Das Mathematische Kolloquium ist eine gemeinsame wissenschaftliche Veranstaltung des gesamten Mathematischen Instituts. Es steht allen Interessierten offen und richtet sich neben den Mitgliedern und Mitarbeitern des Instituts auch an die Studierenden. Das Kolloquium findet dreimal im Semester am Donnerstag um 17:00 s.t. im Hörsaal II, Albertstr. 23b statt. Danach (gegen 16:15) gibt es Kaffee und Kekse, zu dem der vortragende Gast und alle Besucher eingeladen sind.
Pure mathematics in crisis?
Thursday, 25.4.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
What is a rigorous mathematical proof? In mathematics\ndepartments we teach the undergraduates the answer to this question: a\nproof is a series of logical deductions, each one justified by\nprevious conclusions and the axioms of mathematics. In my talk I will\nargue that the "proofs" that we produce in our research are not of\nthis nature at all. The main reason for this is that mathematical\nproofs in the literature are written by humans, and hence contain\nomissions (often) and errors (occasionally). Some of the errors are\nunfixable, and some of the omissions are serious. I will speak about\npractical consequences of this, giving explicit examples of issues\nacross pure mathematics. Many modern proofs rely on ideas which are\n"known to the experts", and sometimes there is no satisfactory\ntreatment of these ideas in the literature. In some cases these\nexperts are dying out and are not being replaced. If our work is not\nreproducible, is it actually mathematics?\n\nI used to be an algebraic number theorist until recently, but after I\nbegan to worry about these issues I spent a year learning how computer\nscientists do formally verified mathematics using computer proof\nsystems. Not only did this change the way I thought about research but\nit also changed the way I taught. I now use these computer tools as\npart of our basic introduction to proof course at Imperial College\nLondon.\n\nI will talk about the problems I believe are facing pure mathematics,\nand to what extent computers can help to solve them.\n
Thursday, 9.5.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Hypoelliptic Laplacian, index theory and the trace formula
Thursday, 23.5.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
The hypoelliptic Laplacian is a family of operators, indexed by b ∈ R^∗_+ ,\nacting on the total space of the tangent bundle of a Riemannian manifold, that\ninterpolates between the ordinary Laplacian as b → 0 and the generator of\nthe geodesic flow as b → +∞ . These operators are not elliptic, they are not\nself-adjoint, they are hypoelliptic.\nThe hypoelliptic deformation preserves subtle invariants of the Laplacian. In\nthe case of locally symmetric spaces, the deformation is essentially isospectral.\nIn a first part of the talk, I will describe the geometric construction of the\nhypoelliptic Laplacian in the context of de Rham theory. In a second part, I\nwill explain applications to the trace formula.
Thursday, 6.6.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Arithmetic of Curves
Thursday, 6.6.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
In 1922, while studying rational solutions to polynomial equations\n \nf(x,y)=0\n\nin two variables, Mordell had the astounding idea that the structure of the problem might be intimately related to the geometry and topology of the complex solution set. This became known as the Mordell conjecture, stating that the equation has only finitely many rational solutions when the genus is at least two.\n\nThis was proved by Faltings in 1983 in a landmark result that developed an astounding array of techniques in arithmetic geometry and led to great advances in numerous areas of number theory and algebraic geometry. This talk will give an eclectic survey of this history and discuss the harder problem of finding all rational solutions to such equations, often called the effective Mordell conjecture.\n\n
Thursday, 4.7.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Around the residue symbol
Thursday, 11.7.19, 16:15-17:15, Hörsaal II, Albertstr. 23b
Everybody knows the “residue" from complex analysis and Cauchy's residue\nformula. One can regard this as a one-dimensional theorem in the sense\nthat the complex plane has complex dimension one. There are several\ndifferent theories of multi-dimensional residues, all essentially\ncompatible, but in complicated ways. I will explain a picture due to A.\nParshin.\nWhereas Cauchy's residue formula implies a statement of the form “the\nsum of residues at all points of a fixed curve is zero", Parshin's\n2-dimensional generalization provides a nice analogous result stating\nthat “the sum of residues along all curves passing through a fixed\npoint" is zero. This talk will focus on the down-to-earth geometric\napproach of the Soviet school to these issues, which is not so\nwell-known in the Western world.\n (I will not talk about the following, because it would be much too\ntechnical, but of course the same result also immediately follows from\nGrothendieck's residue symbol, the approach more popular in the Western\nworld, but only after introducing f!, derived categories, local\ncohomology, etc.; in fact Grothendieck's theory even in the classical\none-dimensional case already relies on local cohomology).
Thursday, 25.7.19, 17:00-18:00, Hörsaal II, Albertstr. 23b