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A live feed of seminars and special events in the upcoming week.

- January 1st, 2013 (10:00am - 10:50am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**Discussion session**Speaker:**TBA, penn State**Location:**MB216- January 8th, 2013 (02:30pm - 03:45pm)
**Seminar:**Logic Seminar**Title:**Implicit definability in arithmetic, part 1.**Speaker:**Stephen G. Simpson, Pennsylvania State University**Location:**MB315**Abstract:**http://www.math.psu.edu/simpson/papers/arith-sing.pdfThe goal of these talks is to present Leo Harrington's unpublished 1975 theorems concerning implicit definability over the natural number system N,+,x,=. In this first talk I shall present some relevant background material including a 1972 theorem of Hisao Tanaka. The 1972 theorem states that every nonempty arithmetical set of reals contains an arithmetical singleton. This session will also include a brief organizational meeting where we will fill out this seminar's spring semester calendar.

- January 8th, 2013 (03:30pm - 05:00pm)
**Seminar:**CCMA PDEs and Numerical Methods Seminar Series**Title:**Numerical Study of Geometric Multigrid Methods on CPU–GPU Heterogeneous Computers**Speaker:**Chensong Zhang, Academy of Mathematics and System Sciences, Chinese Academy of Sciences**Location:**MB023- January 10th, 2013 (02:30pm - 03:30pm)
**Seminar:**Noncommutative Geometry Seminar**Title:**Coherent states, geometry and quantization, 1**Speaker:**Nigel Higson, Penn State University**Location:**MB106- January 15th, 2013 (11:15am - 12:05pm)
**Seminar:**Combinatorics/Partitions Seminar**Title:**An Unexpected Congruence Modulo 5 for 4--Colored Generalized Frobenius Partitions**Speaker:**James Sellers, PSU**Location:**MB106In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\phi_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $c\phi_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. In 2011, Baruah and Sarmah proved a number of congruence properties for $c\phi_4$, all with moduli which are powers of 4. In this brief note, we add to the collection of congruences for $c\phi_4$ by proving this function satisfies an unexpected result modulo 5. The proof is elementary, relying on Baruah and Sarmah's results as well as work of Srinivasa Ramanujan.

- January 15th, 2013 (02:30pm - 03:45pm)
**Seminar:**Logic Seminar**Title:**Implicit definability in arithmetic, part 2.**Speaker:**Stephen G. Simpson, Pennsylvania State University**Location:**MB315**Abstract:**http://www.math.psu.edu/simpson/papers/arith-sing.pdfIn part 2 I shall begin by stating Harrington's unpublished 1975 theorems concerning implicit definability over the natural number system N,+,x,=. After that, I shall prove a simplified version of the theorems. The full version will be proved in part 3.

- January 15th, 2013 (02:30pm - 03:30pm)
**Seminar:**GAP Seminar**Title:**Homology of partition posets and Koszul duality of operads**Speaker:**Benoit Fresse, University of Lille 1**Location:**MB106I consider partitions of a set with r elements {1...r}. This set forms a poset, with the order relation given by the refinement of partitions. Let K(r) be the simplicial complex formed by chains of partitions which starts at the smallest element and ends at the largest element of this poset. The goal of this talk is to give a conceptual proof, in terms of a Koszul duality result for operads, that the cohomology of this poset K(r) is identified with the Lie representation of the symmetric groups in degree r-1, and vanishes otherwise. I will relate partition posets to tree complexes, and I will use this picture to give an intuitive introduction to this application of the theory of operads.

- January 15th, 2013 (03:30pm - 06:00pm)
**Seminar:**Working Seminar: Dynamics and its Working Tools**Title:**Commutator methods with applications to the spectral analysis of dynamical systems, I**Speaker:**Rafael Tiedra de Aldecoa, Pontiﬁcia Universidad Católica de Chile**Location:**MB216- January 15th, 2013 (03:30pm - 04:45pm)
**Seminar:**Job Candidate Talk**Title:**The Inverse Galois Problem for PSL_2(F_p)**Speaker:**David Zywina**Location:**MB114**Abstract:**http://The Inverse Galois Problem asks whether every finite group occurs as the Galois group of some extension of the rationals. This problem is still wide open, even in the special case of simple groups. In this talk, we will explain why each of the simple groups PSL_2(F_p) occurs as a Galois group; these are the "simplest" simple groups for which the Inverse Galois Problem was not completely settled. Our Galois extensions will arise by studying the cohomology of a well-chosen surface. An important role will also be played by the L-functions of certain elliptic curves.

- January 16th, 2013 (04:40pm - 05:30pm)
**Seminar:**Applied Algebra and Network Theory Seminar**Title:**Eigenvectors of tensors and Waring decomposition**Speaker:**Luke Oeding, U.C. Berkeley**Location:**MB106Waring’s problem for polynomials is to write a given polynomial as a minimal sum of powers of linear forms. The minimal number of summands required in a Waring decomposition (the Waring rank) is related to secant varieties. I will explain recent work of Landsberg and Ottaviani that unified and generalized many constructions for equations of secant varieties via vector bundle techniques. With Ottaviani we have turned this construction into effective algorithms to actually find the Waring decomposition of a polynomial (provided the Waring rank is below a certain bound). Our algorithms generalize Sylvester’s algorithm for binary forms, using an essential new ingredient – eigenvectors of tensors. Of course a naive algorithm always exists, but is rarely effective. I will explain how computations using linear algebra make our algorithms effective. Given time, I will demonstrate our Macaulay2 implementations.

- January 17th, 2013 (11:15am - 12:05pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**Hyperdeterminants of polynomials**Speaker:**Luke Oeding, U C Berkeley**Location:**MB106Hyperdeterminants were brought into a modern light by Gelfand, Kapranov, and Zelevinsky in the 1990's. Inspired by their work, I will answer the question of what happens when you apply a hyperdeterminant to a polynomial (interpreted as a symmetric tensor). The hyperdeterminant of a polynomial factors into several irreducible factors with multiplicities. I identify these factors along with their degrees and their multiplicities, which both have a nice combinatorial interpretation. The analogous decomposition for the mu-discriminant of polynomial is also found. The methods I use to solve this algebraic problem come from geometry of dual varieties, Segre-Veronese varieties, and Chow varieties; as well as representation theory of products of general linear groups.

- January 17th, 2013 (02:30pm - 03:30pm)
**Seminar:**Noncommutative Geometry Seminar**Title:**Coherent states, geometry and quantization, 2**Speaker:**Nigel Higson, Penn State University**Location:**MB106- January 17th, 2013 (02:30pm - 03:20pm)
**Seminar:**PMASS Colloquium**Title:**Proofs from the Book**Speaker:**Sergei Tabachnikov, Penn State**Location:**MB113The famous mathematician of the last century, Paul Erdos, often referred to "The Book" in which God keeps the most elegant proofs of mathematical theorems. So, he would say: "This is a proof from the Book", or "This is a correct proof, but not from the Book". In fact, a book called "Proofs from the Book" was written by M. Aigner and G. Ziegler, and this is one of the most popular mathematical books ever. In this talk, I shall present some proofs of great theorems from this book, and some other ones that, in my opinion, belong there.

- January 18th, 2013 (12:20pm - 01:30pm)
**Seminar:**CCMA Luncheon Seminar**Title:**How to search for transition states/saddle points?**Speaker:**Qiang Du, Pennsylvania State University, Mathematics**Location:**MB114Exploring complex energy landscape is a challenging issue in many applications. Besides locating equilibrium states, it is often also important to identify the transition states given by saddle points. In this talk, we will discuss the mathematics and algorithms, in particular, the shrinking dimer dynamics, developed to compute transition states. Some applications will be considered, including the study of critical nuclei morphology in solid state transformations, optimal photonic crystal design and the generalized Thomson problem.

- January 21st, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**MLK DAY**Speaker:**NO SEMINAR**Location:**MB114- January 22nd, 2013 (02:30am - 04:00am)
**Seminar:**CCMA PDEs and Numerical Methods Seminar Series**Title:**A posteriori error estimation, adaptive mixed FEM and convergence for convection-diffusion-reaction equations**Speaker:**Xiaoping Xie, School of Mathematical Sciences Sichuan University Chengdu 610064, China**Location:**MB023A new technique of residual-type a posteriori error analysis is developed for the lowest-order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations in two- or three-dimension. Both centered mixed scheme and upwind-mixed scheme are considered. The a posteriori error estimators, derived for the stress variable error plus scalar displacement error in $L^{2}$-norm, can be directly computed with the solutions of the mixed schemes without any additional cost, and are full robust with respect to inhomogeneities and anisotropy of the diffusion-dispersion tensor. Local efficiency dependent on local or global variations in coefficients is obtained without any saturation assumption, and holds from the cases where convection or reaction are not present to convection-or reaction-dominated problems. An adaptive mixed FEM is also proposed based the a posteriori error estimation. The convergence is analyzed without using any quasi orthogonality for stress and displacement variables and without marking oscillation dependent on discrete solutions and data. Numerical experiments are reported to verify the theoretical results. This is a joint work with Shaohong Du (Chongqing Jiaotong University).

- January 22nd, 2013 (10:00am - 10:50am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**stability of traveling waves I**Speaker:**Toan Nguyen, Penn State**Location:**MB216- January 22nd, 2013 (02:30pm - 03:45pm)
**Seminar:**Logic Seminar**Title:**Random continuous functions.**Speaker:**Adrian Maler, Pennsylvania State University**Location:**MB315This talk is an introduction to the theory of random continuous functions on Cantor space and random closed subsets of that space. Our source is the paper "Algorithmic randomness of continuous functions," Archive for Mathematical Logic, 2008, http://link.springer.com/article/10.1007%2Fs00153-007-0060-4, by Barmpalias, Brodhead, Cenzer, Remmel, and Weber.

- January 22nd, 2013 (02:30pm - 03:30pm)
**Seminar:**GAP Seminar**Title:**Twistor theory for generalized complex manifolds**Speaker:**Justin Sawon, University of North Carolina**Location:**MB106A generalized complex structure (in the sense of Hitchin) on a manifold is an endomorphism of $T\oplus T^*$ with square $-Id$, satisfying a certain integrability condition. Complex structures and symplectic structures yield natural generalized complex structures, and generalized complex geometry locally looks roughly like a product of complex and symplectic geometry. In the case of a hyperkahler manifold $M$, there is an $S^2\times S^2$-family of generalized complex structures compatible with the metric. We show that these structures can be assembled into a generalized complex structure on $Z=M\times S^2\times S^2$, which we call the ``generalized twistor space''. Developing a generalized twistor correspondence remains an open problem. This is joint work with Rebecca Glover.

- January 22nd, 2013 (03:30pm - 06:00pm)
**Seminar:**Working Seminar: Dynamics and its Working Tools**Title:**Commutator methods with applications to the spectral analysis of dynamical systems, II.**Speaker:**Rafael Tiedra de Aldecoa, Pontiﬁcia Uni- versidad Católica de Chile,**Location:**MB216ATTENTION: there will be third lecture of this series on MONDAY January 28 from 3:35 to 5:30 pm.

- January 23rd, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**Time delay and Calabi invariant in classical scattering theory**Speaker:**Rafael Tiedra de Aldecoa, Pontiﬁcia Universidad Católica de Chile**Location:**MB114We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H_0,H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud-Wigner formula of quantum mechanics. Using recent results of V. Buslaev and A. Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincaré scattering map. Our results are applied to dispersive Hamiltonians and to Hamiltonians on the Poincaré ball. This is a joint work with Antoine Gournay (University of Neuchâtel).

- January 24th, 2013 (10:00am - 10:50am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**stability of traveling waves II**Speaker:**Toan Nguyen, Penn State**Location:**MB216- January 24th, 2013 (11:15am - 12:05pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**Semipositivity in positive characteristics**Speaker:**Zsolt Patakfalvi, Princeton University**Location:**MB106Results of Griffiths, Fujita, Kawamata, Viehweg, Kollar, etc. stating semipositivity of relative canonical bundles and of the pushforwards of their powers were crucial in the development of modern algebraic geometry. Most of these results required the characteristic zero assumption, partially due to the use of Hodge theory. In this talk I present semi-positivity results in positive characteristics. The main focus is moduli theoretic situations, in which the best known results in positive characteristics were for families of stable curves by Szpiro and Kollar and for K3 surfaces by Maulik. I present results for arbitrary fiber dimensions allowing sharply F-pure (char p equivalent of log canonical) singularities and semi-ample or ample canonical sheaves for the fibers. I will also discuss some applications: projectivity of proper coarse moduli spaces, characteristic zero implications and a special case of subadditivity of Kodaira dimension in the above mentioned moduli setting.

- January 25th, 2013 (12:20pm - 01:30pm)
**Seminar:**CCMA Luncheon Seminar**Title:**Tsunamis**Speaker:**Harvey Segur, University of Colorado Boulder, Applied Mathematics**Location:**MB114This is an introduction to the afternoon talk with the same title.

- January 25th, 2013 (03:35pm - 04:25pm)
**Seminar:**Computational and Applied Mathematics Colloquium**Title:**Tsunamis**Speaker:**Harvey Segur, University of Colorado Boulder, Applied Mathematics**Location:**MB106Tsunamis have gained worldwide attention over the past decade, primarily because of the destruction caused by two tsunamis: one that killed more than 200,000 people in coastal regions surrounding the Indian Ocean in December 2004; and another that killed 15,000 more and triggered a severe nuclear accident in Japan in March 2011. This talk has three parts. It begins with a description of how tsunamis work: how they are created, how they propagate and why they are dangerous. This part involves almost no mathematics, and should be understandable to everyone. The second part of the talk is about the operational models now being used to provide tsunami warnings and forecasts. These models predict some features of tsunamis accurately, and other features less accurately, as will be discussed. The last part of the talk is more subjective: what public policies could be enacted to mitigate some of the dangers of tsunamis? Much of the material in this talk appeared in a paper by Arcas & Segur, Phil. Trans. Royal Soc. London, 370, 2012.

- January 28th, 2013 (09:00am - 10:00am)
**Seminar:**Job Candidate Talk**Title:**A new twist on the Carleson operator**Speaker:**Lillian Pierce**Location:**MB114Must the Fourier series of an L^2 function converge pointwise almost everywhere? In the 1960's, Carleson answered this question in the affirmative, by studying a particular type of maximal singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will introduce the Carleson operator and survey several of its generalizations, and then describe new joint work with Po Lam Yung that introduces curved structure to the setting of Carleson operators.

- January 28th, 2013 (02:15pm - 03:15pm)
**Seminar:**Job Candidate Talk**Title:**Equidistribution and Counting for Polygonal Billiards**Speaker:**Jayadev Athreya**Location:**MB114We will survey some results on equidistribution and counting for billiards in Euclidean polygons. These results are united by the the fact they use the geometry and dynamics of the SL(2, R) action on the moduli space of holomorphic (and sometimes meromorphic) quadratic differentials on Riemann surfaces, which serves as a renormalization system for billiards. This talk will include joint work with G. Forni, joint work with J. Chaika and S. Lelievre, and joint work with A. Eskin and A. Zorich.

- January 28th, 2013 (03:30pm - 05:00pm)
**Seminar:**Probability and Financial Mathematics Seminar**Title:**Optimal investment with high-watermark performance fee**Speaker:**Mihai Sirbu, University of Texas, Austin, Mathematics, ***NOTE CHANGE OF ROOM***: This seminar will be held in **122 BUSINESS BUILDING****Location:**MB106We consider the problem of optimal investment and consumption when the investment opportunity is represented by a hedge-fund charging proportional fees on profit. The value of the fund evolves as a geometric Brownian motion and the performance of the investment and consumption strategy is measured using discounted power utility from consumption on infinite horizon. The resulting stochastic control problem is solved using dynamic programming arguments. We show by analytical methods that the associated Hamilton-Jacobi-Bellman equation has a smooth solution, and then obtain the existence and representation of the optimal control in feedback form using verification arguments. Joint work with Karel Janecek.

- January 28th, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**Large deviations for coded systems**Speaker:**Dan Thompson, Ohio State University**Location:**MB114We establish a Large Deviations Principle (LDP) for a large class of symbolic systems previously studied by Climenhaga and myself. The techniques are suitable for adaptation beyond the symbolic setting, and provide a new framework to establish LDP for systems with non-uniform structure. I will begin with a brief summary of the theory of Large Deviations in dynamical systems. I will explain the results via the application to the S-gap shifts: a natural family of symbolic spaces which are easy to define but which can be challenging to study! This is joint work (in progress) with Vaughn Climenhaga (Houston) and Kenichiro Yamamoto (Tokyo Denki).

- January 28th, 2013 (04:00pm - 05:00pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**Quadratic forms, primes, and the circle method**Speaker:**Lillian Pierce, University of Oxford**Location:**MB216This talk will describe joint work with Roger Heath-Brown on a new application of the circle method to pairs of quadratic forms, via a novel two-dimensional analogue of Kloosterman's version of the circle method. As a result, we prove (under a mild geometric constraint) that any two quadratic forms with integer coefficients, in 5 variables or more, simultaneously attain prime values infinitely often.

- January 29th, 2013 (02:30am - 04:00am)
**Seminar:**CCMA PDEs and Numerical Methods Seminar Series**Title:**A Preconditioner For Ensemble Kalman Filter**Speaker:**Yicun Zhen, Dept. of Maths, Penn State University**Location:**MB023- January 29th, 2013 (09:30am - 11:00am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**stability of traveling waves III**Speaker:**Toan Nguyen, Penn State University**Location:**MB216- January 29th, 2013 (02:30pm - 03:45pm)
**Seminar:**Logic Seminar**Title:**Partial randomness and strong separations, part 1.**Speaker:**Phil Hudelson, Pennsylvania State University**Location:**MB315**Abstract:**http://www.personal.psu.edu/wmh129/abstract_for_logic_seminar.pdfAlgorithmic randomness and Kolmogorov complexity respectively provide recursion-theoretic frameworks for the study of probability theory and information theory. In this talk we will prove a new strong separation for partial randomness concepts. Two partial randomness definitions, pwt-f-random and dwt-f-random, can be characterized in terms of Kolmogorov complexity by generalizations of Schnorr's theorem: X is dwt-f-random if and only if KP(X_n)>f(n)+O(n) for all n, and X is pwt-f-random if and only if KA(X_n)>f(n)+O(n), where KP and KA mean prefix-free and a priori complexity respectively, and X_n means the length n initial segment of X. We will prove using a forcing argument that under suitable conditions on the function f, there is an X such that KP(X_n)>f(n) for all n, but no Y recursive in X satisfies KP(Y_n)>f(n)+2log_2(f(n))+O(1) for all n (also the analogous result for KA). This theorem, which will be published in The Journal of Symbolic Logic, generalizes the theorem of Miller that there exists a Turing degree of effective Hausdorff dimension exactly 1/2. The new theorem also implies that there exists an X of effective Hausdorff dimension 1 which does not compute a Martin-Lof random, a result originally due to Greenberg and Miller.

- January 29th, 2013 (03:30pm - 06:00pm)
**Seminar:**Working Seminar: Dynamics and its Working Tools**Title:**What we know and do not know about slow entropy and related notions for group actions.**Speaker:**Anatole Katok**Location:**MB216- January 29th, 2013 (03:45pm - 05:00pm)
**Seminar:**Probability and Financial Mathematics Seminar**Title:**Stochastic Perron's method in linear and non-linear problems**Speaker:**Mihai Sirbu, University of Texas, Austin, Mathematics**Location:**MB114We introduce a stochastic version of the classical Perron's method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub and super) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a martingale along the solutions of the stochastic differential equation. The unique viscosity solution is actually equal to the expected payoff. This amounts to a verification result (Ito's Lemma) for non-smooth viscosity solutions of the linear parabolic equation. We show how the method can be extended to non-linear problems, like free boundary problems associated to optimal stopping or Dynkin games and Hamilton-Jacobi-Bellman equations in stochastic control. The presentation is based on joint work with Erhan Bayraktar.

- January 31st, 2013 (09:30am - 11:00am)
**Seminar:**Ph.D. Thesis Defense**Title:**"Representation of Integers by a Family of Cubic Forms in Seven Variavbles"**Speaker:**Manoj Verma, Adviser: Robert C. Vaughan, Penn State**Location:**322 Sackett Building**Abstract:**http://We derive asymptotic formulas for the number of representations of zero inside a box and the number of representations of a large positive integer inside a box of suitable dimensions by cubic forms that can be written as $L_1(x_1,x_2, x_3)Q_1(x_1,x_2, x_3) +L_2 (x_4,x_5, x_6)Q_2 (x_4,x_5, x_6) + a_7x_7^3$ where $L_1$ and $L_2$ are linear forms, $Q_1$ and $Q_2$ are quadratic forms and $a_7$ is a non-zero integer, under certain conditions on the coefficients.

- January 31st, 2013 (10:00am - 10:50am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**Large solutions for 1-D compressible Euler equations**Speaker:**Geng Chen, Penn State University**Location:**MB216Although the well-posedness of solutions with small total variation for the hyperbolic conservation laws in one space dimension, including compressible Euler equations as an example, has been fairly well understood, the large BV solutions are still wide open. In this talk, we focus on two topics on the large BV solutions for Euler equations: Singularity formation and smooth solutions; Wave interactions.

- January 31st, 2013 (11:15am - 12:05pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**Some problems of `Partitio Numerorum': Hybrid expressions**Speaker:**Robert C. Vaughan, Penn State University**Location:**MB106In the 1920s Hardy and Littlewood produced a series of seminal papers with the generic title "Some problems of `Partitio Numerorum'". The third of these has the subtitle "On the expression of a number as a sum of primes". In addition to primes they also consider their new method as a heuristic leading to a range of conjectures on a spread of additive problems. Many of these concern hybrid questions in which the summands can be combinations of primes and powers. In part this talk is a rather eclectic survey of some of these questions and their relatives and in part it is a description of some recent work of the speaker and his collaborators on questions of a "hybrid" kind.

- January 31st, 2013 (02:30pm - 03:20pm)
**Seminar:**PMASS Colloquium**Title:**Game Theory 101: numerical modeling of life events**Speaker:**Eii Byrne, Penn State**Location:**MB113This talk will illustrate the process of assigning numerical payoffs to model outcomes of multi-agent decision profiles that are not necessarily numerical in nature. The emphasis will be on modeling; i.e. the logic of how inequalities between the numerical payoffs within a game model agent preferences over outcomes. I will present example problems requiring no technical background.

- January 31st, 2013 (03:35pm - 04:25pm)
**Seminar:**Department of Mathematics Colloquium**Title:**Generalized exponents, family algebras and representations with simple spectrum**Speaker:**Alexandre Kirillov, University of Pennsylvania**Location:**MB114In representation theory there are several "evergreen" subjects such as symmetric functions, multiplicity of weights, tensor invariants, etc. One of such subject, less known among non-experts, is the theory of generalized exponents, introduced by Kostant half a century ago and still resisting all endeavors towards explicit computation. The collection of generalized exponents is the so-called q-analogue of the multiplicity of the zero weight. As one possible approach to the problem, I introduced in 2000 the notion of a family algebra associated to any irreducible representation of a simple Lie algebra. The structure of these algebras is closely related to generalized exponents. For representations with simple spectrum, family algebras have relatively a simple structure, and this allows to compute explicitly generalized exponents for many irreducible representations.