twice an invited speaker at the ICM (plenary speaker in 1994). He is a member of the French Academy of Sciences (1994) and the Brazilian Academy of Sciences (1994). Other awards include the IBM prize of mathematics in 1985 and the Jaffe prize of the Academy of Sciences in 1991. Prof. Yoccoz's lecture will occur in a special location as he is visiting us as the plenary speaker in the Ergodic theory workshop. Please see the conference web-site for further details. `

`Colloquium Committee`` `

**ABSTRACT:***The class of standard interval exchange maps is a very natural extension of that of circle rotations. In the same way, the class of generalized interval exchange maps extends that of circle diffeomorphisms. After reviewing some of the highpoints in the study of circle diffeomorphisms (rotation number, wandering intervals, KAM-theoretical results), we will present some results on interval exchange maps which emphasize both the similarities and the differences with the circle case.*` `

*March 29, 2012 *

Xiaoyi Zhang

`Xiaoyi Zhang is an Assistant Professor at the University of Iowa. Upon completing her Ph.D. at the China Academy of Engineering Physics in 2003, Prof. Zhang spent time at MSRI and the Institute for Advanced Study at Princeton before starting her position at Iowa in 2009. She has been named a Sloan Fellow (2010-2013) and a Von Neumann Early Career Fellow at IAS. She has produced extremely sharp results in the areas of harmonic analysis and nonlinear dispersive partial differential equations, as well as doing some work in fluid dynamics. See for instance her works "Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions" with M. Visan and T. Tao from 2007, as well as "The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher" with R. Killip and M. Visan from 2008. `

`Colloquium Committee`` `

**ABSTRACT:***We consider the defocusing energy critical NLS in three dimensions. In the whole space case, the problem was extensively studied and it was shown that the global solution exists and scatters. For the problem on the exterior domain with Dirichlet boundary condition, it is still open whether the global wellposedness and scattering hold true. In this talk, I will introduce our recent result toward this open problem. This talk is based on the joint work with Rowan Killip and Monica Visan.*` `

*April 12, 2012 *

Jared Bronski

`Jared Bronski is a professor at the University of Illinois, Urbana-Champaign. Upon completing his Ph.D. at Princeton with David McLaughlin in 1994, Prof. Bronski was a Szëgo postdoc at Stanford and a postdoctoral fellow at the IMA in Minneapolis before arriving in Illinois. An expert in the intersection of spectral theory, dynamical systems and semiclassical analysis, Prof. Bronski's work has helped rigorously develop stability theory in nonlinear partial differential equations with applications to the mathematical studies of soliton theory in optics and Bose-Einstein condensates, as well as other Hamiltonian models in mathematical physics. He has also done work on uncertainty and randomness. In addition, Prof. Bronski won the Sloan Research Fellowship in 2001 and currently serves as an editor of the journal Physica D, which has published many important developments in mathematical physics. Among his notable works are "Nonlinear scattering and analyticity of solitons" in 1998, "Soliton dynamics in a potential" with R. Gerrard in 2000, "Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem" in 1996 and "Uncertainty estimates and L2 bounds for the Kuramoto-Sivashinsky equation" with T. Gambill in 2006.`

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**ABSTRACT:***The phenomenon of synchronization, the tendency of weakly coupled oscillators to frequency lock, was first observed by Huygens in his "Horologium Oscillatorium", and similar phenomenon have been observed in:*

*1. The synchronization of the flashing of fireflies.*

*2. The unstable oscillations of the Millenium Bridge.*

*3. The synchronization of the cardiac pacemaker.*

*4. Oscillations in large power transmission networks.*

*The Kuramoto model*

*\frac{d\theta_i}{dt} = \omega_i +\gamma \sum_{j=1}^N \sin(\theta_j-\theta_i)*

*is a canonical model for the synchronization of weakly coupled oscillators. We develop an index theorem for stationary solutions of this model, allowing us to precisely count the dimensions of the unstable manifolds. This allows us to prove rigorously the existence of a phase transition in this model. Along the way we will encounter:*

*1. A high dimensional polytope, and an associated lattice.*

*2. An unusual combinatorial identity.*

*3. A couple of strange norms.*

*4. An extreme value statistic.*` `

*April 19, 2012 *

Gunther Uhlmann

`Cloaking via Transformation Optics`

`Gunther Uhlmann is currently a professor at both the University of California, Irvine and the University of Washington. Upon completing his Ph.D. under the direction of Professor Victor Guillemin at MIT in 1976, Prof. Uhlmann did postdoctoral work at the Courant Institute and was an assistant professor at MIT before moving to the University of Washington, where he has been from 1984-present. In 2010, he took up the Excellence in Teaching Chair in Mathematics Professorship at UC-Irvine. Prof. Uhlmann has won numerous awards for his work on inverse problems and partial differential equations using microlocal analysis, including the Kleinman Prize (2011), Bocher Prize (2011), election to the American Academy of Arts and Sciences (2009), a Guggenheim Fellowship (2001-02) and a SIAM Fellowship (2010). He has delivered many plenary talks throughout the world, among them at several ICIAM and AMS meetings. He was also an invited ICM speaker in Berlin in 1998. Among his notable works are "A global uniqueness theorem for an inverse boundary value problem" with J. Sylvester in 1987, "The Calderón problem with partial data" with C. Kenig and J. Sjöstrand in 2007 and a groundbreaking work on cloaking "Full-wave invisibility of active devices at all frequencies" with A. Greenleaf, Y. Kurylev and M. Lassis in 2007. `

`Colloquium Committee`` `

**ABSTRACT:***We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and quantum waves. We emphasize the method of transformation optics. For the case of electromagnetic waves, Maxwell's equations have transformation laws that allow for design of electromagnetic materials that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. We recount some of the history of the subject and discuss some of the mathematical issues involved.*` `

### *** FALL 2011: ***

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*December 1, 2011 *

Alexei Oblomkov

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`University of Massachusetts Amherst`

`Khovanov-Rozansky invariants of knots, plane curve singularities and Cherednik algebras`

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`Alexei Oblomkov is a promising young mathematician working at the intersection of algebraic geometry, representation theory, 'quantum' topology and physics. He got his Ph.D. at M.I.T. under the direction of Pavel Etingof. Currently he is a Sloan Fellow at the UMass Amherst. He is one of the best experts in double affine Hecke algebras; his papers on quantization of cubic and del Pezzo surfaces (with Etingof and Rains) are among the best achievements of this theory. He is engaded now in the study of cohomology of Hilbert schemes in collaboration with Maulik, Okounkov and Pandharipande. It led him towards exploring the relation between Hilbert schemes of a singular curve and HOMFLY polynomial of the torus knot. Working with Shende, Oblomkov extends the relation between the Euler characteristic of a Hilbert scheme and the HOMFLY polynomial to that between the cohomology of Hilbert schemes and the triply graded homology of torus knots. This new theory attracts a lot of attention now of mathematicians and physicists. `

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`Colloquium Committee`` `

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**ABSTRACT:***In 1985 V. Jones discovered a new remarkable knot invariant which revolutionized the field of low-dimensional topology.***Jones's invariant**was interpreted as Euler characteristics of some natural complex by M. Khovanov in 2000; the Poincare polynomial of this complex is now known under the name**Khovanov invariant.**Later Khovanov and Rozansky added to the picture a new invariant called the**Khovanv-Rozansky invariant**; it is related to the Khovanov invariant by means of the spectral sequence discovered by J. Rasmussen. Lead by the recent discoveries in enumerative geometry, speaker together with V. Shende and J. Rasmussen proposed a formula for the Khovanov-Rozansky invariant of the link of a plane curve singularity as generating functions of some natural blow-ups of symmetric powers of the curves. In the case of curve x^{p}=y^{q}, the corresponding knot is the torus knot and its Khovanov-Rozansky homology are expected to carry natural action of the spherical subalgebra of the rational Cherednik algebra.*November 3, 2011 *

Ken McLaughlin

`University of Arizona`

`Oscillatory phenomenon in a scaling limit for the periodic linear Schrödinger equation``Ken McLaughlin is one of the leading experts in random matrix theory, approximation theory, orthogonal polynomials and the analytical theory of integrable systems. He contributed significantly to the intricate ties between these fields, which have been identified and exploited with very interesting results. Among quite a few his major invited lectures, he was a speaker of the 14th Congress on Mathematical Physics (Lisbon, 2003); Percy Dieft reported his joint work with Kriecherbauer, McLaughlin, Venakides and Zhou at the ICM-98 in Berlin (an invited lecture). Ken is currently working on universality in partial differential equations and random matrix theory. The lecture, centering on rough periodic solutions to the linear (yes, linear!) Schrödinger equation, will also discuss the history of the Gibbs phenomenon, the strange correspondence and slow pace of the mail which led to Michelson not receiving credit where credit was due, the Talbot effect and the precision of experimentation during the 1840s. Ken moved to Arizona University 7 years ago from our dept, one of the best our researches and a great colleague; we are very happy to see him! `

`Colloquium Committee`

**ABSTRACT:***In the 1830s Henry Fox Talbot discovered a self-imaging phenomenon in coherent illumination of a periodic diffraction grating. Since then, studies of the Talbot effect (as it is called) have themselves recurred frequently. One such incarnation was initiated by Michael Berry; eventually some interesting fractal dimension results were proven by subsequent researchers. I will explain some of these connections, and some recent developments concerning oscillatory phenomena reminiscent of Gibbs' phenomenon. In passing I will explain some possible misconceptions concerning the discovery of Gibbs' phenomenon. The main mathematical connection is that the Fourier series solution is quite directly related to exponential sums appearing in classical analytic number theory. Joint work with Nigel Pitt.**October 27, 2011 *

Boris Khesin

`University of Toronto`

`Optimal Transport and Geodesics on Diffeomorphism Groups`

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`Boris Khesin is a well-known scientist with truly universal interests in mathematics, from hydrodynamics and global analysis to the infinite-dimensional Lie theory; not very surprising for a student of Vladimir Arnold! Indisputably, he is one of the best lecturers in mathematics with great international connections; France and Germany are among his favorite places when working abroad (10+7 visiting positions there from 1992). He was a Clay fellow and an invited speaker of Royal Society (London) among other honors; 3 Canadian awards must be mentioned: Premier Research Excellence Award (Ontario), the McLean Award (University of Toronto) and the Andre-Aisenstadt Mathematics Prize (Montreal). `

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**ABSTRACT:***In 1965 Vladimir Arnold described the hydrodynamical Euler equation as an equation of geodesics on the group of volume-preserving diffeomorphisms with respect to the right-invariant L*^{2}metric. In the talk we describe the corresponding extrinsic geometry by regarding the group of volume-preserving diffeomorphisms as a Riemannian submanifold in the group of all diffeomorphisms. Various applications of the latter approach include a relation to L^{2}and H^{1}optimal mass transport problems, a non-holonomic version of the Moser theorem, and integrable PDEs with several space variables.*October 13, 2011 *

Michele Vergne

`Michele Vergne is a prominent scholar, a member of the French Academy of Sciences and the American Academy of Arts and Sciences. Académie des Sciences is one of the oldest and the most respected such bodies in the world (only 150 full members and 120 foreign associates); Prix Ampère preceded her election in 1998. She was a professor at MIT during 1977-86 and Directeur de Recherches, CNRS from 1981 till 2008 (promoted to Classe Exceptionelle in 1997). She was a plenary speaker at International and European Congresses of Mathematicians (2006, 1992), delivered Emmy Noether Lecture series in Goettingen (2008) among many others talks and lectures. Her fundamental contributions to geometry, representation theory and harmonic analysis are well known, including important books "The Weil representation, Maslov index and theta series" (with G. Leon) and "Heat Kernels and Dirac Operators" (with Berline and Getzler). `

`Colloquium Committee`

**ABSTRACT:***The lecture will be a report on work with Nicole Berline. Following Varchenko, we consider the "analytic continuation of a polytope". We compute explicitly the shape of the analytic continuation when crossing a wall. I will also state an analogous wall crossing formula for Duistermaat-Heckman in the context of Hamiltonian geometry.**September 8, 2011 *

Andras Szenes

`The lecturer is a professor of mathematics from the Geneve Univesity, a well-known specialist in a variety of fields including topology, algebraic geometry, mathematical physics and related representation theory. His lecture will be about one of the most interesting interdisciplinary topics in mathematical physics and beyond, the Y-system, which is connected, among quite a few other lines, with the Hirota identities, the Zamolodchikovs conjecture, proven via the cluster algebras, and the AdS-CFT correspondence (the previous colloquium by Vladimir Kazakov). We are sure that the lecture will be understandable and helpful. `

`Colloquium Committee`**ABSTRACT:** *In 1991, Zamolodchikov discovered an intriguing rational recursion called the Y-system. The recursion has some remarkable periodicity properties, and relations to number theory, representation theory, and several other fields of Mathematics. The Y-systems gave rise to identities among the values of the dilogarithm function, and also served as one of the first examples of an important new notion of modern algebra: the cluster algebras. In this talk, we will review recent results and developments in the field. *

### *** SPRING 2011: ***

*April 21, 2011 *

Vladimir Kazakov

`The lecturer, a brilliant physicist, is one of the best experts in Matrix Models and integrability of physics theories. He will speak about the breakthrough development called the AdS/CFT correspondence , which resulted in the integrability of important "really" 4-dimensional Super-Yang Mills theories (in the planar limit). Counting zillions of Feynman graphs needed for finding anomalous dimensions and similar important physics quantaties can be replaced by an essentially algebraic and very elegant theory based on the quantum Hirota identities, Bethe equations and the Y-system for certain super Lie algebras of non-compact type. The lecture is expected understandable for mathematicians; certain experience with classical "integrability" is desirable. `

`Colloquium Committee`**ABSTRACT:** *I will speak about the recently found solution of the problem of spectrum of anomalous dimensions in a four dimensional conformal field theory - the Yang-Mills theory with N=4 super-symmetries at any strength of the interaction. Due to the AdS/CFT correspondence, this gauge theory can be mapped to a two-dimensional string sigma model, which appeared to be integrable classically, as well as quantum mechanically. This allowed to apply the full set of the integrability tools, such as the thermodynamic Bethe ansatz and the so called Y-system (equivalent to the famous Hirota bi-linear finite difference equation). The lowest anomalous dimension, for the Konishi operator, was successfully found by numeric solution of the resulting equations, and their analytic analysis allowed to easily reproduce the results of summation of Feynman diagrams. *

*March 31, 2011 *

Alexander Braverman

`Alexander Braverman is one of the best specialists in the homogeneous spaces of loop groups, the geometric Langlands program and related algebraic geometry. His interests are truly universal, in the range from quantum cohomology to double arithmetic. Whatever his current research field is, he always obtains advanced sharp results. Undoubtedly, he is one of the best lecturers in this highly developed and quickly progressing area. His lecture will be devoted to the Satake isomorphism, which is one of the key topics in the classical and modern representation theory. `

`Colloquium Committee`**ABSTRACT:** *In the first part of the talk I am going to recall the formulation of the so called Satake isomorphism for a group G over a p-adic field K and explain why and how it can be regarded as the starting point for the notion of automorphic L-function and for the Langlands conjectures. I will also explain some interesting combinatorics around the Satake isomorphism. In the second part of the talk I am going to describe the generalization of the Satake isomorphism to (infinite-dimensional) loop groups groups as well as a (much more involved) generalization of the above combinatorics (joint work with David Kazhdan and Manish Patnaik). *

*March 17, 2011 *

Yakov Sinai

`The speaker is known virtually to everyone in the world of professional mathematics and physics. His major groundbreaking results are in the theory of dynamical systems, mathematical physics and in probability theory; Kolmogorov-Sinai entropy, Sinai's billiards, Sinai's random walk, Sinai-Ruelle-Bowen measures, just to mention some. Among his awards are the Boltzmann Medal (1986), Dirac Medal (1992), the Wolf Prize in Mathematics (1997), most recently, his election to the Royal Society of London (2009) and Henri Poincare Prize (2009). We are honored by his visit initiated by the Ergodic Theory Workshop (March 17-20). The talk will be on a new approach proposed in a work by Cellarosi and Sinai that allows to study random properties of the classical Moebius function. `

`Colloquium Committee`*February 24, 2011 *

Andrei Caldararu

`Andrei Caldararu is a relatively young mathematician; he got his PhD from Cornell University in 2000. However he is already a recognized expert in algebraic geometry, including the algebraic aspects of string theory and homological algebra. Many of his works fit into the derived algebraic geometry, a new branch of commutative and non-commutative algebraic geometry that provides powerful tools for the deformation theory and intersection theory. Maxim Kontsevich, Carlos Simpson and, recently, Jacob Lurie are among the major contributors to this field. It brings the idea of homotopy equivalence into the core of scheme theory, with the goal of explaining such phenomena as stacks that really should have a tangent space that is not a vector space, but a complex of vector spaces (up to homotopy). Understanding derived categories of sheaves, fundamental invariants of complex manifolds, is one of the key objectives of Andrei Caldararu's research. `

`Colloquium Committee`**ABSTRACT:** *String theory relates certain seemingly different manifolds through a relationship called mirror symmetry. Discovered about 25 years ago, this story is still very mysterious from a mathematical point of view. Despite the name, mirror symmetry is not entirely symmetric -- several distinct spaces can be mirrors to a given one. When this happens it is expected that certain invariants of these "double mirrors" match up. For a long time the only known examples of double mirrors arose through a simple construction called a flop, which led to the conjecture that this would be a general phenomenon. In joint work with Lev Borisov we constructed the first counterexample to this, which I shall present. Explicitly, I shall construct two Calabi-Yau threefolds which are not related by flops, but are derived equivalent, and therefore are expected to arise through a double mirror construction. The talk will be accessible to a wide audience, in particular to graduate students. *

*February 17, 2011 *

Michael Weinstein

Analysis and Applications

`Michael Weinstein is a professor in the Applied Physics and Mathematics Department at Columbia University. He has rare deep insights both into the physically interesting models to study, as well as the mathematical techniques required to analyze them. His contributions to the theory of nonlinear waves are well known, including modulational stability for solitary wave solutions of the equations like KdV and NLS, dispersive PDE, cloaking, fluid dynamics and geophysics, in addition to holding two patents related to theoretical optical engineering devices. His lecture will introduce problems from nonlinear waves and discuss some of the mathematical ideas needed to understand the nature of the fascinating underlying nonlinear dynamics. `

`Colloquium Committee`**ABSTRACT:** *This talk will overview results and open problems on the dynamics of coherent structures for certain nonlinear dispersive PDEs, a class of infinite dimensional Hamiltonian systems. Many such systems have spatially localized solutions, describing coherent structures (hydrodynamic, electrodynamic, quantum...) such as soliton pulses or vortices, with remarkable stability properties. The general PDE dynamics can be viewed in terms of the nonlinear interaction of such coherent structures with linear dispersive waves. The infinite-time behavior is the subject of nonlinear scattering. Intermediate but very long-time transients, e.g. metastable states, play an important role in the analysis and a central role in applications. A detailed understanding involves ideas from dynamical systems (Hamiltonian theory of normal forms,...) and scattering theory (wave operators, non-self adjoint spectral theory,...), variational and harmonic analysis. We will consider these questions in the context of the nonlinear Schroedinger - Gross Pitaevskii equation, a class of PDEs having wide applications to classical and quantum systems. Applications to the control of soliton-like states in nonlinear optical and quantum systems will be discussed. *

*February 10, 2011 *

Raman Parimala

`Raman Rarimala, a speaker at the latest ICM in Hyderabad and the one in 1994, is a fellow of all three Indian academies of science. She received Srinivasa Ramanujan Birth Centenary Award in 2003 and the 2005 prize in mathematics from the Academy of Sciences for the Developing World (TWAS) among her other awards. She was the first woman working in the fields of mathematics and physics honored with the TWAS prize during its 20 years of history for her work on the quadratic analogue of Serre's conjecture, the triviality of principal homogeneous spaces of classical groups over fields of cohomological dimensions 2 and the m-invariant of p-adic function fields. In 2005, Raman Parimala was appointed the Asa Griggs Candler Professor of Mathematics at Emory University in Atlanta. `

`Colloquium Committee`**ABSTRACT:** *Hasse-Minkowski's theorem says that a quadratic form over a number field is isotropic if it is isotropic over completions at all places of the number field. One could look for a Hasse principle in the function field setting with respect to all discrete valuations of the function field. This is particularly interesting for Q _{p}(t); a Hasse principle for isotropy of quadratic forms would lead to the fact: every quadratic form over Q_{p}(t) in at least nine variables has a nontrivial zero. We explain the Hasse principle over function fields of nondyadic p-adic curves with respect to a set of divisorial discrete valuations of the field. *

*January 27, 2011 *

Nicolas Burq

`Nicolas Burq, a member of Institut universitaire de France, made fundamental contributions to a wide range of modern problems in analysis and physics, from scattering theory and eigenfunction properties to control theory and nonlinear dispersive equations. His works on dispersive estimates for the Schrödinger equation on manifolds, eigenfunction restriction properties, defect measures in various geometric settings, scattering theory and resonances are well known. Impressive diversity of his research interests is recognized; three (!) times he spoke at Séminaire Bourbaki. His lecture will be based on joint works with N. Tzvetkov (Université de Cergy). `

`Colloquium Committee`**ABSTRACT:** *The strating point of my talk will be a result by Paley and Zygmund (1930): Consider any l^{2} sequence, (α_{n}), nÎN. Then of course Parseval's theorem shows that the trigonometric series on the torus ∑_{n} α_{n}e^{inx} is in L^{2}(T). It is also known (and quite easy to prove) that generically (for the l^{2} topology, i.e. for (α_{n}) in a dense Gδ set), the series is in no L^{p}(T) space (p>2). However, Paley-Zygmund's theorem ensures that if one simply changes the signs of the coefficients α_{n} randomly and independently, then the series is almost surely in every L^{p}(T) space! This striking phenomenon was later widely studied in the context of harmonic analysis, giving rise to a huge amount of results (works by Kahane, Pisier and many others). However, surprisingly, this phenomenon was until recently not exploited in the context of partial differential equations. The purpose of this talk is precisely to present some occurences of this phenomenon in non linear partial differential equations. For simplicity, I will focus my talk on one of the simplest models: the cubic non linear wave equation, and give examples where it is possible to exhibit much better behaviour for solutions to such a PDE when the initial data are randomly chosen than the behaviour predicted by the classical deterministic theory. *

### *** FALL 2011: ***

*November 11, 2010 *

Zhiwei Yun

`This colloquium lecture will be mainly devoted to the Fundamental Lemma in finite characteristic, an important part of the Langlands program, verified recently by Bao-Chau Ngo in complete generality. It resisted all attempts to prove it for almost 30 years. The speaker is a (very) young mathematician, an absolute winner of the International Mathematics Olympiad in 2000, a recognized expert in a bunch of highly regrarded directions around the Fundamental Lemma. We are delighted that he agreed to deliver an introductory lecture for us on this topic, which attracts a lot of attention (to be continued at "Geometric Methods in RT", Nov. 12, 4:15pm, Ph367). `

`Colloquium Committee`**ABSTRACT:** *This summer, Bao-Chau Ngo got a Fields medal for his proof of the Fundamental Lemma, FL. I will try to explain what FL is about and the idea of his proof from a geometric point of view. The FL as formulated by Langlands and Shelstad was an identity between integrals. We will first see that these integrals actually count lattices. Then, in the function field case, one can "globalize" the picture to count vector bundles. Towards the end of the talk, Hitchin moduli spaces will show up (hopefully) naturally. *

*November 4, 2010 *

Vikram Mehta

`Vikram Mehta is a professor of Tata Institute of Fundamental Research. Jointly with A. Ramanathan, he introduced the notion of Frobenius Splitting, which is a tool of fundamental importance in algebraic geometry and representation theory. It guaranties that the cohomology ( i>0) vanishes for all ample line bundles on an algebraic variety (with a possibility to use it in characteristic zero); many Schubert varieties are of this type. His colloquium lecture is expected understandable to non-specialists; some familiarity with algebraic manifolds and the cohomology is assumed. `

`Colloquium Committee`**ABSTRACT:** *The talk will be mainly about the local criterion for splitting and compatible splitting, with special emphasis on Schubert varieties in G/B. We will discuss the diagonal splitting, the splitting of the cotangent bundle, the splitting of the moduli spaces of bundles on curves and applications to varieties in characteristic p>0 with trivial tangent bundle. *

*October 28, 2010 *

Mikhail Khovanov

*A special lecture *by the creater of Khovanov cohomology, a striking discovery, which goes well beyond the low-dimensional topology. The quantum *SL(2) *and the related categorification constructions will be the key theme; certain familiarity with simple Lie algebras is assumed.

`Colloquium Committee`**ABSTRACT:** *Quantum groups, discovered by Drinfeld and Jimbo, are Hopf algebra deformations of the universal enveloping algebras of simple Lie algebras. Various forms of quantum groups can be realized as Grothendieck groups of certain rings given by planar diagrammatical generators and relations. These constructions will be explained in the talk.*

List of past colloquia.