Speaker(s): Jakub Witaszek (University of Michigan)

]]>Given a structure, one can form a tree whose nodes are tuples from the structure, ordered by extension, and with each tuple labeled by its atomic type. This structure encodes the back-and-forth information of the structure and hence, by a back-and-forth argument, its isomorphism type. The tree of tuples appeared implicitly in the seminal Friedman-Stanley paper on Borel reducibility. With Montalban I showed that there are structures which cannot be recovered computably from their tree of tuples. I will talk about why we care about the tree of tuples, and about this result and what it has to say about coding and Borel reducibility.

Speaker(s): Matthew Harrison-Trainor (UM)

Speaker(s): Malavika Mukundan (University of Michigan)

]]>Why do organelles have their particular sizes, and how does the cell maintain them given the constant turnover of proteins and biomolecules? To address these fundamental biological questions, we formulate and study mathematical models of organelle size control rooted in the physicochemical principles of transport, chemical kinetics, and force balance. By studying the mathematical symmetries of competing models, we arrive at a hypothesis describing general principles of organelle size control. In particular, we consider flagellar length control in the unicellular green algae Chlamydomonas reinhardtii, and develop a minimal model in which diffusion gives rise to a length-dependent concentration of depolymerase at the flagellar tip. We show how noise may be used to fit model parameters and explain how similar principles may be applied to other examples of organelle size and scaling such as the ratio of nucleus to cell volume. Speaker(s): Thomas Fai (Brandeis University)

]]>I will talk about the recent work of Alpert and Manin on the homology groups of the ordered configuration space of open unit disks on an infinite strip of finite width. We will see that for a fixed width w these groups form a twisted (non-commutative) algebra, and begin the set up for the next talk where we will see that this algebra is finitely generated.

Speaker(s): Nick Wawrykow (UM)

Skandera showed that all dual canonical basis elements of C[SL_m] can be written in terms of Kazhdan-Lusztig immanants, which were introduced by Rhoades and Skandera. We use this result as well as Lewis Carroll's identity (also known as the Desnanot-Jacobi identity) to show that a broad class of dual canonical basis elements are positive when evaluated on k-positive matrices, matrices whose minors of size k and smaller are positive. This is joint work with Melissa Sherman-Bennett.

Speaker(s): Sunita Chepuri (University of Michigan)

When a compact complex manifold is given as the vanishing locus of polynomial equations, its singular cohomology groups possess natural direct-sum decompositions called Hodge structures. These Hodge structures strongly influence the topology of the variety; for instance, their mere existence implies that the odd-degree cohomology groups have even rank. I will explain what these structures are and how they lead to nice discrete invariants of algebraic varieties, such as Hodge numbers and Hodge diamonds. I will also discuss connections to the Grothendieck ring of varieties. Speaker(s): James Hotchkiss (UM)

]]>https://arxiv.org/abs/2101.09246 by Ahmadinezhad and Zhuang Speaker(s): Yueqiao Wu

]]>A famous conjecture by Michel stated that all simple Riemannian manifolds are boundary rigid. In this talk, we will first introduce Gromov's filling minimality and its relation to the boundary rigidity problem. Then we will introduce Burago-Ivanov's approach to prove both filling minimality and boundary rigidity for almost Euclidean and almost hyperbolic metrics. If time permits, I will briefly explain how to generalize their argument to almost rank-1 metrics of non-compact type. Speaker(s): Yuping Ruan (U Michigan)

]]>In the famous 1983 paper, when studying the heuristic distribution of class groups of imaginary quadratic fields, Cohen and Lenstra considered the weighting of a finite abelian group G with a weight proportional to 1/#Aut(G). More generally, for a given Dedekind domain R, they studied the statistics of finite-cardinality R-modules under the 1/#Aut weighting. They defined a "zeta" function \sum_M 1/#Aut(M) |M|^{-s} summing over all finite-cardinality R-modules, and they showed that it is an infinite product involving the Dedekind zeta function of R. In this talk, we discuss this Cohen--Lenstra zeta function defined for other families of rings, where the known results are organized in terms of the Krull dimension. The "nodal singularity" R=Fq[u,v]/(uv) is a surprisingly interesting example that gives rise to a peculiar q-series, which we will describe in more detail. Speaker(s): Yifeng Huang (University of Michigan)

]]>An orbispace is a "space" which is locally the quotient of a topological space by a continuous action of a finite group. Familiar examples of orbispaces include orbifolds, (the analytifications of) Deligne--Mumford stacks over C, and moduli spaces of solutions to elliptic partial differential equations, as they appear in low-dimensional and symplectic topology. The fibers of a vector bundle over an orbispace are representations of its stabilizer groups. When do there exist vector bundles all of whose fiber representations are faithful? This condition is called having "enough" vector bundles, and plays an important role in the stable homotopy theory of orbispaces. Speaker(s): John Pardon (Princeton)

]]>Speaker(s): Patrik Nabelek (Oregon State University)

]]>Speaker(s): Calvin Yost-Wolff (UM)

]]>Speaker(s): Christopher Stith (University of Michigan)

]]>Speaker(s): Alexander Bauman (UM)

]]>How many steps does it take to shuffle a deck of n cards, if at each step we pick two cards uniformly at random and swap them? Diaconis and Shahshahani proved that 1/2 n log n steps are necessary and sufficient to mix the deck. Using the representation theory of the symmetric group, they proved that this random transpositions card shuffle exhibits a sharp transition from being unshuffled to being very well shuffled. This is called the cutoff phenomenon. In this talk, I will explain how to use the spectral information of a Markov chain to study cutoff. As an application, I will briefly discuss the random to random card shuffle (joint with M. Bernstein) and the non-backtracking random walk on Ramanujan graphs (joint with P. Sarnak). Speaker(s): Evita Nestoridi (Princeton University)

]]>The increasing supermartingale coupling, introduced by Nutz and Stebegg (Canonical supermartingale couplings, Annals of Probability, 46(6):3351--3398, 2018) is an extreme point of the set of `supermartingale' couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively.

Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglböck et al. (The potential of the shadow measure, arXiv preprint, 2020) obtained in the martingale setting.

Joint work with Erhan Bayraktar and Shuoqing Deng. Speaker(s): Dominykas Norgilas (UM)

For a smooth surface S, the Hilbert scheme of points S^(n) is a well studied smooth parameter space. In this talk I will consider a natural generalization, the nested Hilbert scheme of points S^(n,m) which parameterizes pairs of 0-dimensional subschemes X \supseteq Y of S with deg(X) = n and deg(Y) = m. In contrast to the usual Hilbert scheme of points, S^(n,m) is almost always singular and it is known that S(n,1) has rational singularities. I will discuss some general techniques to study S^(n,m) and apply them to show that S^(n,2) also has rational singularities. This relies on a connection between S^(n,2) and a certain variety of matrices, and involves square-free Gröbner degenerations as well as the Kempf-Weyman geometric technique. This is joint work with Alessio Sammartano. Speaker(s): Ritvik Ramkumar (UC Berkeley)

]]>Speaker(s): Jonghyun Lee (University of Michigan)

]]>In this talk, I will describe a systematic approach for extending a class of integro-differential operators, defined on hypersurfaces, to ones defined in the tubular neighborhoods in the ambient Euclidean space. Surface integrals become volume integrals with identical evaluations. Such extensions facilitate the development of numerical methods for boundary integral methods and partial differential equations in applications where parameterization of surfaces is costly or difficult (e.g. inverse problems involving shapes). I will show applications of the implicit boundary integral method and relate to other existing methods. Speaker(s): Richard Tsai (University of Texas, Austin)

]]>Speaker(s): Nick Wawrykow (UM)

]]>Speaker(s): Swaraj Pande (UM)

]]>Cross-ratio dynamics is a well known dynamical system in discrete differential geometry. It was recently shown to be integrable (in the sense of Liouville-Arnold) by Arnold, Fuchs, Izmestiev and Tabachnikov. We relate it to the cluster integrable system of Goncharov and Kenyon associated with the dimer model on a certain class of graphs. In particular, we find a cluster algebra structure describing cross-ratio dynamics. This is joint work with Niklas Affolter and Sanjay Ramassamy. Speaker(s): Terrence George (University of Michigan)

]]>Speaker(s): Chuhao Sun (University of Michigan)

]]>https://arxiv.org/abs/2105.11009 by Ertl, Shiho, and Sprang Speaker(s): Lukas Scheiwiller

]]>TBA Speaker(s): Shubhodip Mondal (University of Michigan)

]]>Speaker(s): Sahana Vasudevan (MIT)

]]>Speaker(s): Fudong Wang (University of Central Florida)

]]>Speaker(s): Andy Gordon (UM)

]]>Title: Tame geometry and Hodge Theory

Hodge theory, as developed by Deligne and Griffiths, is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties. It is an essential fact that at heart, Hodge theory is NOT algebraic. On the other hand, according to both the Hodge conjecture and the Grothendieck period conjecture, this transcendence is severely constrained.

Tame geometry, whose idea was introduced by Grothendieck in the 80s, seems a natural setting for understanding these constraints. Tame geometry, developed by model theorists as o-minimal geometry, has for prototype real semi-algebraic geometry, but is much richer. It studies structures where every definable set has a finite geometric complexity.

The aim of these lectures is to present a number of recent applications of tame geometry to several problems related to Hodge theory and periods. Speaker(s): Bruno Klingler (Humboldt University)

Speaker(s): Moon Duchin (Tufts University)

]]>We present pathwise Ito-Tanaka theory, completely devoid of any probability structure, with help from the relevant notion of local time. Applying this theory to the functional generation of trading strategies, we discuss how to generate trading strategies in a pathwise way. A relevant definition of arbitrage and sufficient conditions leading to such arbitrage follow with examples.

Speaker(s): Donghan Kim (UM)

Speaker(s): Joe Waldron (MSU)

]]>Speaker(s): Mirko Mauri (University of Michigan)

]]>Speaker(s): Aida Maraj (UM)

]]>Speaker(s): Thomas Anderson (University of Michigan)

]]>Speaker(s): Christopher Fraser (Michigan State)

]]>Speaker(s): Yueqiao Wu (UM)

]]>Speaker(s): Chris Connell (Indiana University)

]]>Speaker(s): Som Phene (University of Michigan)

]]>https://arxiv.org/abs/2102.08347 by Villadsen Speaker(s): Sridhar Venkatesh

]]>TBA Speaker(s): Alex Dobner (University of Michigan)

]]>Speaker(s): Alex Kapiamba (U(M))

]]>Speaker(s): Calvin Yost-Wolff (UM)

]]>Speaker(s): Tejaswi Tripathi (University of Michigan)

]]>Speaker(s): Shelby Cox (UM)

]]>Speaker(s): tentatively reserved

]]>Speaker(s): Yilin Wang (MIT)

]]>We consider a mean field control problem with regime switching in the state dynamics. The corresponding value function is characterized as the unique viscosity solution of a HJB master equation. We prove that the value function is the limit of a finite agent centralized optimal control problem as the number of agents go to infinity. In the process, we derive the convergence rate and a propagation of chaos result for the optimal trajectory of agent states. Speaker(s): Prakash Chakraborty (UM)

]]>Speaker(s): Mihnea Popa (Harvard University)

]]>Speaker(s): David Stapleton (University of Michigan)

]]>Speaker(s): Emanuel Gull (University of Michigan, Physics)

]]>Speaker(s): Matthew (UM)

]]>Speaker(s): Bradley Zykoski (Michigan)

]]>Speaker(s): Jack Wakefield (University of Michigan)

]]>https://arxiv.org/abs/1706.06662 by Neeman Speaker(s): Andy Jiang

]]>Speaker(s): Fall Break

]]>Speaker(s): no talk

]]>Speaker(s): Huyen Pham (Universite Paris Diderot)

]]>Speaker(s): Shuoqing Dengg (UM)

]]>Speaker(s): Xiaolei Zhao (UC Santa Barbara)

]]>Speaker(s): David Speyer (University of Michigan)

]]>Speaker(s): Viktoria Taroudaki (Eastern Washington University)

]]>Speaker(s): Shiliang Gao (University of Illinois)

]]>https://arxiv.org/abs/2106.08381 by Esnault, Srinivas, and Stix Speaker(s): Gleb Terentiuk

]]>Speaker(s): Ondrej Maxian (New York University)

]]>Music is a universal language that has the power to influence our moods and inspire. All music is ultimately due to the physics of different kinds of vibrating objects. Professor Aidala will discuss how natural modes of vibration lead to musical tones and timbres.

]]>TBA Speaker(s): Patrick Daniels (University of Michigan)

]]>Speaker(s): Malavika Mukundan (U(M))

]]>Speaker(s): Brian Simanek (Baylor University)

]]>Speaker(s): Ilia Nekrasov (UM)

]]>Speaker(s): Som Phene (University of Michigan)

]]>Speaker(s): Karthik Ganapathy (University of Michigan, Ann Arbor)

]]>Speaker(s): Akshay Venkatesh (IAS)

]]>Speaker(s): Matthew Stoffregen (Michigan State University)

]]>Speaker(s): Ilesnami Adeboye (Wesleyan University)

]]>Speaker(s): Arthur Bik (Max Planck)

]]>Speaker(s): TBA

]]>Speaker(s): Will Dana (University of Michigan)

]]>Speaker(s): Preetham Mohan (University of Michigan)

]]>https://arxiv.org/abs/2101.01075 by Voisin Speaker(s): Jonghyun Lee

]]>TBA Speaker(s): Karol Koziol (University of Michigan)

]]>Speaker(s): Ethan Farber (Boston College)

]]>TBA

]]>Speaker(s): Huang & Reinecke

]]>Speaker(s): Alapan Mukhopadhyay (University of Michigan, Ann Arbor)

]]>Speaker(s): Mark de Cataldo (Stony Brook University)

]]>We consider an ergodic harvesting problem with model ambiguity that arises from biology. The problem is constructed as a stochastic game with two players: the decision-maker (DM) chooses the `best' harvesting policy and an adverse player chooses the `worst' probability measure. The main result is establishing an optimal control of the DM and showing that it is a threshold policy. The optimal threshold and the optimal payoff are obtained by solving a free-boundary problem emerging from the HJB equation. As part of the proof, we fix a gap that appeared in the HJB analysis of previous papers, which analyzed the risk-neutral version of the ergodic harvesting problem. Finally, we study the dependence of the optimal threshold and the optimal payoff on the ambiguity parameter and show that if the ambiguity goes to 0, the problem converges to the risk-neutral problem. Speaker(s): Chuhao Sun (UM)

]]>Speaker(s): James Hotchkiss (University of Michigan)

]]>Speaker(s): Paul Apisa (U Michigan)

]]>Speaker(s): Jason Kaye (Flatiron Institute)

]]>Speaker(s): Andy Gordon (UM)

]]>Speaker(s): Claudia Yun (Brown University)

]]>https://arxiv.org/abs/2104.01218 by Ein, Ha, and Lazarsfeld Speaker(s): Swaraj Pande

]]>Speaker(s): Scott Weady (New York University)

]]>TBA Speaker(s): Stephanie Chan (University of Michigan)

]]>Speaker(s): Caroline Davis (Indiana)

]]>Speaker(s): Geordie Williamson (University of Sydney)

]]>Speaker(s): Geordie Williamson

]]>Speaker(s): Jennifer Reiser (Emory University)

]]>Speaker(s): Zachary Hamaker (University of Florida)

]]>Speaker(s): Gary Hu (UM)

]]>Speaker(s): Sebastian Olano (University of Michigan)

]]>Speaker(s): Guanhua Sun (University of Michigan)

]]>TBA Speaker(s): Nate Harman (University of Michigan)

]]>Speaker(s): Danny Stoll (U(M))

]]>Speaker(s): Guanjie Huang (UM)

]]>Speaker(s): Katja Vassilev (University of Michigan)

]]>Speaker(s): Teresa Yu (University of Michigan, Ann Arbor)

]]>Speaker(s): Yu Deng (University of Southern California)

]]>Speaker(s): Yu Deng (University of Southern California)

]]>Speaker(s): Philip Ernst (Rice University)

]]>Speaker(s): Bhargav Bhatt (University of Michigan)

]]>Speaker(s): Wee Liang Gan (UC-Riverside)

]]>Speaker(s): TBA

]]>Speaker(s): Lena Ji (UM)

]]>Speaker(s): Karl Winsor (Harvard)

]]>https://arxiv.org/abs/2005.02338 by Ma Speaker(s): Alapan Mukhopadhyay

]]>Speaker(s): no talk

]]>Speaker(s): Havi Ellers (UM)

]]>Speaker(s): Anna Brosowsky (University of Michigan, Ann Arbor)

]]>Speaker(s): Hong Wang (UCLA)

]]>Speaker(s): Nguyen Huu Kien (KU Leuven)

]]>Speaker(s): Matthew Hedden (Michigan State University)

]]>Speaker(s): TBA (University of Michigan)

]]>Speaker(s): Youngsoo Choi (LLNL)

]]>Speaker(s): Alapan Mukhopadhyay (UM)

]]>Speaker(s): Bena Tshishiku (Brown)

]]>https://arxiv.org/abs/2107.13584 by Beheshti and Riedl Speaker(s): Lena Ji

]]>Speaker(s): Carsten Peterson (UM)

]]>Speaker(s): Jonghyun Lee (University of Michigan, Ann Arbor)

]]>Speaker(s): Will Sawin (Columbia University)

]]>Speaker(s): TBA (University of Michigan)

]]>Speaker(s): Inna Entova-Aizenbud (Ben Gurion University)

]]>Speaker(s): Saket Shah (UM)

]]>https://arxiv.org/abs/2105.06242 by Kollár Speaker(s): Mirko Mauri

]]>https://arxiv.org/abs/1912.10932 by Cesnavicius and Scholze Speaker(s): Bhargav Bhatt

]]>Speaker(s): Bourdin Blaise (McMaster University)

]]>Speaker(s): Dexuan Xie (University of Wisconsin-Milwaukee)

]]>Title: Tame geometry and Hodge Theory

Hodge theory, as developed by Deligne and Griffiths, is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties. It is an essential fact that at heart, Hodge theory is NOT algebraic. On the other hand, according to both the Hodge conjecture and the Grothendieck period conjecture, this transcendence is severely constrained.

Tame geometry, whose idea was introduced by Grothendieck in the 80s, seems a natural setting for understanding these constraints. Tame geometry, developed by model theorists as o-minimal geometry, has for prototype real semi-algebraic geometry, but is much richer. It studies structures where every definable set has a finite geometric complexity.

The aim of these lectures is to present a number of recent applications of tame geometry to several problems related to Hodge theory and periods. Speaker(s): Bruno Klingler (Humboldt University)

Speaker(s): Bruno Klingler (Humboldt University)

]]>