Colloquium
To receive updates to the colloquium schedule by email, email Marco Varisco.
The colloquium usually meets on Friday at 3:00 pm in room ES 143.
*** Please notice the unusual day & time. ***
Vijay Ravikumar (Chennai Mathematical Institute, India)
The Cohomology of the Complex Grassmannian
*Tuesday*, May 2, 2017
*1:15* p.m. in ES143
Abstract. The Grassmannian is a simple example of a moduli space, and its geometry is very well understood. We give an introduction to the geometry of the complex Grassmannian X by studying its cohomology ring H*(X). We also describe our work on the more general Ktheory ring of X and the equivariant cohomology ring of X.
First we describe the Schubert varieties of X and show how the corresponding Schubert classes form an additive basis for H*(X). By relating the intersections of Schubert varieties to the products of Schubert classes, we determine the multiplicative structure on H*(X) with respect to the Schubert basis. In particular, we describe Pieri’s rule for multiplying arbitrary Schubert classes with certain special Schubert classes, and show how it can be used to solve classical problems in enumerative geometry. Finally, we describe our work generalizing Pieri’s rule to the settings of Ktheory and equivariant cohomology.
Earlier colloquia and abstracts:
April 21
*Sixth Maheshwari Colloquium*
Robert Ghrist (University of Pennsylvania)
http://www.albany.edu/~mv312143/coll/
Xi Chen (University of Alberta)
On Vojta's "1 plus Epsilon" Conjecture
Friday, April 7, 2017
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES153)
Alex Iosevich (University of Rochester)
Finite point configurations in Euclidean space and vector spaces over finite fields
Friday, October 21, 2016
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES153)
Abstract. The basic question we ask is, how "large" does a subset of Euclidean space or a vector space over a finite field needs to be to ensure that it contains a point configuration of a give type. For example, how large does the Hausdorff dimension of a subset of ${\Bbb R}^d$ need to be to ensure that it contains vertices of an equilateral triangle? We shall describe some recent developments pertaining to these problem and stress the connections between analytic, combinatorial and number theoretic considerations.
Kirill Zainoulline (University of Ottawa)
Motives Versus Modules
Friday, October 7, 2016
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract. In this talk we explain how a classical category of Grothendieck’s motives of twisted flag varieties can related to the category of representations of the associated Hecketype algebras. The talk is based on a series of recent joint works with B. Calmes, C. Lenart, A. Neshitov, and C. Zhong.
Željko Čučković (University of Toledo)
From the @Neumann Operator to Hankel Operators and Back Friday,
September 9, 2016 3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract. Compactness of the @Neumann operator is closely connected to the compactness of Hankel operators acting on Bergman spaces. At first we use the @ methods to relate the compactness of a Hankel operator to the boundary behavior of its symbol. In the absence of compactness, we give the essential norm estimates of Hankel operators on convex domains. This, in turn, led us to obtain the essential norm estimates for the @Neumann operator on convex domains. Throughout the talk we will provide several examples to illustrate our results. (This is joint work with Sonmez Sahutoglu).
Fifth Annual Maheshwari Colloquium
Friday, April 15, 2016
Michael Christ (University of California, Berkeley)
Sharpened Inequalities for the Fourier Transform via Additive Combinatorics
Luca Capogna (Worcester Polytechnic Institute)
Regularity for subelliptic PDE through uniform estimates in multi scale geometries
Friday, April 8, 2016
3:00  3:45 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
AND
Loredana Lanzani (Syracuse University)
Harmonic Analysis techniques in Several Complex Variables
Friday, April 8, 2016
4:00  4:45 p.m. in ES143
(tea & coffee at 3:45 p.m. in ES152)
Richard Rimanyi (University of North Carolina at Chapel Hill)
Global Singularity Theory
Friday, March 11, 2016
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract. The topology of the spaces A and B may force every map from A to B to have certain singularities. For example, a map from the Klein bottle to 3space must have double points. A map from the projective plane to the plane must have an odd number of cusp points.
To a singularity one may associate a polynomial (its Thom polynomial) which measures how topology forces this particular singularity. In this lecture, we will explore the theory of Thom polynomials and their applications to enumerative geometry. Along the way, we will meet a wide spectrum of mathematical concepts from geometric theorems of the ancient Greeks to the theory of diagrams of linear maps (quivers).
Guozhen Lu, Wayne State University
Sharp Geometric Inequalities and Applications to PDEs
Friday, February 26, 2016
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract. In this talk, we will discuss some wellknown sharp geometric inequalities such as the Sobolev inequality, the MoserTrudinger and Adams inequality, etc. These inequalities play an important role in analysis, geometry and PDEs. We are mainly interested in the best constants and existence and forms of their maximizers. Identifying the best constants and maximizers is very important for applications, but nevertheless challenging. We are dealing with cases where the classical Pólya–Szegő inequality does not hold and thus there is no symmetrization principle. These cases include the high order Sobolev spaces and the simplest nonabelian nilpotent Lie group such as the Heisenberg group. Harmonic analysis method is often used in attacking these problems. We will describe some recent progress and open problems in this direction.
Malabika Pramanik, University of British Columbia and Cornell University
Needles, Bushes, Hairbrushes, and Polynomials
Friday, December 11,, 2015
3:00 p.m. in ES143
(NB: there will be no "tea & coffee" because of the University Gala)
Abstract. Points, lines, and circles are among the most primitive and fundamental of mathematical concepts, yet few geometric objects have generated more beautiful and nontrivial mathematics. Deceptively simple in their formulation, many classical problems involving sets of lines or circles remain open to this day. I will begin with a sample of problems that has spearheaded much of modern research, and explore their connections with analysis, geometry, and combinatorics.
R. Michael Range
A New Look at Calculus: How an Old Idea Naturally Leads from Simple Algebra to the Heart of Analysis
Friday, October 30, 2015
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract. We discuss a novel approach to calculus and analysis that builds upon an old idea of René Descartes. We begin with simple algebra to solve the tangent problem for all algebraic curves without using any limits. By adding an elementary estimate one is naturally led to the idea of continuity and, more generally, of limits. In particular, one recognizes that the algebraic derivative can also be captured by a nonalgebraic approximation process that opens the door to handling transcendental functions such as E_2 (x) = 2^x. The corresponding generalization of the algebraic derivative leads to the traditional concept of differentiability, in a formulation introduced by Constantin Carathéodory over 60 years ago, and whose advantages should be recognized more widely. We hope that this approach will stimulate discussions about alternatives to the standard introduction to calculus.
Amanda Folsom (Amherst College)
Mock Modular and Quantum Modular Groups
Friday, September 18, 2015
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract. The subject of mock modular forms has seen many advances over the course of the last decade. While relatively new, the origins of the subject date back to Hardy and Ramanujan. In this talk we will first discuss the general history and theory that has developed. We will then address one of Ramanujan’s remaining claims about mock theta functions from his last letter to Hardy. In our work, quantum modular forms, defined by Zagier in 2010, play a key role, as do certain wellknown combinatorial functions. This is joint work with Ken Ono (Emory U.) and Rob Rhoades (CCRPrinceton).
SPECIAL COLLOQUIUM to Celebrate Mark Steinberger’s 65th Birthday
Peter B. Shalen (University of Illinois at Chicago)
ThreeDimensional Topology and the Enumeration of Arithmetic Groups
Friday, September 4, 2015
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract. Quaternion algebras are natural objects in number theory, analogous to number fields. An arithmetic lattice in a quaternion algebra is roughly analogous to the group of units in the integers of a number field. Remarkably, certain arithmetic lattices (those defined over fields with exactly one complex place, and ramified at all real places) can be studied via the geometry and topology of 3manifolds, or more generally (if they have elements of finite order) of objects called 3orbifolds.
The manifolds (or orbifolds) associated with arithmetic lattices are hyperbolic manifolds, locally modelled on the 3dimensional nonEuclidean geometry of Gauss, Bolyai and Lobachevsky (or quotients of the latter by finite groups). In particular, each of these manifolds has a welldefined volume.
A theorem of Borel’s asserts that for any positive real number V, there are at most finitely many arithmetic lattices of covolume at most V. Determining all of these for a given V is algorithmically possible thanks to work by Chinburg and Friedman, but appears to be impractical except for very small values of V, say V = 0.41. (The smallest covolume of a hyperbolic 3orbifold is about 0.39.) It turns out that the difficulty in the computation for a larger value of V can be dealt with if one can find a good bound on dim H_1(O,Z/2Z), where O is a hyperbolic 3orbifold of volume at most V.
In the case of a hyperbolic 3manifold M, not necessarily arithmetic, joint work of mine with Marc Culler and others gives good bounds on the dimension of H_1(M,Z/2Z) in the presence of a suitable bound on the volume of M. In this talk I will discuss some analogous results for hyperbolic 3orbifolds, and the prospects for applying results of this kind to the enumeration of arithmetic lattices. A feature of the work that I find intriguing is that while it builds on my geometric work with Culler, the new ingredients involve primarily purely topological arguments about manifolds—the underlying spaces of the orbifolds in question—and have a classical, combinatorial flavor.
May 1
Eric Sommers (UMass Amherst)
April 24
Fourth Annual Maheshwari Colloquium
Jill Pipher (Brown University, ICERM)
April 17
Bogdan Ion (University of Pittsburgh)
April 10
KyuHwan Lee (University of Connecticut)
February 27
Anna Mazzucato (Penn State)
John Schmitt (Middlebury College)
Warning’s Second Theorem with Restricted Variables
Friday, January 30, 2015
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: The polynomial method is a successful and promising approach for solving combinatorial problems. We will discuss this method via a theorem of Alon and Furedi and offer a new (and short) proof of a numbertheoretic theorem of Ewald Warning from 1935, which concerns the number of zeros of a polynomial system over a finite field. We also offer a broad generalization of Warning’s theorem. Further, we will discuss applications of this generalization to various zerosum problems in additive combinatorics.
This is joint work with Pete L. Clark (U. Georgia) and Aden Forrow (M.I.T.)
Bernard Leclerc (Université de Caen, France)
KirillovReshetikhin Modules and Their qCharacters
Monday, November 3, 2014
10:25–11:25 in ES139
Abstract. KirillovReshetikhin modules are a class of irreducible finite
dimensional representations of quantum affine algebras (or Yangians). Ini
tially introduced in mathematical physics in relation with certain integrable
models, they appear as the most “accessible” irreducible representations. I
will explain recent new methods to describe their qcharacters: (1) an algo
rithmic method which allows to calculate them by successive approximations;
(2) a geometric method which expresses them in terms of Euler characteristics
of Grassmannians of submodules for a quiver with potential. Both methods
come from the interpretation of the qcharacters of KirillovReshetikhin mod
ules as cluster variables in the Grothendieck ring. They work in a uniform
way for all untwisted quantum affine algebras. This is joint work with David
Hernandez (arXiv:1303.0744).
David CruzUribe (Trinity College)
The Rise, Fall and Rebirth of the MuckenhouptWheeden Conjectures
Friday, November 7, 2014
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Kirill Zainoulline (University of Ottawa)
Oriented Equivariant Cohomology, Formal Group Laws, And Generalized Schubert Calculus
Friday, October 10, 2014
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract. Oriented equivariant cohomology theories and the associated formal groups laws have been a subject of intensive investigations since the 60’s, mostly inspired by the theory of complex cobordism in topology. In the present talk we discuss several recent developments in the study of algebraic analogues of such theories, e.g., algebraic cobordism of LevineMorel or algebraic elliptic cohomology, of projective homogeneous varieties. In particular, we address the problem of constructing the Schubert and the BottSamelson classes for such theories.
ManDuen Choi (University of Toronto)
What On Earth Does A Quantum Computer Mean?
Friday, March 7, 2014
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: Suddenly, there arises the new era of real quantum computers, with all sorts of information process in the setting of noncommutative analysis. From the point of view of a pure mathematician, I will explain how matrix method can be used intensively in the recent development of quantum information joining quantum mechanics and computer science. This is an expository talk with rudiments of matrix theory; no background knowledge of physics or computer will be assumed.
Massimiliano Pontil (University College London)
Learning from Data
Monday, February 17, 2014
4:15 p.m. in ES143
(tea & coffee at 3:15 p.m. in ES152)
Abstract: Machine learning is an exciting and rapidly expanding area of computer science with cross links to computational mathematics and statistics and to the emerging fields of big data and data science. Machine learning has expanded dramatically during the past fifteen years or so, becoming a mature area with rich theory and applications. Modern learning techniques draw substantial ideas and inspirations from numerous areas of mathematics. Approximation theory, functional analysis and numerical optimization provide important tools to formalise the problem of learning from data. At the same time, probability and statistics offer important concepts which can be used to bound the performance (generalization error) of a learning algorithm in a principled way. A main goal of machine learning is to learn functional relationships and representations from multiple and high dimensional data sources. In this talk, I will give an overview of recent progress on this problem, highlighting the interplay between ideas and tools from maths, statistics and computer science. A crucial step for effective and efficient learning is to incorporate prior knowledge on the model underlying the data. I will present a comprehensive framework to achieve this goal, which allows us to incorporate constraints in the learning problem using certain convex regularization functionals. They drive the learning algorithm towards functions which fit the data well and satisfy the desired properties. In particular, I will describe families of norms which encourage smoothness and sparsity, as well as relationships between multiple functions. Furthermore I will discuss extensions of these ideas to learn tensors, using multilinear algebra. Finally, I will present numerical simulations which illustrate the practical value of this framework in applications arising in various domains, including user modeling, computer vision and affective computing.
Jin Wang (Old Dominion University)
Computation and Analysis of Multiphase Fluid Flows with Moving Interfaces
Thursday, February 13, 2014
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: Multiphase fluid flows with moving interfaces occur in a wide range of natural and technological phenomena. Owing to their strong nonlinearity and inherent multiphysics, our current understanding of the fundamental mechanisms involved remains limited. In the first part of this talk, we focus our attention on the interfacial motion between two incompressible and viscous fluids. After a short discussion of some results from linear analysis, we present a novel numerical method with high accuracy for the simulation of the fully nonlinear interfacial flow. We discuss in detail the algorithm development, simulation results, and treatment and analysis of the numerical data. With this algorithm, traveling waves, such as Stokes waves, can be followed sufficiently in time to reveal the effects of small viscosity. The numerical observation and data analysis are then justified by deriving a formal asymptotic theory. In the second part of this talk, we discuss several related studies, particularly the fluidstructure interaction problems, and their many interdisciplinary applications.
Brett Wick (Georgia Institute of Technology)
The Corona Theorem
Friday, November 8, 2013
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: Carleson’s Corona Theorem from the 1960’s has served as a major motivation for many results in complex function theory, operator theory, and harmonic analysis. In a simple form, the result states that for N bounded analytic functions f1,...,fN on the unit disc such that inf f1+···+fN >= d > 0 it is possible to find N other bounded analytic functions g1,...,gN such that f1g1+···+fNgN = 1. Moreover, the functions g1,...,gN can be chosen with some norm control.
In this talk we will discuss some generalizations of this result to certain vectorvalued functions and connections with geometry and to function spaces on the unit ball in several complex variables.
Peter Shalen (University of Illinois at Chicago)
Quantitative Mostow Rigidity
Friday, October 25, 2013
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: A hyperbolic nmanifold is a space locally modeled on an ndimensional nonEuclidean space in which the sum of the angles of a triangle is less than π. Equivalently, it is a Riemannian manifold of constant sectional curvature −1. It is a special case of the Mostow Rigidity Theorem that for n > 2 a compact hyperbolic nmanifold M is determined up to isometry by its fundamental group. In particular any geometrically defined invariant of M, such as its volume or diameter, is a topological invariant of M (and in fact an algebraic invariant of π_1(M)). This raises the fascinating question of how to relate these geometrically defined quantities to more classical topological invariants of M, such as the rank of its first homology group H_1(M). While qualitative connections of this kind are relatively easy, making them quantitative can be quite challenging; for example, it is not hard to obtain a linear bound for the rank of H_1(M) in terms of the volume of M, but the best constant of linear growth is not known. I will explain how I have been bringing deep topological, geometric and analytic techniques to bear on this problem for the case n = 3. In particular, techniques of 3dimensional topology from the mid20th century play a surprising role. Some of the work I will describe is joint work with Marc Culler and others.
Greg Kuperberg (University of California, Davis)
What is quantum probability?
Friday, September 20, 2013
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: Quantum mechanics is difficult for many people to understand because it is difficult to believe. The heart of the problem is quantum probability, which is an entirely rigorous theory; nonetheless even many working mathematicians have trouble believing it. (Quantum field theory is far from entirely rigorous, but that is a different topic.) In the past 15 years or so, work in quantum probability has greatly expanded in the guise of quantum computation and quantum information theory. In this talk, I will discuss some of the ideas of quantum probability, quantum computation, and quantum information, using the language of pure mathematics.
Holger Reich (Freie Universität Berlin, Germany)
GL(n,Z) and Algebraic KTheory
Friday, September 13, 2013
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: The FarrellJones conjecture gives a conjectural description of the algebraic Ktheory of the group ring of an infinite group G. The conjecture has important consequences in geometric topology. The talk will give an elementary introduction to the conjecture and report on the case G=GL(n,Z).
Željko Čučković (University of Toledo)
Compactness of Hankel Operators on Convex Domains
Friday, March 29, 2012
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: We are interested in the following question: How does compactness of (products of) Hankel operators on the Bergman space relate to the boundary geometry of domains in C^n? We will present some previous results on convex domains as well as our current work on convex Reinhardt domains in C^2.
Avraham Soffer (Rutgers University)
Solitons and Nonlinear Partial Differential Equations
Friday, February 22, 2012
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: This is a general talk about the notion of Soliton and its importance in understanding Dispersive Wave Equations. I will describe the existence and stability of solitons, then the modern theory of Asymptotic Stability and finally open problems and conjectures.
Anders Buch (Rutgers University)
GromovWitten Invariants and Puzzles
Friday, November 30, 2012
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: The development of algebraic geometry has been motivated by enumerative geometric questions where one asks for the number of geometric figures of some type that satisfy a list of conditions. For example, the GromovWitten invariants of a flag manifold count the number of curves that meet a list of Schubert varieties in general position. I will focus on the (3 point, genus zero) GromovWitten invariants of Grassmannians, which are known to be special cases of the multiplicative structure constants of the Schubert polynomials studied in combinatorics. A conjecture of Allen Knutson asserts that certain Schubert structure constants are equal to the number of triangular puzzles that can be created using a list of puzzle pieces. I will discuss a recent proof of this conjecture, how it leads to a positive combinatorial formula for Grassmannian GromovWitten invariants, and generalizations. This talk is based on papers with A. Kresch, L. Mihalcea, K. Purbhoo, and H. Tamvakis.
“Mathematics In Literature” — Is That Really A Math Course?
Friday, November 2, 2012
3:00 p.m. in ES143
(tea & coffee at 2:30 p.m. in ES152)
Abstract: During the last couple of decades, writers of both fiction and nonfiction, dramatists, movie directors and television producers have increasingly turned to mathematics and the lives of mathematicians as a fertile source of material. My own career as a research mathematician and enthusiastic reader has led me recently to ponder the possibility of creating a course that engages students in noteworthy mathematical concepts, results or individuals, primarily through literary fiction and fictional biography. The result is a course that I created (and taught) entitled “Mathematics in Literature.” The promotion for the course was the following:
“Do you like to read? Do you like mathematics? Combine your interest in both in a new and stimulating way. Read historical fiction and fictional biography with a focus on historically important mathematics problems, significant mathematicians or fundamental issues involved in the foundations of mathematical certainty. Read and discuss novels with protagonists who are mathematicians or with the narrative thread weaving mathematics and mathematicians into a web of intrigue.”
This talk will discuss various aspects of the course and will include both a sample of readings from the source material and a brief description of the significant mathematical issues unifying most of the material.
Laurent Baratchart (INRIA Sophia Antipolis, France)
Compacts of Minimum Capacity and Rational Approximation
Friday, October 19, 2012
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract: Approximation of holomorphic functions of one variable on compact sets of their analyticity domain is an old issue in function theory. From the possibility of approximation asserted by Runge’s theorem, the emphasis has gradually moved towards error rates and asymptotics for the poles. For (possibly multivalued) functions with singular set of zero capacity, a theory has emerged where certain extremal problems from logarithmic potential theory, of Chebotarev type, play a central role. We shall survey these developments and present some recent results. For instance, best approximants (in various senses) of degree n on a Jordan curve to a function with branchpoints inside the curve converge as n goes large in the complement of the set of smallest Green capacity outside of which the function is singlevalued.
Michael J. Hopkins (Harvard University)
Symmetry, Homotopy, and Smooth Manifolds
The Inaugural Maheshwari Colloquium , endowed in honor of Man Mohan and Asha Devi Maheshwari by our alumnus Raj Maheshwari ’83.
Friday, April 20, 2012
3:30 p.m. in Lecture Center 4
(Refreshments will be served at 2:45 p.m.)
For details, see http://www.albany.edu/math/coll/
Alex Iosevich (University of Rochester)
MultiLinear Operators, Distribution of Simplexes, and Geometric Combinatorics
Friday, March 23, 2012
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract: We are going to study several Erdős type problems on the distribution of simplexes in finite subsets of the Euclidean space using multilinear operator bounds, geometric measure theory, and a variety of conversion mechanisms. The talks should be accessible to a wide audience.
Victor Ivrii (University of Toronto)
100 Years Of Weyl’s Law
Friday, February 17, 2012
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract: In 1911–1912 Hermann Weyl published two papers (more followed) describing the distribution of the eigenvalues of the Dirichlet Laplacian in a bounded domain. These were among the first publications by Weyl and a new exciting field of mathematics was created.
I will discuss:
* Weyl law with sharper remainder estimates (in particular, Weyl conjecture);
* Generalized Weyl law;
* When the generalized Weyl law works and when it does not and how it
should be modified;
* What should be used instead of the eigenvalue counting function when the
spectrum is not necessarily discrete;
* Weyl law and ThomasFermi theory.
Joshua Isralowitz (University of Göttingen)
Compactness of operators in the Toeplitz algebra of the Fock space
Thursday, February 9, 2012
4:15 p.m. in ES143 (tea & coffee at 3:30 p.m. in ES152)
Abstract. In 2004 D. Suarez showed that a bounded operator on the Bergman space of the ball is compact if and only if the operator is in the Toeplitz algebra and the Berezin transform of the operator vanishes at the boundary of the ball. In this talk, I will discuss an even stronger version of this result for the Fock space. This is joint work with W. Bauer.
David Anderson (University of Washington)
Okounkov bodies: from algebraic to convex geometry
Tuesday, February 7, 2012
4:15 p.m. in ES143 (tea & coffee at 3:30 p.m. in ES152)
Abstract. Building on earlier work of Okounkov, in 2008 Kaveh, Khovanskii, Lazarsfeld, and Mustata showed how to construct a convex body in ndimensional Euclidean space naturally associated to a line bundle on an ndimensional algebraic variety, in such a way that the convex geometry of this body reflects algebrogeometric properties of the line bundle. This construction generalizes a wellunderstood correspondence between toric varieties and polytopes: when one starts with a toric variety and an equivariant line bundle, the associated convex body is the polytope arising from the yoga of toric geometry.
After describing the history and construction of these socalled “Okounkov bodies” from an elementary point of view, I will explain how the toric correspondence can be made tighter: under the right conditions, the Okounkov body is a polytope, and the variety in question deforms to a toric variety with the same Okounkov body. The toric correspondence provides a remarkably useful bridge between several branches of mathematics, and we will see connections between geometry, algebra, combinatorics, and representation theory.
Artem Pulemotov (University of Chicago)
Geometric flows on manifolds with boundary
Thursday, February 2, 2012
4:15 p.m. in ES143 (tea & coffee at 3:30 p.m. in ES152)
Abstract. Geometric flows are partial differential equations that describe evolutions of geometric objects. They are typically used to tackle problems in topology, mathematical physics, and several other fields. The canonical example of a geometric flow is the heat equation on a Riemannian manifold. In the first part of the talk, we will discuss the fundamental features of this equation. We will also speak about two estimates for its positive solutions on manifolds with boundary. A more contemporary example of a geometric flow is the Ricci flow for a Riemannian metric. It is mostly famous for its role in the proof of the Poincaré conjecture. The second part of the talk will be devoted to the main features and the behavior of the Ricci flow on manifolds with boundary. Towards the end, we will give a brief overview of related problems.
Liz Vivas (Purdue University)
Dynamics of holomorphic selfmaps near a fixed point
Tuesday, January 31, 2012
4:15 p.m. in ES143 (tea & coffee at 3:30 p.m. in ES152)
Abstract. The local dynamics of holomorphic selfmaps of C^n around a fixed point has been an object of study since the time of Schröder, Fatou and Julia. In this talk we will explain the results known for n=1 and the partial known results for n>1. We will focus in the case n=2 and of maps tangents to the identity, that is, when the derivative of our selfmap at the fixed point is the identity. In this case the usual tools of linearization introduced by Poincaré are not possible to use and some new techniques are required.
Daniel Ramras (New Mexico State University)
Spaces of Representations and the Topological AtiyahSegal Transformation
Friday, January 27, 2012
4:15 p.m. in ES143 (tea & coffee at 3:30 p.m. in ES152)
Abstract. The relationship between representations of a group and vector bundles was first studied by Atiyah and Segal in the early 1960s, via a construction that associates a vector bundle to each representation. Early work in this direction focused on finite groups, or compact Lie groups. When one considers infinite discrete groups, such as the fundamental group of a closed manifold, continuous families of representations come into play. In particular, the AtiyahSegal construction may be generalized so as to associate a vector bundle to each such family. In this talk, I’ll explain how methods from differential geometry, algebraic geometry, and homotopy theory can be combined to study this construction, yielding concrete results about vector bundles over familiar spaces such as surfaces.
Thomas Banchoff (Brown University)
Triple Points of Surfaces, Immersed and NonImmersed
*Wednesday*, January 25, 2012
*4:15* p.m. in *ES241* (tea & coffee at 3:45 p.m. in ES152)
Abstract. Forty years ago the first proofs were published relating the number of triple points of a surface immersed in threespace to the number of handles of the surface. New proofs then appeared including one by Richard Goldstein and Ted Turner that is particularly good for proving an extension of that result for stable mappings with pinch points. The talk will feature computer graphics images and animations.
Allan Greenleaf (University of Rochester)
Resolution of Singularities for Analysts
Friday, November 4, 2011
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract. A basic problem is to describe the zero set of a polynomial or analytic function. In algebraic geometry, resolution of singularities was established by Hironaka in 1964 and has since developed into a powerful collection of methods. In analysis, there is a need for more concrete and effective approaches, allowing one to find numerical invariants of functions near zeros or critical points, such as the critical integrability, sublevel growth and oscillatory indices. I will describe these invariants, how they arise, and outline an analystfriendly approach to resolution of singularities.
This is joint work with Tristan Collins and Malabika Pramanik.
Mark Shimozono (Virginia Polytechnic Institute and State University)
Parabolic Lusztig qAnalogues and OneDimensional Sums
Thursday, October 6, 2011
1:15 p.m. in ES146 (tea & coffee at 12:45 a.m. in ES152)
Abstract. Parabolic Lusztig qanalogues are a family of polynomials which
include Lusztig’s qanalogues of weight multiplicity, which describe the
intersection cohomology of certain Schubert varieties in the affine flag
manifold. Onedimensional (1d) sums are polynomials which arose in the study
of twodimensional solvable lattice models and in the Kyoto school’s
construction of crystal graphs for highest weight modules over quantum affine
algebras. We show that for G of classical type there is a subfamily called
stable parabolic Lusztig qanalogues, which coincides with the family of
largerank limits of 1d sums.
This is joint work with Cedric Lecouvey and Masato Okado.
Alexander Dranishnikov (University of Florida, Gainesville)
LusternikSchnirelmann category and the fundamental group
Friday, September 9, 2011
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract. The LusternikSchnirelmann category measures the complexity
of manifolds. It gives a low bound for the number of critical points
of any (not necessarily Morse) smooth function. It is known that for
nmanifolds the LScategory does not exceed n. The Whitehead theorem
states that for simply connected nmanifolds it does not exceed n/2.
We extend Whitehead’s theorem for spaces with certain fundamental
groups.
Paul Loya (Binghamton University, SUNY)
An introduction to Witten’s holonomy theorem
Friday, April 29, 2011
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract: In the 1980’s Daniel Quillen introduced determinant line
bundles and about the same time Edward Witten derived a remarkable
formula for the holonomy of the determinant line bundle of a Dirac
operator using something called the “eta invariant” of Atiyah, Patodi,
and Singer. In the physics literature, the holonomy of the determinant
line bundle is called the “global anomaly". Witten’s derivation was
later made rigorous by Bismut and Freed and also by Cheeger.
In this talk I will give an introduction to eta invariants and
Witten’s holonomy theorem, and then I will discuss recent work
concerning generalizations of this theorem to situations quite
different from the original results. This talk will be suitable for a
general audience.
Adaptive Fourier Decompositions (AFDs) and Best Approximation by Rational Functions
Friday, April 8, 2011
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract. The talk will compare three different models of adaptive
Fourier decompositions with illustrative experimental results.
relation to the algorithm of its solution.
Hara Charalambous (University of Thessaloniki)
Betti fibers for binomial ideals and indispensable complexes
Friday, February 4, 2011
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract: In a polynomial ring we consider ideals generated by
binomials. We study the characterization of minimally generating
sets for such ideals even when Nakayama’s Lemma fails.
Alfonso Montes Rodriguez (University of Sevilla, Spain)
Uniqueness Sets for the KleinGordon Equation and the Solution of a
Conjecture of Salem
Friday, November 12, 2010
4:30 p.m. in ES143 (tea & coffee at 4:00 p.m. in ES152)
(please notice unusual time)
Patricia Hersh (North Carolina State University)
Combinatorics and Topology of Stratified Spaces
Friday, October 29, 2010
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract: Anders Björner characterized which finite, graded partially
ordered sets (posets) are closure posets of regular CW complexes, and
he also observed that a finite, regular CW complex is homeomorphic to
the order complex of its closure poset. One might therefore hope to
use combinatorics to determine topological structure of stratified
spaces by studying their closure posets; however, it is possible for
two CW complexes with very different topological structure to have the
same closure poset if one of them is not regular. I will discuss a
criterion for determining whether a finite CW complex is regular (with
respect to a choice of characteristic maps); this will involve a
mixture of combinatorics and topology. Along the way, I will review
the notions from topology and combinatorics we will need. Finally I
will discuss an application: the proof of a conjecture of Fomin and
Shapiro, a special case of which says that the Bruhat cell
decomposition of the neighborhood of the origin in the totally
nonnegative part of the space of upper triangular matrices with 1's on
the diagonal is a regular CW complex homeomorphic to a ball.
Pablo GonzalezVera (University of La Laguna, Spain)
On the Computation of Weighted Integrals: from the real line to the unit circle
Friday, October 22, 2010
3:00 p.m. in ES143 (tea & coffee at 2:30 p.m. in ES152)
Abstract: In this talk quadrature formulas to approximately calculate
integrals both on intervals of the real line and the unit circle are
revisited and their basic features exposed. Different connections
between the real line and the unit circle are exploited by showing
that the computation of the quadrature formulas essentially reduce to
the case of the unit circle.
Raul Curto (University of Iowa)
Cubic Column Relations in Truncated Moment Problems
Friday, October 15, 2010
3:00 p.m. in ES143
(Tea at 2:30 p.m. in ES152)
Spring 2010

Fall 2009
Wednesday, September 16, 2009 4:15 pm (note unusual time!), ES 147 Friday, October 30, 2009 3:00 pm, ES 143 Thursday, November 12, 2009 3:00 pm, ES 146 Friday, December 4, 2009 3:00 pm, ES 143 
Spring 2009
Wednesday, February 4, 2009 3:00 pm, ES 143 Wednesday, February 25, 2009 4:00 pm, ES 147 Friday, February 27, 2009 4:00 pm, ES 143 Friday, April 24, 2009 3:00 pm, ES 143 Friday, May 1, 2009 3:00 pm, ES 143 
Fall 2008

Spring 2008

Fall 2007
