[McNinch-web]Tag: algebraic-groups2019-08-09T12:53:23-0400two talks in the NIU mathematics department ColloquiumGeorge McNinchgeorge.mcninch@tufts.edu2019-03-23T11:00:00-0400<p>I give two talks at the <a href="http://www.math.niu.edu/colloq/">Northern Illinois University mathematics colloquium</a>.</p><p>Here are the abstracts, and links for the slides (as pdf documents):</p><ul><li><p><strong>Group cohomology and Levi decompositions for linear groups</strong> (Mar 21)</p><p><em>Abstract</em>: If G is a linear algebraic group over a field F, we
describe what the Hochschild cohomology of G with coefficients in
linear representations of G says about those algebraic groups
which are extensions of G by connected unipotent algebraic groups
over F. If G is reductive and if F has characteristic zero -- say,
if F is the field of complex numbers -- one knows that every such
extension is trivial. But if F has positive characteristic, there
are non-trivial extensions -- i.e. there are linear groups with no
Levi decomposition. The talk will give details and examples about
these notions and results.</p><p>Here are <a href="assets/slides/2019-03---NIU--Talk-1--transparencies.pdf">the slides</a></p></li></ul><ul><li><p><strong>Some tools for the study of reductive groups over local fields</strong> (Mar 22)</p><p><em>Abstract</em>: Let K be a local field -- i.e. the field of fractions of a
complete discrete valuation ring A. The study of linear algebraic
groups G over such fields K has applications in number theory and
algebraic geometry. Some reductive groups (“split groups”) have
models over A which are reductive. But e.g. if G does not become
split upon base change with any unramified extension of K, it can
happen that G has no reductive model. Our interest here is in an
interesting family of models for G -- the so-called parahoric
group schemes P. If k denotes the residue field of A, then by
“base-change”, P determines a linear algebraic group P_k over k.
When P is not reductive, we investigate the question: does P_k
have a Levi decomposition (as in the first talk)? This second
talk will include a good bit of example-oriented background
discussion.</p><p>Here are <a href="assets/slides/2019-03---NIU--Talk-1--transparencies.pdf">the slides</a>.</p></li></ul>Workshop for Jens Carsten Jantzen’s 70th birthdayGeorge McNinchgeorge.mcninch@tufts.edu2018-11-25T08:00:00-0500<p>I gave one of the lectures at the workshop on <a href="https://www.mpim-bonn.mpg.de/node/8209"><em>Algebraic Groups, Lie
Algebras and their
Representations</em></a> held at the
<a href="https://www.mpim-bonn.mpg.de">Max Planck Institute</a> (Bonn, Germany)
to celebrate the 70th birthday of <a href="https://wikipedia.org/wiki/Jens_Carsten_Jantzen">Jens Carsten
Jantzen</a>.</p><p><img src="assets/images/2018-11-Poster-Jantzen70.jpg" alt="image" /></p>Southeast Lie Theory meeting -- UGeorgia (Athens)George McNinchgeorge.mcninch@tufts.edu2018-06-14T08:00:00-0400<p>I gave a talk on <em>Reductive subgroup schemes of a parahoric group
scheme</em> at the 10th annual <a href="https://www.math.lsu.edu/~pramod/selie/10/">Southeast Lie Theory
workshop</a> at the
University of Georgia (Athens) during June 10–12, 2018.</p><p>Here are the <a href="assets/slides/2018-05---Athens---transparencies.pdf">slides for my talk</a>.</p><p>And here is a conference photo (I seem to be ’way at the top of the stairs!):</p><p><img src="assets/images/2018-06-SELieConfPic.jpg" alt="conference photo" title="photo" /></p>Lyon meeting on algebraic groupsGeorge McNinchgeorge.mcninch@tufts.edu2018-05-30T08:00:00-0400<p>I attended some workshops during the <a href="https://geolang.sciencesconf.org/">Trimestre
thématique</a> on <em>Groupes algébriques
et géométrisation du programme de Langlands</em> in Lyon (France).</p><p>During this time, I followed the <em>Mini-cours sur la correspondence de
Langlands locale pour les corps locaux d'égale caractéristique,
d'après Genestier-Lafforgue</em> given by <a href="https://web.math.princeton.edu/~smorel/">Sophie
Morel</a> and <a href="https://webusers.imj-prg.fr/~benoit.stroh/">Benoît
Stroh</a>.</p><p>Following the minicourse, there was a conference during May 22-25, 2018:</p><p><img src="assets/images/2018-05-Lyon-poster-small.jpg" alt="" /></p><p>In this conference, I gave a lecture on <em>Reductive subgroup schemes of
a parahoric group scheme</em>.</p>Special Session -- AMS sectional meeting at Northeastern UnivGeorge McNinchgeorge.mcninch@tufts.edu2018-04-23T08:00:00-0400<p>The Spring Eastern Sectional Meeting of the AMS
was held at Northeastern University (Boston, MA)
April 21-22, 2018.</p><p>At this meeting, there was a special session on <a href="http://www.ams.org/meetings/sectional/2252_program_ss15.html#title">Combinatorial Aspects
of Nilpotent
Orbits</a>
organized by Anthony Iarrobino (Northeastern University), Leila
Khatami (Union College) and Juliana Tymoczko (Smith College).</p><p>In this session, I gave a presentation on <em>Centralizers of nilpotent
elements</em>; here are the <a href="assets/slides/2018-04---Northeastern---nilpotent-centralizers---transparencies.pdf">slides for my
talk</a>.</p>MSRI -- Representations of Finite & Algebraic GroupsGeorge McNinchgeorge.mcninch@tufts.edu2018-04-14T08:00:00-0400<p>I attended the <a href="http://www.msri.org/web/cms">MSRI</a> program
<a href="http://www.msri.org/workshops/820">Representations of Finite and Algebraic
Groups</a> Apr 9 - 13, 2018, organized
by Robert Guralnick (University of Southern California), Alexander
Kleshchev (University of Oregon), Gunter Malle (Universität
Kaiserslautern), Gabriel Navarro (University of Valencia), and Pham
Tiep (Rutgers University)</p><p><img src="assets/images/2018-MSRI-building.jpg" alt="image" /></p>ARTIN workshop -- University of ManchesterGeorge McNinchgeorge.mcninch@tufts.edu2017-09-15T12:00:00-0400<p>The <a href="https://sites.google.com/view/artin51-manchester/home">Workshop on Lie Theory, Representation Theory and Algebraic
Groups</a> was
held at the University of Manchester (UK) Sept 11-14, 2017.</p><p>I contributed a lecture on <em>Nilpotent orbits of a reductive group over
a local field</em>.</p>Workshop at Newcastle UniversityGeorge McNinchgeorge.mcninch@tufts.edu2017-09-09T12:00:00-0400<p>My colleague <a href="https://www.staff.ncl.ac.uk/david.stewart/">David
Stewart</a> organized a
<a href="https://sites.google.com/view/prgs-newcastle/home">workshop on Pseudo-reductive
groups</a> in
September 2017, which was partially funded by the Heilbronn Institute.</p><p>In this workshop, Gopal Prasad gave a mini-course on his work with
Conrad and Gabber on pseudo-reductive groups.</p><p>I contributed a lecture on <em>Reductive subgroups of parahoric group schemes</em>.
Here is the abstract for my talk:</p><blockquote><p>Let K be the field of fractions of a complete discrete valuation ring
A with residue field k, and let G be a connected reductive algebraic
group over K. Suppose P is a parahoric group scheme attached to G. In
particular, P is a smooth affine A-group scheme having generic fiber
P_K = G; the group scheme P is in general not reductive over A. Assume
that G splits over an unramified extension of K.</p><p>The talk will give an overview of two results about G.</p><p>First, there is a closed and reductive A-subgroup scheme M of P for
which the special fiber M_k is a Levi factor of P_k. Moreover, the
reductive subgroups of G=P_K of the form M_K may be characterized.</p><p>Second, let X be a nilpotent section in Lie(P). We say that X is
balanced if the fibers C_K and C_k are smooth group schemes of the
same dimension, where C=C_P(X) is the scheme theoretic centralizer of
X. If X_0 is a given nilpotent element in the Lie algebra of the
reductive quotient of the special fiber P_k, we give results on the
possible <em>lifts</em> of X_0 to a balanced nilpotent section X of Lie(P).</p></blockquote>Special Session -- AMS sectional meeting at Bowdoin CollegeGeorge McNinchgeorge.mcninch@tufts.edu2016-09-26T12:00:00-0400<p>The Fall Eastern AMS sectional meeting was held at Bowdoin College in
Brunswick ME on Sept 24-25, 2016. Tony Iarrobino (Northeastern Univ),
Leila Khatami (Union College) and Julianna Tymoczko (Smith College)
organized a Special Session on <a href="http://www.ams.org/meetings/sectional/2238_program_ss18.html#title">Combinatorial Aspects of Nilpotent
Orbits</a>
at this meeting, and I contributed a talk on <em>Nilpotent elements and
sections</em>.</p><p>Here are the <a href="assets/slides/2016-09---Bowdoin---comparing-centralizers.pdf">slides to my talk</a>.</p>Worshop at the Centre Intrafacultaire Bernoulli (EPFL)George McNinchgeorge.mcninch@tufts.edu2016-07-30T12:00:00-0400<p>I was an academic visitor for about 3 weeks in July 2016 at the <a href="http://cib.epfl.ch/">Centre Intrafacultaire
Bernoulli</a>, l'École Polytechnique Fédérale de
Lausanne (EPFL, Switzerland) during the program on <em>Local
Representation Theory and Simple Groups</em> organized by Donna Testerman
(EPFL), Gunter Malle (TU Kaiserslautern), and Radha Kessar (City
University London).</p><p>I enjoyed following lectures by Markus Linckelmann (City University
London), Meinolf Geck (Universität Stuttgart), Marc Cabanes
(Université Paris Diderot), Britta Späth (Bergische Universität
Wuppertal), and Olivier Dudas (Université Paris Diderot) on - among
other things - block theory, and aspects of the
Deligne-Lusztig description of representations of finite reductive
groups.</p>Institut Mittag-Leffler workshopGeorge McNinchgeorge.mcninch@tufts.edu2016-06-01T08:00:00-0400<p>In May 2016, I attended a week-long workshop on <a href="http://www.mittag-leffler.se/workshop/branching-problems-reductive-groups"><em>Branching Problems
for Reductive
Groups</em></a>
at the <a href="http://www.mittag-leffler.se">Institut Mittag-Leffler</a> (Djursholm, Sweden).</p><p>I gave a lecture on <em>An overview of representations of reductive
algebraic groups</em>; essentially everything I discussed can be found in
the text of Jens Jantzen on the representation theory of algebraic
groups. I tried to focus on things that seemed relevant to the
difficulties presented by modular representation theory to “branching
problems”: (Hochschild) cohomology, the description of simple
representations for reductive groups, reduction mod \(p\) and the
Jantzen filtration of Weyl modules.</p><p>Here are the <a href="assets/slides/2016-05---Mittag-Leffler---reductive-reps--transparencies.pdf">slides from my talk</a>.</p>Skip Garibaldi -- Norbert Wiener Lectures at TuftsGeorge McNinchgeorge.mcninch@tufts.edu2014-04-23T12:00:00-0400<p><a href="http://www.garibaldibros.com/">Skip Garibaldi</a> gave the <a href="http://math.tufts.edu/seminars/lecturesWiener.htm">Wiener
Lectures in the Tufts University Math
Department</a> in
April 2014.</p><ul><li><strong>Some people have all the luck</strong> (public lecture)</li></ul><p><strong>Abstract:</strong> Winning a prize of at least $600 in the lottery is a
remarkable thing - for a scratcher ticket the odds are worse than
1-in-1200 and 1-in-9000 is a more typical figure. Some people have
won many of these large prizes, and clearly they are very lucky or
they buy a ton of lottery tickets. When we investigated records of
all claimed lottery prizes, we discovered that some people had won
hundreds of these prizes! Such people seem to be not just lucky, but
suspiciously lucky. I will explain what we thought they might have
been up to, what mathematics says about it, and what further
investigations revealed. This talk is about joint work with Lawrence
Mower, an investigative reporter for the Palm Beach Post, and Philip
B. Stark, professor and chair of the UC Berkeley Department of
Statistics.</p><ul><li><strong>Topological and generic methods in algebra</strong> (math major lecture)</li></ul><p><strong>Abstract:</strong> In calculus we take limits and think about graphs all
the time, and it would be nice to be able to use the same sorts of
techniques in courses like abstract linear algebra and abstract
algebra, even when you aren’t working with real or complex
numbers. I’ll explain how these techniques can be used and give some
examples of their application.</p><ul><li><strong>Simple algebraic groups and polynomial
invariants</strong> (colloquium lecture)</li></ul><p><strong>Abstract:</strong> The classical “linear preserver problem” asks: Given a
polynomial in finitely many variables, what is the group of linear
transformations that preserve it? This problem has been solved for
many interesting polynomials, usually by means that are special to the
particular polynomial under consideration. We turn this problem on
its head by starting with a polynomial that is preserved by a simple
algebraic group and observe that the full preserver can be described
by a general theorem. The results are new even in the case where the
field is the complex numbers, and as an application we shed some light
on a 125+ year old problem. This is joint work with Bob Guralnick.</p><p><img src="assets/images/2014---Wiener-Lecture-Poster.jpg" alt="poster" title="poster" /></p>