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Abstract: We consider a PDE model consisting of a nonlinear vibrating structure immersed in a flow of gas moving through 3-d space with a supersonic velocity. The latter is described by the so called modified wave equation whose resolvent exhibits the loss of ellipticity.
Long time behavior of the resulting interaction is discussed. It is shown that structural solutions converge asymptotically to an attracting set which is both finite dimensional and smooth. The convergence is uniform with respect to the natural topology induced by finite energy. Thus, the ultimate long time behavior of the hyperbolic- like evolution becomes both smooth and finite-dimensional.
In contrast to other studies on the subject the result described holds (i) without imposing any dissipation on the structure and (ii) without assuming smoothing effects within the structure (such as thermoelasticity or strong damping which both induce regularizing and damping effects ).
The talk will describe some history of the problem, with past developments in the area, and the sketch of the main ideas leading to the results recently obtained. These are based on exploiting (1) ”hidden regularity” associated with the delay term describing the effect of the flow; (2) natural dispersive effects associated with the flow which then translate into ”weak” dissipation affecting the structural component; and (3) compensated compactness resulting from the analysis of the interaction on the interface.

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Abstract: The theme of Mathematics Awareness Month 2014, which launched on 1 Apr, is
"Mathematics, Magic, and Mystery", which is closely patterned after the title of
a classic 1956 Dover paperback by the legendary Martin Gardner (1914--2010).

Martin was without a doubt the best friend mathematics ever had, and it's fitting
that in his centennial year we seize the opportunity to leverage his extensive
written legacy---over 100 books---to turn new generations on the magic and mystery
of mathematics, and the joys of problem solving and rational thinking. The goal
is to inspire many "Aha!" moments, and add to Gardner's record of turning innocent
youngsters into mathematics professors (and mathematics professors into innocent
youngsters).

Mathematics Awareness Month will provide people with multimedia opportunities to
explore the kinds of topics Martin made famous via his famous "Mathematical Games"
columns for Scientific American, and associated books. These range from
hexaflexagons, magic squares, geometric vanishes, mobius bands, and mathemagic,
to juggling, Penrose tiles and the connection between card shuffling and fractals.

We'll give an overview of the Mathematics Awareness Month activities, while
surveying Martin Gardner's achievements, and highlighting the potential for major
outreach into the nation's youth.

Twitter users may enjoy following @MathAware (Mathematics Awareness Month),
@WWMGT (What Would Martin Gardner Tweet?) and @MGardner100th (Martin Gardner
Centennial).

Bio: Colm Mulcahy is a professor of mathematics at Spelman College, in Atlanta, where
he has taught since 1988. He's currently visiting The American University in
Washington, DC. Over the last decade, he has been at the forefront of publishing
new mathemagical principles and effects for cards, particularly in his
long-running bi-monthly Card Colm for the MAA. Some of his card effects have
been featured in the New York Times Numberplay blog. His book Mathematical Card
Magic: Fifty-Two New Effects was published by AK Peters/CRC Press in 2013.
http://www.crcpress.com/product/isbn/9781466509764.

Colm gave the MAA Lecture for Students at Mathfest 2009 in Portland, Oregon, and
is a recipient of MAA's Allendoerfer Award for excellence in expository writing,
for an article on image compression using wavelets.

He's part of the team heading up Mathematics Awareness Month 2014, which is
inspired by the legacy of Martin Gardner, whom he was fortunate to know for the
last decade of his life. He also chairs the Martin Gardner Centennial Committee.

His interests are broad, ranging from algebra and number theory to geometry. He
earned a B.Sc. and M.Sc. in mathematical science from University College Dublin
in his native Ireland, and a PhD from Cornell University for research in the
algebraic theory of quadratic forms, under Alex F.T.W. Rosenberg.

He tweets at @CardColm and highly recommends that Gardner fans follow @WWMGT and
@MGardner100th.

New websites: http://www.mathaware.org and http://www.martin-gardner.org and h
ttp://www.maa.org/news/mam-2014-mathematics-magic-and-mystery

View Abstract

Abstract: The notion of distributional or weak solutions has played an important role in the PDE theory for a long time and it has become indispensable in many situations. However, in certain situations very naturally looking definitions may come with surprises. We will discuss some examples, focusing mostly on non-linear elliptic systems and some conjectural behavior of the Navier-Stokes equations.

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Abstract: http://www2.math.umd.edu/~pbrosnan/Pages/ColVeech2014.pdf

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Abstract: I will discuss the classical problem of simplifying polynomials in one variable by Tschirnhaus transformations and explain how it naturally leads to the notion of essential dimension.

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Abstract: A homology theory designed to understand fixed points of Hamiltonian systems was introduced
by Floer several decades ago. Floer's homology is a version of Morse homology on the loop space of a symplectic manifold. The Lagrangian version of this theory plays a role of Konstevich's mirror symmetry conjecture. Often most Lagrangians turn out to have vanishing Floer homology, but the mirror symmetry conjecture suggests that there is always at least one with non-vanishing Floer homology. After a review of symplectic geometry, I will sketch Floer's theory and discuss some advances on the existence conjecture above and a conjectural estimate for how many for smooth projective varieties, which is related to the minimal model program.

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Abstract: Upon reading Hermann Minkowski’s The Geometry of Number, Charles Hermite exclaimed “I believe I see the promised land,” but little is generally known about this underappreciated mathematical Moses. If he is remembered for anything, it is his light-cone geometric structure of the spacetime of special relativity, but the truth is that Minkowski was a crucially important figure in determining what questions mathematicians asked in the 20th century, how they went about solving them, and how we have come to understand what mathematical propositions in applied contexts are supposed to mean.

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Abstract: Given a plane complex curve C and a (possibly singular) point z in C the link of C at z is
the intersection of C with a small three sphere around z. For example a link of cusp curve x²=y³ at z=(0,0)
is a trefoil knot. In my talk I will explain how one can extract topological invariants of curve singularity (Milnor number, Betti numbers of
the compactified Jacobian of the curve and so on) from the knot invariants of the links its singularities. I will also explain
some applications of the above mentioned relations to the combinatorics of q,t-Catalan numbers and the character
theory of the double affine Hecke algebras. The talk is based on the joint projects with Gorsky, Shende, Rasmussen and Yun.

View Abstract

Abstract: The group of operators on L2(R) generated by a single translation f(t) → f(t+1) and a single modulation f(t) → e2πiωtf(t) is a familiar object of study in signal analysis. It forms a unitary representation of the so-called discrete Heisenberg group. As such, its analysis yields easily to classical methods when ω is rational. But when ω is irrational, it provides instructive concrete examples of several pathological representation-theoretic phenomena. We shall introduce the relevant concepts from representation theory, discuss the rational case briefly, and then explore the irrational case in more detail.

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Abstract: Hypergeometric functions frequently describe the variation of periods in families of Calabi-Yau manifolds, the classical case being that of the Gauss hypergeometric function and elliptic curves in Legendre normal form. Motivated by mirror symmetry conjectures, a decade ago John Morgan and I classified the possible generalized hypergeometric variations which could underlie families of Calabi-Yau threefolds over the thrice-punctured sphere. The question of the geometric origin of the one “missing case” has proved enormously fruitful, yielding a general approach to the study of a broad class of Calabi-Yau threefolds, each fibered by K3 surfaces which are themselves built from pairs of elliptic curves. A perfect analogue of the Weierstrass normal form for elliptic curves in the case of K3 surfaces provides the critical Hodge-theoretic/geometric dictionary fiberwise, reducing much of the analysis to a class of rational functions on the base of each fibration – a (Kodaira-style) generalized functional invariant. The theory finds immediate application to a recent question of Sarnak involving the geometric meaning of arithmetic versus thin monodromy representations.

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Abstract: We discuss constraints on the Poisson brackets coming from symplectic topology, applications to symplectic intersections and Lagrangian knots,as well as links to quantum mechanics.

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Abstract: Many scientific and engineering problems involve interconnected moving
interfaces separating different regions, including dry foams, crystal grain
growth and multi-cellular structures in man-made and biological materials.
Producing consistent and well-posed mathematical models that capture the
motion of these interfaces, especially at degeneracies, such as triple points and triple lines where multiple interfaces meet, is challenging.

Joint with Robert Saye of UC Berkeley, we introduce an efficient and robust mathematical and computational methodology for computing the solution to two and three-dimensional multi-interface problems involving complex junctions and
topological changes in an evolving general multiphase system. We demonstrate the method on a collection of problems, including geometric coarsening flows under curvature and incompressible flow coupled to multi-fluid interface problems.
Finally, we compute the dynamics of unstable foams, such as soap bubbles, evolving under the combined effects of gas-fluid interactions, thin-film lamella drainage, and topological bursting.

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Abstract: In physics, supersymmetry is a pairing between bosons and fermions appearing in theories of subatomic particles. One may study supersymmetry mathematically by using Adinkras, which are graphs with vertices representing the particles in a supersymmetric theory and edges corresponding to the supersymmetry pairings. In combinatorial terms, Adinkras are N-regular, edge N-colored bipartite graphs with signs assigned to the edges and heights assigned to the vertices, subject to certain conditions. We will see how to capture some of the structure of an Adinkra using binary linear error-correcting codes, and all of it using a very special case of a geometric construction due to Grothendieck. The talk is designed to be accessible to an undergraduate audience.

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Abstract: A Calabi-Yau manifold is a complex Kahler manifold with vanishing first Chern class. This seemingly innocent topological condition has many remarkable consequences. Of particular interest is Yau's 1978 landmark theorem showing that such manifolds, if compact, admit Ricci-flat Kahler metrics - a feat of nonlinear analysis, and one of very few general sources of solutions to the Einstein equations in Riemannian geometry. A longstanding question in the field asks for extensions of Yau's theorem to noncompact settings and particularly for the possible asymptotics of noncompact Ricci-flat manifolds at infinity or at a singularity. I will survey recent progress on this question, mainly based on the point of view that a detailed understanding of the Riemannian geometry of the problem is key to proving useful estimates.

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Abstract: We shall discuss an isoperimetric conjecture regarding capacities of convex sets (called Viterbo's conjecture) and explain how it implies the well known Mahler conjecture from convexity, regarding the minimizers, among symmetric convex bodies, of the volume product of a body and its polar. The connection uses some new results on billiard dynamics, which we shall explain as well. Based of joint work with Yaron Ostrover and with Roman Karasev.

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Abstract: Let X be your favorite Banach space of continuous functions on R^n. Let f be a function defined on an (arbitrary) given subset E of R^n. How can we decide whether f extends to a function F in X? If such an F exists, then how small can we take its norm? What can we say about the derivatives of F at a given point? Can we take F to depend linearly on f? What if we relax the requirement that F=f on E exactly, and just require that F-f must be small (in some sense) on E?

Suppose E is a large finite set. Can we compute an F as above with norm close to least possible? If so, how many computer operations does it take? What if we are allowed to discard a few points of E as "outliers"? Which points should we discard?

These questions have fascinated me for about the last ten years. In this talk I'll sketch the current state of the art.

View Abstract

Abstract: The talk will consist of four points:

1. The basic definition of K-theory
2. A brief history of K-theory
3. Algebraic versus topological K-theory
4. The unity of K-theory

This is an expository talk intended for a general mathematical audience.

View Abstract

Abstract: I will give a basic introduction to principal bundles over different kinds of spaces, and I will discuss the various spaces that classify these bundles. A key insight is that even topologically these bundles can be classified by maps to complex algebraic varieties. This led Atiyah and Hirzebruch to the first counterexamples to the original, integral Hodge conjecture. I will discuss how a different approach to these algebraic classifying spaces, due to Totaro, led Ben Williams and myself to a method for answering concrete questions about division algebras over fields, and I will explain how these ideas resulted in our answer to a question in algebra from 1960.

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Abstract: Algebraic varieties (zero sets of families of multivariate polynomials) are notoriously "rigid". One manifestation of this is that a typical surface (2-dimensional variety) does not admit any surjective map from a product of two curves. I will describe a simple construction of large families of surfaces that do admit such maps, and explain some of the wonderful arithmetic consequences that follow. One nice application is to the construction of elliptic curves (and higher genus Jacobians) with large rank over function fields.

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Abstract: Scaling limits of Smoluchowski's coagulation equation are related to probability theory in numerous remarkable ways. E.g., such an equation governs the merging of ancestral trees in critical branching processes, as observed by Bertoin and Le Gall. A simple explanation of this relies on how Bernstein functions relate to a weak topology for Levy triples. From the same theory, we find the existence of `universal' branching mechanisms which generate complicated dynamics that contain arbitrary renormalized limits. I also plan to describe a remarkable application of Bernstein function theory to a coagulation-fragmentation model introduced in fisheries science to explain animal group size.

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Abstract: We will survey some recent work on Allen-Cahn phase transition models.
Earlier work involved the construction of solutions with transitions occurring in one spatial direction.
For the newer models, much more general transition regions can be constructed.

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Abstract: Oriented cohomology theories and the associated formal groups laws have
been a subject of intensive investigations since 60's, mostly inspired by
the theory of complex cobordism. In the present talk we discuss several recent developments in the study of algebraic analogues of such theories, e.g. algebraic cobordism of Levine-Morel or algebraic elliptic cohomology, of projective homogeneous spaces.

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Abstract: For a natural number m, let f_m denote multiplication by m mod 1 on T =
R/Z. Let p,q denote two multiplicatively independent numbers (no power of p equals a power of q) such as 2,3. An old result (1967) states that if E is a closed subset invariant under both f_p and f_q, then either E is all of T or it consists of a finite set of rationals. (This is far from true for a set invariant under only one transformation.) The question has been raised whether the analogous result for probability measures on T is true (known as the x2,x3 conjecture!) This question is still open, but a partial answer given by D. Rudolph treating the positive entropy case has been fruitful and has led to significant advances in homogeneous dynamics and diophantine questions. A new
conjecture is that the f_p orbit closure and f_q orbit closure of any irrational point cannot both be small - specifically , that the sum of the Hausdorff dimensions is never < 1. We shall connect this problem with a question regarding fractals, making plausible the modified conjecture that the "dimension
inequality" holds for all but a set of dimension 0.

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Abstract: We will discuss some results on the geometry of the
moduli space of Riemann surfaces, and applications, obtained using
meromorphic differentials with all periods real.
These constructions are motivated by Whitham perturbation theory of
integrable systems, and somewhat parallel to those used to study
Teichmueller dynamics for abelian differentials.
Based on joint work with Igor Krichever.

View Abstract

Abstract: Estimating the Tree of Life will likely involve a two-step procedure, where in the first step trees are estimated on many genes, and then the gene trees are combined into a tree on all the taxa. The computational problems involved here are substantial and interesting – even if the gene trees are correctly estimated, because the individual gene trees may not include all the species, finding the true tree that contains all the species is NP-hard. Furthermore, estimated gene trees may not be correct, making the estimation problem additionally challenging. Finally, the true gene trees may not agree with each other or with the species tree, due to biological processes such as deep coalescence, gene duplication and loss, and horizontal gene transfer.
In this talk, I will present new algorithms for these problems. The first algo- rithm is a method called SuperFine (Swenson et al., Syst Biol 2012 and Nguyen et al., J Alg Mol Biol 2012) that handles the case where there should not be con- flict between true gene trees. SuperFine is a “booster” for supertree methods, and produces highly accurate supertrees on large datasets. We then address the case where gene trees can differ form the species tree (and from each other) due to incomplete lineage sorting. For this problem, we present algorithms for the MDC (minimize deep coalescence) problem, taking gene tree estimation error into account (Yu et al., RECOMB and J Comp Biol 2011, Bayzid et al., J Comp Biol 2012). We also explore the use of “bin-and-conquer” to improve the ac- curacy and/or scalability of coalescent-based species tree methods (Bayzid and Warnow, Bioinformatics 2012).

(By invitation of the hiring committee)

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Abstract: Consider a rational homogeneous variety X. (For example, take X to be the Grassmannian Gr(k,n) of k-planes in complex n-space.) The Schubert classes of X form a free additive basis of the integral homology of X. Given a Schubert class [S], represented by a Schubert variety S in X, Borel and Haefliger asked: aside from the Schubert variety, does [S] admit any other algebraic representatives? I will discuss this, and related questions, in the case that X is Hermitian symmetric.

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Abstract: Crystals play a critical role in the design of novel devices.
In particular, the surface of a crystal can evolve with time and give rise to
a variety of interesting structures used in applications. In this evolution, several length and time scales,
from the atomistic to the continuum, are implicated; the description of their linkages poses challenging
mathematical questions.
How can surface evolution at large scales, possibly described by continuum laws, emerge from the motion of
mesoscale crystal defects? And how does the description of such defects
arise from atomistic motion? In this talk, I will address recent progress and open challenges
in answering these questions.