The current understanding of nature finds in the „Standard Model“ the most complete and verified theory (for now). The mathematics it involves heavily relies on Lie theory (Lie groups and Lie algebras). To better describe the universe, phisicists have come up with a „Supersymmetry“ theory (among others). This theory is described in terms of Lie superalgebras. The goal of this talk is to give an overview of which Lie algebras/superalgebras are used in Physics and why they help describing nature.
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25 entries found
 December

07. December17:00  18:00
Location: 48436AG Algebra, Geometrie und ComputeralgebraEmilio Rotilio, TU Kaiserslautern: Lie Superalgebras in Physics

05. December15:30  17:00
Location: 48436AG Algebra, Geometrie und ComputeralgebraDan Roche/US Naval Academy Annapolis: nteger Polynomial Sparse Interpolation with NearOptimal Complexity
We investigate algorithms to discover the nonzero coefficients and exponents of an unknown sparse polynomial, provided a way to evaluate the polynomial over any modular ring. This problem has been of interest to the computer algebra community for decades, and its uses include multivariate
polynomial GCD computation, factoring, and sparse polynomial arithmetic. Starting with the early works of Zippel, BenOr and Tiwari, and Kaltofen, one line of investigation has a key advantage in achieving the minimal number of evaluations of the polynomial, and has received considerable attention and improvements over the years. It is closely related to problems in coding theory and exponential analysis. The downside, however, is that these methods are not polynomialtime over arbitrary fields. A separate line of work starting with Garg and Schost and continuing with a few papers by the speaker and coauthors, has developed a different approach that works over any finite field. After years of improvements, the
complexity of both approaches over ZZ[x] is currently the same. They scale well in most aspects except for the degree; the bit complexity in both cases is currently cubic in the bitlengths of the exponents. By careful combination of the techniques in both approaches and a few new tricks, we are now able to overcome this hurdle. We present an algorithm whose running time is softlylinear in the size of the output and performs nearly the minimal number of evaluations of the unknown polynomial. Preliminary
implementation results indicate good promise for practical use when the unknown polynomial has a moderate number of variables and/or large exponents.  November

23. November17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraReda Chaneb, Université Paris Diderot: Basic sets for unipotent blocks of finite reductive groups
Let G(q) be a finite group of Lie type and l be a prime number not dividing q. Geck and Hiss have proved that, if l is good, the restriction of unipotent characters to lregular elements is a basic set for unipotent blocks of G(q). When l is bad, there are usually not enough unipotent characters to form a basic set. An alternative strategy to get a basic set is to construct projective characters satisfying some properties. For classical groups with connected center and l=2, by using a family of projective representations constructed by Kawanaka, Geck proved the existence of a basic set such that the decomposition matrix of the unipotent block has a unitriangular shape for this basic set. I will introduce those results and present generalizations of the results of Geck for the case of groups with disconnected center.

16. November17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraOlivier Dudas, Université Paris Diderot: On the unitriangular shape of decomposition matrices for finite reductive group
In order to classify the unipotent characters of finite reductive groups, Kawanaka constructed a family of representations induced from 'almost' unipotent groups, and conjectured a multiplicity formula for the unipotent constituents of these representations. I will explain how one can prove this formula providing some mild assumptions on unipotent classes. This can be used to show that the decomposition matrices of unipotent blocks have unitriangular shape. This is a joint work with O. Brunat and J. Taylor.

02. November17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraCaroline Lassueur: Lifting Morita equivalences with an endopermutation source
Motivated by current work of BoltjeKessarLinckelmann on the Picard group of Morita selfequivalences with endopermutation source, we consider the problem of whether Morita equivalences with an endopermutation source are liftable from positive characteristic to characteristic zero in general. The first part of the talk will give a gentle introduction to the concepts of vertices, sources and multiplicity modules in modular representation theory, while the second part of the talk will focus on the aforementioned lifting problem.
 October

26. October17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraAndrew Mathas (University of Sydney): Jantzen filtrations and graded Specht modules
The Jantzen sum formula is a classical result in the representation theory of the symmetric and general linear groups that describes a natural filtration of the Specht modules over any field. Analogues of this result exist for many algebras including the cyclotomic Hecke algebras of type A. Quite remarkably, the cyclotomic Hecke algebras of type A are now know to admit a Zgrading because they are isomorphic to cyclotomic KLR algebras. I will explain how to give an easy proof, and stronger formulation, of Jantzen sum formula for the cyclotomic Hecke algebras of type A using the KLR grading. I will discuss some consequences and applications of this approach.
 August

25. August14:00  15:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraShigeo Koshitani, University of Chiba: Brauer indecomposability of AlperinScott modules for finite nonabelian 2groups
We will be discussing the socalled AlperinScott module for the group algebra kG of a finite group G over a field k of characteristic p>0. It is a ppermutation indecomposable kGmodule such that it has the trivial module k in the socle with multiplicity one. Brauer indecomposabitily is considered in the 2011 paper by KessarKunugiMitsuhashi, and it is useful to get for instance a stable equivalence of Morita type in order to prove Broue's abelian defect group conjecture. In the above paper they prove the Brauer indecomposability for abelian pgroups. So as next step we will consider nonabelian 2groups cases.
(*) This is a part of joint work with Caroline Lassueur. 
09. August11:30  13:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraDaniel Schultz, TU Kaiserslautern: Modular Equations in Two Variables
By adding certain equianharmonic elliptic sigma functions to the
coefficients of the Borwein cubic theta functions, an interesting set of six
twovariable theta functions may be derived. These theta functions
invert a special case of Appell’s hypergeometric function and
satisfy several identities akin to those satisfied by the Borwein
cubic theta functions. In this talk I will discuss the modular
properties of these functions as well as the computation of their
modular equations, which turn out to be algebraic surfaces. An
application of these results is a new twoparameter family X^9 −
3*X^8 + 4*t*X^6 − 6*s*X^5 − 6*s*X^4 + 4*s*t*X^3 − 3*s^2*X + s^2 = 0
of solvable nonic equations.  July

25. July11:00  12:30
Location: 48210AG Algebra. Geometrie und Computer AlgebraMichael Monagan/Simon Fraser University: Toward High Performance Factorization
To factor a multivariate polynomial Wang's incremental
design must solve many multivariate polynomial diophantine equations
of the form sigma A + τ B = C for sigma and τ in Zp[x_{1},...,x_{n}].
We present a new efficient sparse approach.
We also modify the main Hensel lift may always be done modulo a machine prime.
Our goal is a high performance algorithm.
This is joint work with Baris Tuncer. 
20. July17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraYujiao Sun, Universität Stuttgart: Supercharacter theories for Sylow psubgroups of finite exceptional groups of Lie type
Classifying the conjugacy classes of the full unitriangular groups U_n(q) for all n and q is known to be a "wild" problem, where q is a power of some prime p. C.A.M. Andre using Kirillov's orbit method, and later N. Yan using a more elementary method constructed supercharacters and superclasses for U_n(q) as approximations of irreducible characters and conjugacy classes. One may conjecture that similar results can be obtained for Sylow psubgroups of all finite groups of Lie type. My PhD thesis was concerned with testing this for some exceptional groups of Lie type. I will present some of the results in this talk.