HCM Graduate Colloquium, Summer Term 2024
Organisers: Michel Alexis, Regula Krapf, Fred Lin, Luise Puhlmann, Christoph Thiele, Radu Toma
This seminar is organised as a BIGS event. The goal is to present topics from all areas of mathematics in an elementary and informal way. The talks should be accessible to a general mathematical audience and are mainly aimed at BIGS students.
Everybody (students, postdocs, faculty, guests) is welcome to attend.
If you would like to give a talk please contact us. The seminar will take place Wednesdays 15:15 - 16:45 in the Lipschitzsaal. The talks will usually take about one hour and there is the subsequent possibility to ask questions. Coffee, tea and cake will be served beforehand between 15:00 and 15:15 in the Plückerraum. A predecessor of the HCM Graduate Colloquium is the Basic Notions Seminar which took place until 2017:
Basic Notions Seminar Summer Term 2017
Date | Speaker | Topic |
---|---|---|
17.04.2024 | Sid Maibach (IAM) | An Exposition to Random Conformal Geometry |
24.04.2024 | Antonia Ellerbrock (DM) | Cost Allocation for Set Covering: the Happy Nucleolus |
29.05.2024 | Sil Linskens (MI) | Grothendieck's Homotopy Hypothesis |
12.06.2024 | Alexander West (IAM) | Minimizing the Willmore energy under a total mean curvature constraint |
03.07.2024 | Hendrik Baers (IAM) | Instability of the Fractional Calderón Problem |
Abstracts
April 17, 2024: Sid Maibach (IAM)
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Title: An Exposition to Random Conformal Geometry
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Abstract. In this talk I will present an overview of the research area sometimes
titled "random conformal geometry". This is the study of random objects
that are symmetric under holomorphic coordinate changes, making them
well-defined objects on Riemann surfaces. The key objects I will
introduce are "Schramm–Loewner evolution" random curves, "Gaussian free
field" random functions, and concepts from conformal field theory.
These appear universally in the study of systems at critical
temperature on 2D lattices as the spacing of the lattice goes to zero.
As an example of such a system I will discuss the Ising model.
I will also briefly touch upon my own research questions about the
correspondence between Kähler structures on the moduli spaces of the
underlying Riemann surfaces and large deviation principles for the
aforementioned random objects. However, instead of going into detail I
will illustrate more examples of probabilistic constructions from which
these universal objects emerge.
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Title: Cost Allocation for Set Covering: the Happy Nucleolus
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Abstract. Imagine you were a delivery service operator who wants to visit a certain set of customers.
There is a given set of possible tours. Each tour serves a subset of the customers and has a certain cost.
You can use as many of these tours as you like.
Your task is to set a price for each customer.
Of course, you want to charge as much money as possible, but without losing customers.
We assume that any group could leave your delivery service and self-fund one of the given tours.
Thus, the summed prices of customers in this group should not exceed the cheapest cost for a tour of their own.
From here, we will build on previous work in the field of cooperative game theory and develop a fair cost allocation concept with efficient computation.
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Title: Grothendieck's Homotopy Hypothesis
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Abstract. In 1983, Grothendieck wrote the influential manuscript Pursuing stacks. In
this work he formulated the famous homotopy hypothesis: “homotopy types =
infinity-groupoids”. This was a deep insight, which completely changed our
understanding of the place of homotopy theory in broader mathematics. In
this talk I will motivate and contextualise the homotopy hypothesis, and
then explain its development since 1983.
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Title: Minimizing the Willmore energy under a total mean curvature constraint
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Abstract. The Willmore energy of a closed surface is the integral of the square of the mean curvature. It appears for example as the main term in the Helfrich energy, used to describe the bending energy of lipid bilayer cell membranes. Consequently, the minimization of the Willmore energy under various constraints has been studied extensively in the past few decades. In this talk, we consider the minimization of the Willmore energy in the class of surfaces with prescribed genus, while keeping a constraint on the total mean curvature and the area of the surface. This problem admits smooth minimizers for an arbitrary genus and a large class of constraints and we will talk about how this existence result can be obtained.
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Title: Instability of the Fractional Calderón Problem
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Abstract. The Calderón problem is one of the classic examples of an inverse problem. It is about determining the conductivity of a medium by making voltage and current measurements on its boundary. We consider the fractional formulation of the problem and prove exponential instability. Physically that means, small differences in the measurements can lead to very different conclusions about the conductiviy.
We will start by introducing the classical formulation of the Calderón problem and then, motivated by this, formulate the fractional version. We will compare some known results and then, finally, we will proceed to the proof of the instability result. There we will see two rather very different mechanisms leading to instability.
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