Department of

# Mathematics

Seminar Calendar
for events the year of Sunday, July 3, 2022.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, January 24, 2022

5:00 pm in 343 Altgeld Hall,Monday, January 24, 2022

#### Boundary of quantum circuits 2

Abstract: We will start the semester by clarifying some aspects of the paper on Entanglement spectrum and boundary theories with projected entangled-pair states by Cirac et al.

Friday, January 28, 2022

1:00 pm in 147 AH,Friday, January 28, 2022

#### Airy Strichartz and Pain

###### Ryan McConnell (UIUC)

Abstract: Abstract: While dispersive PDE's is known for the harmonic analysis that it requires as an input, periodic dispersive PDEs goes a step beyond and often requires a bit of number theoretic input. In particular, most of the known proofs for periodic Strichartz estimates comes from divisor counting arguments. In this talk, I will discuss the L^8 Strichartz estimate (an open problem) and the proofs surrounding easier cases of this estimate. The proofs will be somewhat introductory, and are excellent tools for understanding basic techniques in estimating exponential sums.

3:00 pm in 341 Altgeld Hall,Friday, January 28, 2022

#### Organizational Meeting

Abstract: This is the organizational meeting for this semester's homotopy theory seminar.

Monday, January 31, 2022

5:00 pm in Altgeld Hall,Monday, January 31, 2022

#### Organizational Meeting

###### Marius Junge (UIUC)

Abstract: We will discuss interesting topics for this semseter.

Thursday, February 3, 2022

2:00 pm in zoom: https://illinois.zoom.us/j/84500990503?pwd=Szhhb0xIUktoM2wxQ29xaERmUVpMZz09.,Thursday, February 3, 2022

#### Robust Mean Estimation

###### Grigory Terlov (UIUC Math )

Abstract: One of the most fundamental problems of statistics is to estimate the expected value of a random variable given $n$ i.i.d. samples. It is natural to do so by considering the empirical mean, however when one works with heavy-tailed distributions empirical mean would give less than ideal accuracy and confidence intervals for finite $n$ and thus one has to consider other estimators. In the first talk I will elaborate on this problem; present several other estimators such as median-of-means, Catoni’s estimator, and Trimmed median; and prove some basics results about them. My talks will be based on the survey paper by Lugosi and Mendelson https://arxiv.org/abs/1906.04280. Here is a zoom link: https://illinois.zoom.us/j/84500990503?pwd=Szhhb0xIUktoM2wxQ29xaERmUVpMZz09.

Monday, February 7, 2022

5:00 pm in 343 Altgeld Hall,Monday, February 7, 2022

#### Time as an additional resource

###### Marius Junge (UIUC)

Abstract: We study quantum evolution processes interrupted by measurements and their connection to large deviation theory.

5:00 pm in 343 Altgeld Hall,Monday, February 7, 2022

#### Time as an additional resource

###### Marius Junge (UIUC)

Abstract: We study quantum evolution processes interrupted by measurements and their connection to large deviation theory.

Thursday, February 10, 2022

2:00 pm in Altgeld Hall 347 ,Thursday, February 10, 2022

#### Robust Mean Estimation (part 2)

###### Greg Terlov (UIUC Math )

Abstract: One of the most fundamental problems of statistics is to estimate the expected value of a random variable given $n$ i.i.d. samples. It is natural to do so by considering the empirical mean, however when one works with heavy-tailed distributions empirical mean would give less than ideal accuracy and confidence intervals for finite $n$ and thus one has to consider other estimators. In the last talk we saw that taking median-of-mean yields promising results in 1 dimension but the construction depends on the given confidence interval. In this talk we will discuss this dependency and show that it cannot be improved without further assumptions on the distributions. Following that we will discuss how to extend this procedure to a multidimensional setting. These talks are be based on the survey paper by Lugosi and Mendelson https://arxiv.org/abs/1906.04280.

Friday, February 11, 2022

1:00 pm in Altgeld Hall 147,Friday, February 11, 2022

#### Einstein metrics with prescribed conformal infinity

###### Xinran Yu (UIUC)

Abstract: Given a conformal class of metrics on the boundary of a manifold, one can ask for the existence of an Einstein metric whose conformal infinity satisfies the boundary condition. In 1991, Graham and Lee studied this boundary problem on the hyperbolic ball. They proved the existence of metrics sufficiently close to the round metric on a sphere by constructing approximate solutions to a quasilinear elliptic system. In his monograph (2006), Lee discussed the boundary problem on a smooth, compact manifold-with-boundary. Using a similar construction, he proved the existence and regularity results for metrics sufficiently close to a given asymptotically hyperbolic Einstein metric. The proof is based on a linear theory for Laplacian and the inverse function theorem.

3:00 pm in 341 Altgeld Hall,Friday, February 11, 2022

#### An Introduction to Exodromy

###### Brian Shin (UIUC Math)

Abstract: One of the first major topics we learn about in algebraic topology is the classification of locally constant sheaves of sets (i.e. covering spaces) of a sufficiently nice topological space in terms of its fundamental group. This classification is mediated by an equivalence of categories known as the monodromy equivalence. An insight of Kan was that, in order to classify locally constant sheaves of more interesting objects, one must pass from fundamental groups to fundamental infinity-groupoid. In this expository talk, I'd like to talk about work of Barwick-Glasman-Haine pushing this circle ideas further into the realm of stratified spaces. The main result is the exodromy equivalence, which classifies constructible sheaves on a stratified space in terms of its profinite stratified shape.

4:00 pm in 347 Altgeld Hall,Friday, February 11, 2022

#### Surfaces in Knot Complements

###### Brannon Basilio (UIUC)

Abstract: We shall survey the role that surfaces play in knot complements. Many surfaces can be embedded in the complement of a knot in S^3. However, we will only concern ourselves with certain surfaces called incompressible surfaces. These surfaces play a major role in studying knots. For example, it turns out that the existence of certain surfaces tells us what kind of geometry the knot complement admits. As this is a survey talk, we will only go over definitions, main ideas, foundational results, new results, and open questions. We will not assume previous knowledge of knots nor will we explicitly prove any theorems, but rather give the overall main ideas and the tools used in the proofs.

Monday, February 14, 2022

5:00 pm in 343 Altgeld Hall,Monday, February 14, 2022

#### The hom-space of the Connes' embedding problem

###### David Gao (UIUC)

Thursday, February 17, 2022

2:00 pm in Altgeld Hall 347 ,Thursday, February 17, 2022

#### Introduction to large deviation theory

###### Peixue Wu (UIUC math )

Abstract: In the first part of my talk, I will review large deviation principles and the well-known Gartner-Ellis theorem. The main technique is to make a detailed analysis of the log moment generating function and its Fenchel dual. As an application, we recover some large deviation principles we established before.

Friday, February 18, 2022

1:00 pm in Altgeld Hall 147,Friday, February 18, 2022

#### An introduction to p-adic analysis

###### Robert J. Dicks (UIUC)

Abstract: This talk is meant to be an introduction p-adic number and p-adic analysis. These arise via completing the rational numbers using a metric (relative to a prime number p) which is very different from the ordinary euclidean metric. The first part of the talk will introduce this number system and focus on ways in which it differs from our "Euclidean" intuition. We then will discuss p-adic integration. We will then define the p-adic zeta function. The goal will be to describe (without proof) how these are related to the Riemann zeta function. This talk is aimed at 1st year graduate students.

3:00 pm in 341 Altgeld Hall,Friday, February 18, 2022

#### Formalism of six operations and derived algebraic stacks

###### Timmy Feng (UIUC Math)

Abstract: The formalism of six operations was originally introduced by A.Grothendieck and his collaborators in the study of \'etale cohomology. It naturally leads to many well-known results in cohomology theory like duality and Lefschetz trace formula. This partially justifies the slogan that the formalism of six operations are enhanced cohomology theories. In this talk, I will introduce the formalism of six operations. I will explain the relation between it and some cohomology theories (Topological, coherent, l-adic). Moreover, I will talk about the application of it to a nice class of derived algebraic stacks. And show that this leads to some nontrivial results of algebraic (homotopy) K-theory for stacks.

4:00 pm in 347 Altgeld Hall,Friday, February 18, 2022

#### An Advert for Generalized Complex Geometry

###### Sambit Senapati (UIUC)

Abstract: For many years physicists and mathematicians have studied the mysterious links between complex and symplectic geometry predicted by mirror symmetry. Generalized Complex geometry is a framework explicitly unifying complex and symplectic geometry and is hoped to allow for a formulation of this curious bridge. It is however interesting in its own right, as a differential geometric framework generalizing and specializing other differential geometric settings. I will try to give a broad introduction to this topic, focusing on an overview rather than on details. In particular, I’ll talk about concepts from symplectic and complex geometry that extend to GC structures as well as aspects that are unique to it.

Monday, February 21, 2022

5:00 pm in 343 Altgeld Hall,Monday, February 21, 2022

#### Curvature-dimension conditions for symmetric quantum Markov semigroups

###### Haonan Zhang (IST )

Abstract: The curvature-dimension condition consists of the lower Ricci curvature bound and upper dimension bound of the Riemannian manifold, which has a number of geometric consequences and is very helpful in proving many functional inequalities. The Bakry--Émery theory and Lott--Sturm--Villani theory allow to extend this notion beyond the Riemannian manifold setting and have seen great progress in the past decades. In this talk, I will first review several notions around lower Ricci curvature bounds in the noncommutative setting and present our work on gradient estimates. Then I will speak about two noncommutative versions of curvature-dimension conditions for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet--Myers theorem, and concavity of entropy power in the noncommutative setting. I will also give some examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz-Schur multipliers over group algebras, and depolarizing semigroups. Our work also gives new proofs and new results in the discrete setting. This is based on arXiv:2007.13506 and arXiv:2105.08303, joint work with Melchior Wirth (IST Austria).

Thursday, February 24, 2022

2:00 pm in Altgeld Hall 347 ,Thursday, February 24, 2022

#### Large Deviations Part II

###### Peixue Wu (UIUC math )

Abstract: I will continue the proof of Gartner-Ellis theorem and show the applications to dynamical Large deviations of Markov processes. Moreover, I will mention some further research problems.

Friday, February 25, 2022

1:00 pm in Zoom,Friday, February 25, 2022

#### Dimension distortion and applications

###### Efstathios Konstantinos Chrontsios Garitsis (UIUC Math)

Abstract: Since Hausdorff dimension was first introduced in 1918, many different notions of dimension have been defined and used throughout many areas of Mathematics. An interesting topic has always been the distortion of said dimensions of a given set under a specific class of mappings. More specifically, Gehring and V\"ais\"al\"a proved in 1973 a theorem concerning the distortion of Hausdorff dimension under quasiconformal maps, while Kaufman in 2000 proved the analogous result for Box-counting dimension. In this talk, an introduction to the different types of dimensions will be presented, along with the results of of Gehring, V\"ais\"al\"a and Kaufman. We will then proceed to discuss analogous theorems we proved for the Assouad dimension and spectrum, which describe how K-quasiconformal maps change these notions of a given subset of $\mathbb{R}^n$. We will conclude the talk by demonstrating how said theorems can be applied to fully classify polynomial spirals up to quasiconformal equivalence. Contact: ryanm12@illinois.edu for Zoom coordinates.

1:00 pm in Zoom,Friday, February 25, 2022

#### Dimension distortion and applications

###### Efstathios Konstantinos Chrontsios Garitsis (UIUC Math)

Abstract: Since Hausdorff dimension was first introduced in 1918, many different notions of dimension have been defined and used throughout many areas of Mathematics. An interesting topic has always been the distortion of said dimensions of a given set under a specific class of mappings. More specifically, Gehring and V\"ais\"al\"a proved in 1973 a theorem concerning the distortion of Hausdorff dimension under quasiconformal maps, while Kaufman in 2000 proved the analogous result for Box-counting dimension. In this talk, an introduction to the different types of dimensions will be presented, along with the results of of Gehring, V\"ais\"al\"a and Kaufman. We will then proceed to discuss analogous theorems we proved for the Assouad dimension and spectrum, which describe how K-quasiconformal maps change these notions of a given subset of $\mathbb{R}^n$. We will conclude the talk by demonstrating how said theorems can be applied to fully classify polynomial spirals up to quasiconformal equivalence. Contact: ryanm12@illinois.edu for Zoom coordinates.

3:00 pm in 341 Altgeld Hall,Friday, February 25, 2022

#### Introduction to Equivariant Homotopy Theory and RO(G)-graded Cohomology

Abstract: In its simplest form, equivariant homotopy theory is the study of homotopy theory with the addition of actions by a group G. By mixing representation theory into homotopy theory, we create additional structure and complexities to consider. To every representation of our group, we may take the one point compactification and the group will then act on the resulting sphere. In so called “genuine” G-spectra, these representation spheres are invertible, so in particular we may grade cohomology theories on (virtual) G-representations. In this introductory talk, I will go over some of the additional properties and structures of equivariant homotopy theory and if time permits illustrate these structures with a cohomology computation.

4:00 pm in 347 Altgeld Hall,Friday, February 25, 2022

#### The Jones Polynomial and its Big Brother, Khovanov Homology

###### Joseph Malionek (UIUC)

Abstract: In this talk, I will acquaint the audience with a link invariant called the Jones Polynomial and one of way of making it much more complicated (but potentially much stronger) called Khovanov Homology. I will go over some key properties of each of these invariants, some example calculations, some big results, and some open problems.

Monday, February 28, 2022

5:00 pm in 343 Altgeld Hall,Monday, February 28, 2022

#### Embedding in R^{omega}, Part 2

###### David Gao

Thursday, March 3, 2022

2:00 pm in Altgeld Hall 347 ,Thursday, March 3, 2022

#### Witten deformation of Laplacian

###### Kesav Krishnan (UIUC math )

Abstract: I will introduce the notion of Witten deformation of Laplacian, and discuss probabilistic interpretations of the same, I will then discuss some applications to the study of lattice models in statistical mechanics.

Friday, March 4, 2022

1:00 pm in 147 Altgeld Hall,Friday, March 4, 2022

#### Traveling wave solutions for discrete and nonlocal diffusive Lotka-Volterra system

###### Ting-Yang Hsiao (UIUC)

Abstract: We will introduce a biological model called Lotka-Volterra competition system. We construct an N-shaped constrained region. By using this region we give an a priori estimate for this system and show that the total population of this 2-species system has a nontrivial lower bound. Finally, we know that one of the important issues in ecology is biodiversity. By using this estimate, we find necessary conditions for coexistence of 3-species Lotka-Volterra system.

4:00 pm in 347 Altgeld Hall,Friday, March 4, 2022

#### The flux homomorphism for groups of symplectomorphisms

###### Wilmer Smilde (UIUC)

Abstract: On a symplectic manifold, one has two types of symmetries. Namely, the symplectomorphisms, which are diffeomorphisms preserving the symplectic form, and Hamiltonian diffeomorphisms, which are generated by Hamiltonian functions. Every Hamiltonian diffeomorphism is also a symplectomorphism. The flux homomorphisms is a tool that enables us to write the symplectomorphism group as an extension of the Hamiltonian group by a (finite-dimensional) abelian group. In this talk, I will go over the definition of the flux homomorphism, and show some basic properties. The aim is to arrive at some nice and pretty exact sequences. In the end I also hope to explain some important and deep results related to it. The nice thing about the flux homomorphism is that it requires very little experience with symplectic structures, so I hope it is accessible for anyone familiar with differential geometry.

Monday, March 7, 2022

1:00 pm in Zoom,Monday, March 7, 2022

#### The Number of Closed Essential Surfaces in Montesinos Knot Complements

###### Brannon Basilio

Abstract: In this prelim, we will discuss new results concerned with the number of closed, connected, essential, orientable surfaces in Montesinos knot complements. We give explicit counts of these surfaces by genus, up to isotopy. We will introduce the necessary background and definitions, the main tools of the proof, and discuss a future project. Even though this is a preliminary exam, all are welcome! If you are interested in attending, please email basilio3 (at) illinois (dot) edu for the Zoom link.

5:00 pm in Altgeld Hall,Monday, March 7, 2022

#### Quantum money

Abstract: We follow the papers by Scott Aaronson for a first understanding of quantum money.

Friday, March 11, 2022

3:00 pm in 341 Altgeld Hall,Friday, March 11, 2022

#### Applications of Dualizing Complex in Commutative Algebra

###### Likun Xie (UIUC Math)

Abstract: Starting with a finitely generated module of finite projective dimension over the completion of a Noetherian local ring A, a natural question is when does this module descend, i.e. when is this module the completion of a finite A-module of finite projective dimension? We will need the theorem of local duality to show it happens over some “good” rings. We will introduce dualizing complex both in the language of derived categories, and in an explicit form for commutative Noetherian ring. We will apply them to study some descent problems for module of finite projective dimension. The techniques employed also allow one to recover a theorem of Horrocks about vector bundles over a punctured spectrum of a local ring. We follow Section I.5 in Dimension projective finie et cohomologie locale by Peskine and Szpiro.

Thursday, March 24, 2022

2:00 pm in Altgeld Hall 347 ,Thursday, March 24, 2022

#### witten deformation of Laplacian part II

###### Kesav Krishnan (UIUC math )

Abstract: I will introduce the notion of Witten deformation of Laplacian, and discuss probabilistic interpretations of the same, I will then discuss some applications to the study of lattice models in statistical mechanics.

Friday, March 25, 2022

1:00 pm in 147 Altgeld Hall,Friday, March 25, 2022

#### Index theorems and boundary conditions

###### Gayana Jayasinghe (UIUC)

Abstract: The study of the Fredholm Index of operators is of great interest in analysis. The 60's saw the development of the theory of the index of elliptic operators as a topological invariant on compact manifolds, a vast generalization of the Gauss Bonnet theorem. Now there are generalizations of all sorts of spaces including spaces with boundary, singularities and in non commutative geometry. I will begin by introducing Index theory, motivating the key ideas with the example of the Gauss Bonnet theorem, on manifolds with boundary. The index in this example has a topological contribution as well as a contribution coming from the asymmetry of the boundary, which can be described in terms of geodesic curvature. I will then describe how for general operators, this asymmetry can be expressed using boundary conditions for the operator, and in the case of spectral boundary conditions, one deals with the spectral asymmetry. I won't assume any prior knowledge on Index theory, and it should be accessible for analysts

3:00 pm in 341 Altgeld Hall,Friday, March 25, 2022

#### An introduction to 2-categories

###### Yigal Kamel (UIUC Math)

Abstract: This talk will be a survey of some of the basic notions and facts about 2-categories. After introducing 2-categories and various types of functors between them, we will consider constructions that map 2-categories to more familiar objects. On one hand, viewing a 2-category as a “poor” higher category, we can “reduce” it to an ordinary category, via a homotopy construction. On the other hand, viewing a 2-category as a “rich” ordinary category, we can “uplift” it to a simplicial set, via the duskin nerve. I will talk about these and related ideas, indicate some quirks of the theory, and provide examples along the way.

4:00 pm in 347 Altgeld Hall,Friday, March 25, 2022

#### Modular Forms in Geometry and Physics

###### Saaber Pourmotabbed

Abstract: Modular forms appear as generating functions of curve counts and BPS states, partition functions of conformal field theories, in moonshine phenomena, sections of line bundles, spectrums of cohomology theories, elliptic genera of K3 surfaces, and many other interesting cases. In this talk we will look at some examples of where modular forms occur in physics and geometry, why they occur, and their interactions between these different fields.

Monday, March 28, 2022

5:00 pm in 343 Altgeld Hall,Monday, March 28, 2022

#### Asymptotics of Young diagrams and quantum central limit theorem

###### Felix Leditzky (UIUC)

Abstract: We follow the work of Greg Kuperberg who proves a central limit theorem for noncommuting random random variables in von Neumann algebras.

Thursday, March 31, 2022

2:00 pm in Altgeld Hall 347 ,Thursday, March 31, 2022

#### An introduction to Pirogov-Sinai Theory

###### Bob Krueger (UIUC MATH)

Abstract: I will introduce a few spin models on lattices and discuss the usual polymer models. Then I will describe the polymer model at the basis of Pirogov-Sinai Theory. I hope to get to a recent result of Helmuth, Perkins, and Regts which uses Pirogov-Sinai Theory to efficiently sample from these spin models at low temperatures. I assume no knowledge of statistical physics.

Friday, April 1, 2022

3:00 pm in 341 Altgeld Hall,Friday, April 1, 2022

#### Splitting $BP<2> ⋀ BP<2>$ at primes $p \ge 5$

###### Liz Tatum (UIUC Math)

Abstract: In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo ⋀ bo$ and $l ⋀ l$. These splittings helped make it feasible to do computations using the $bo$- and $l$-based Adams spectral sequences. In this talk, we will construct an analogous splitting for $BP<2> ⋀ BP<2>$ at primes $p \ge 5$.

4:00 pm in 347 Altgeld Hall,Friday, April 1, 2022

#### (Linear) analysis at singularities, infinities, and other things

###### Gayana Jayasinghe

Abstract: How does one make sense of a singularity which can be described in terms of a degenerate Riemannian metric? How does one study operators on a Riemannian manifold with a specific growth at infinity? How can we put this all together to study all sorts of spaces with degeneracies and growth rates and study (pseudo) differential operators, for instance to prove elliptic regularity? How does one study the wave equation with the Weyl Peterson metric and other such fantasies? I'll try and explain how we do this with geometric microlocal analysis.

Monday, April 4, 2022

5:00 pm in 343 Altgeld Hall,Monday, April 4, 2022

#### Asymptotics of Young diagrams and quantum central limit theorem

###### Felix Leditzky And Cloe Kim (UIUC)

Abstract: Continue from last time

Thursday, April 7, 2022

2:00 pm in Altgeld Hall 347 ,Thursday, April 7, 2022

#### An introduction to Pirogov-Sinai Theory

###### Bob Krueger (UIUC Math )

Abstract: I will introduce a few spin models on lattices and discuss the usual polymer models. Then I will describe the polymer model at the basis of Pirogov-Sinai Theory. I hope to get to a recent result of Helmuth, Perkins, and Regts which uses Pirogov-Sinai Theory to efficiently sample from these spin models at low temperatures. I assume no knowledge of statistical physics.

Friday, April 8, 2022

1:00 pm in 147 Altgeld Hall,Friday, April 8, 2022

###### Aric Wheeler (Indiana University)

Abstract: Generalizing results of Matthews-Cox/Sukhtayev for a model reaction-diffusion equation, we derive and rigorously justify weakly nonlinear amplitude equations governing general Turing bifurcation in the presence of conservation laws. In the nonconvective, reaction-diffusion case, this is seen similarly as in Matthews-Cox, Sukhtayev to be a real Ginsburg-Landau equation weakly coupled with a diffusion equation in a large-scale mean-mode vector comprising variables associated with conservation laws. In the general, convective case, by contrast, the amplitude equations consist of a complex Ginsburg-Landau equation weakly coupled with a singular convection-diffusion equation featuring rapidly-propagating modes with speed $\sim 1/\epsilon$ where $\epsilon$ measures amplitude of the wave as a disturbance from a background steady state. Applications are to biological morphogenesis, in particular vasculogenesis, as described by the Murray-Oster and other mechanochemical/hydrodynamical models. This work is joint with Kevin Zumbrun.

3:00 pm in 341 Altgeld Hall,Friday, April 8, 2022

#### Towards a universal property of the ∞-equipment of enriched (∞,1)-categories

###### Samuel Hsu (UIUC Math)

Abstract: One way or another, enriched 1-category theory has held an important spot in the study of homological and homotopical phenomena practically since the very start of ordinary category theory. For many purposes, enriched 1-categories or their model 1-categorical counterparts are simply too rigid, or they might not even exist at all. In recent years various models of enriched (∞,1)-categories have been introduced, and some comparisons at differing levels have been made e.g. the underlying parameterizing ∞-operads or their ∞-categories (with a closed left action over Cat_∞). We are interested in a universal property that can compare these theories at a level which can detect pointwise Kan extensions for example. Part of one approach to this involves upgrading the underlying machinery appearing in Gepner and Haugseng to the scaled simplicial setting. This talk will be heavily focused on examples and justifying why we would want such theories anyway. The only prerequisite is some knowledge of enriched 1-category theory and an appetite for homotopy theory. Time permitting, we may discuss the situation with enriched (∞,1)-operads and (∞,1)-properads, or other possible uses of intermediate results.

Monday, April 11, 2022

5:00 pm in 343 Altgeld Hall,Monday, April 11, 2022

#### More on quantum central limit theorem

###### Cloe Kim (UIUC)

Abstract: Part 2 on quantum CLT

Thursday, April 14, 2022

2:00 pm in Altgeld Hall 347 ,Thursday, April 14, 2022

#### The ant on a rubber rope paradox

###### Ting-Yang Hsiao (UIUC Math )

Abstract: In this talk, we consider a puzzle called "Ant on a rubber rope”. It is a stochastic process on a random growing domain. To be more specific, an ant is at the left endpoint of a rubber band, which is 1 kilometer long and the ant crawls along with the rubber band at a pace having an expected value of 1 centimeter per second. After the first second, the rubber band stretches extra length L such that the expected value of L is 1 kilometer. Repeat the steps above. It seems that there is a high probability that this ant will not achieve the right endpoint of the rubber band. In this talk, however, we prove that the ant will reach the right end in finite time almost surely.

Friday, April 15, 2022

3:00 pm in 341 Altgeld Hall,Friday, April 15, 2022

###### Doron Grossman-Naples   [email] (UIUC Math)

Abstract: One of the most important roles played by topological spaces is being a base for geometry, i.e. “something to have sheaves on”. As is often the case, however, this classical notion falls short when it comes to describing homotopical geometry. The correct generalization is that of an ∞-topos. In this talk, I will describe the theory of ∞-topoi, how they generalize classical objects from topology and geometry, and several applications. No prior knowledge of 1-topoi or presentable ∞-categories will be assumed.

Monday, April 18, 2022

5:00 pm in Altgeld Hall,Monday, April 18, 2022

###### Piexue Wu (UIUC)

Abstract: Peixue will discuss stability of Lindbladians.

Friday, April 22, 2022

3:00 pm in 341 Altgeld Hall,Friday, April 22, 2022

#### Rational Homotopy Theory

###### Langwen Hui (UIUC Math)

Abstract: Rational homotopy theory is homotopy theory modulo torsion. This simplification reduces topology to algebra. More precisely, Quillen proved that the rational homotopy theory of 2-connected spaces is equivalent to that of (1) 1-connected dg Lie algebras (2) 2-connected dg cocommutative coalgebras. This is subsequently augmented by Sullivan, who provides a dg commutative algebra model of rational homotopy theory with computational strength. Time permitting, I will also discuss interesting applications to geometry and local algebra.

Friday, April 29, 2022

4:00 pm in 245 Altgeld Hall,Friday, April 29, 2022

#### Descriptive Set Theory and generic measure preserving transformations

###### Slawomir Solecki (Cornell )

Abstract: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation $T$ has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this was the case. We will describe the background touched on above, including the relevant definitions and the connections with Descriptive Set Theory. Further, we will indicate a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is {\bf not} topologically isomorphic to $L^0$. The proof rests on an analysis of unitary representations of the non-locally compact group $L^0$. PS: There will be a reception at 3 (common room)

Monday, May 2, 2022

5:00 pm in 343 Altgeld Hall,Monday, May 2, 2022

###### Piexue Wu (UIUC)

Abstract: We will prove the stability of CLSI.

Friday, May 6, 2022

3:00 pm in 341 Altgeld Hall,Friday, May 6, 2022

#### Calculus for Algebraic Topologists

###### Johnson Tan (UIUC Math)

Abstract: Early on in our mathematical studies, we learn that instead of studying a problem directly it is useful to study a linearization of the problem. For example, to understand a smooth map between manifolds we can look at the resulting linear map between the tangent spaces at a point. In this talk we will be looking at a categorification of this idea through the lens of Goodwillie Calculus. In particular, given a map between sufficiently nice infinity categories we will define what it means for such a map to be "linear" and furthermore how one can approximate by such maps. Time permitting, we will explore further generalizations of this idea.

Saturday, June 11, 2022

9:00 am in TBD,Saturday, June 11, 2022

#### In Honor of Tom Nevins

Abstract: TBA

Sunday, June 12, 2022

9:00 am in TBD,Sunday, June 12, 2022

Abstract: TBA