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DTSTART;TZID=America/New_York:20171113T090000
DTEND;TZID=America/New_York:20171117T160000
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SUMMARY:Workshop on Algebraic Methods in Combinatorics
DESCRIPTION:The workshop on Algebraic Methods in Combinatorics will take place November 13-17\, 2017 at the Center of Mathematical Sciences and Applications\, located at 20 Garden Street\, Cambridge\, MA. \nThe main focus of the workshop is the application of algebraic method to study problems in combinatorics.  In recent years there has been a large number of results in which the use of algebraic technique has resulted in significant improvements to long standing open problems. Such problems include the finite field Kakeya problem\, the distinct distance problem of Erdos and\, more recently\, the cap-set problem. The workshop will include talks on all of the above mentioned problem as well as on recent development in related areas combining combinatorics and algebra. \nConfirmed participants include: \n\nAbdul Basit\, Rutgers\nBoris Bukh\, Carnegie Mellon University\nPete L. Clark\, University of Georgia\nDavid Conlon\, University of Oxford\nFrank de Zeeuw\, EPFL\nThao Thi Thu Do\, MIT\nNoam Elkies\, Harvard University\nJordan Ellenberg\, University of Wisconsin\nDion Gijswijt\, Delft Institute of Technology\nSivankanth Gopi\, Princeton University\nVenkatesan Guruswami\, Carnegie Mellon University\nMarina Iliopoulou\, University of California\, Berkeley\nRobert Kleinberg\, Cornell University\nMichael Krivelevich\, Tel Aviv University\nVsevelod Lev\, University of Haifa at Oranim\nLászló Miklós Lovász\, UCLA\nBen Lund\, Rutgers\nPéter Pach\, Budapest University of Technology and Economics\nJános Pach\, New York University\nZuzana Patáková\, Institute of Science and Technology Austria\nOrit Raz\, Institute for Advanced Study\nOliver Roche-Newton\, Johannes Kepler University\nMisha Rudnev\, University of Bristol\nAdam Sheffer\, California Institute of Technology\nAmir Shpilka\, Tel-Aviv University\nNoam Solomon\, Harvard CMSA\nJozsef Solymosi\, University of British Columbia\nBenny Sudakov\, ETH\, Zurich\nAndrew Suk\, University of California\, San Diego\nTibor Szabó\, Freie Universität Berlin\nChris Umans\, California Institute of Technology\nAvi Wigderson\, Princeton University\nJosh Zahl\, University of British Columbia\n\nCo-organizers of this workshop include Zeev Dvir\, Larry Guth\, and Shubhangi Saraf. \nMonday\, Nov. 13 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n\n\n\n9:30-10:30am \nVideo\nJozsef Solymosi \n \n\nOn the unit distance problem \nAbstract: Erdos’ Unit Distances conjecture states that the maximum number of unit distances determined by n points in the plane is almost linear\, it is O(n^{1+c}) where c goes to zero as n goes to infinity. In this talk I will survey the relevant results and propose some questions which would imply that the maximum number of unit distances is o(n^{4/3}).  \n\n\n\n10:30-11:00am\nCoffee Break\n\n\n\n11:00-12:00pm \nVideo \n \nOrit Raz\nIntersection of linear subspaces in R^d and instances of the PIT problem  \nAbstract: In the talk I will tell about a new deterministic\, strongly polynomial time algorithm which can be viewed in two ways. The first is as solving a derandomization problem\, providing a deterministic algorithm to a new special case of the PIT (Polynomial Identity Testing) problem. The second is as computing the dimension of the span of a collection of flats in high dimensional space. The talk is based on a joint work with Avi Wigderson.\n\n\n12:00-1:30pm\nLunch\n\n\n\n1:30-2:30pm \nVideo\nAndrew Hoon Suk\n\nRamsey numbers: combinatorial and geometric \nAbstract:  In this talk\, I will discuss several results on determining the tower growth rate of Ramsey numbers arising in combinatorics and in geometry.  These results are joint work with David Conlon\, Jacob Fox\, Dhruv Mubayi\, Janos Pach\, and Benny Sudakov. \n\n\n\n2:30-3:00pm\nCoffee Break\n\n\n\n3:00-4:00pm \nVideo\nJosh Zahl\n\nCutting curves into segments and incidence geometry \n\n\n\n4:00-6:00pm\nWelcome Reception\n\n\n\n\nTuesday\, Nov. 14 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n\n\n\n9:30-10:30am \nVideo\nPéter Pál Pach\n\nPolynomials\, rank and cap sets \nAbstract: In this talk we will look at a new variant of the polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like $\mathbb{Z}_4^n$ and $\mathbb{F}_q^n$ are exponentially small (compared to the size of the group). We will discuss lower and upper bounds for the size of the extremal subsets and mention further applications of the method. \n\n\n\n10:30-11:00am\nCoffee Break\n\n\n\n11:00-12:00pm\nJordan Ellenberg\n\nThe Degeneration Method \nAbstract:  In algebraic geometry\, a very popular way to study (nice\, innocent\, nonsingular) varieties is to degenerate them to (weird-looking\, badly singular\, nonreduced) varieties (which are actually not even varieties but schemes.)  I will talk about some results in combinatorics using this approach (joint with Daniel Erman) and some ideas for future applications of the method. \n\n\n\n12:00-1:30pm\nLunch\n\n\n\n1:30-2:30pm \nVideo\nLarry Guth\nThe polynomial method in Fourier analysis \nAbstract: This will be a survey talk about how the polynomial method helps to understand problems in Fourier analysis.  We will review some applications of the polynomial method to problems in combinatorial geometry.  Then we’ll discuss some problems in Fourier analysis\, explain the analogy with combinatorial problems\, and discuss how to adapt the polynomial method to the Fourier analysis setting.\n\n\n  \n2:30-3:00pm\nCoffee Break\n\n\n\n3:00-4:00pm\nOpen Problem\n\n\n\n\nWednesday\, Nov. 15 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n\n\n\n9:30-10:30am \n \nAvi Wigderson\n\nThe “rank method” in arithmetic complexity: Lower bounds and barriers to lower bounds \nAbstract: Why is it so hard to find a hard function? No one has a clue! In despair\, we turn to excuses called barriers. A barrier is a collection of lower bound techniques\, encompassing as much as possible from those in use\, together with a  proof that these techniques cannot prove any lower bound better than the state-of-art (which is often pathetic\, and always very far from what we expect for complexity of random functions). \nIn the setting of  Boolean computation of Boolean functions (where P vs. NP is the central open problem)\,  there are several famous barriers which provide satisfactory excuses\, and point to directions in which techniques may be strengthened. \nIn the setting of Arithmetic computation of polynomials and tensors (where  VP vs. VNP is the central open problem) we have no satisfactory barriers\, despite some recent interesting  attempts. \nThis talk will describe a new barrier for the Rank Method in arithmetic complexity\, which encompass most lower bounds in this field. It also encompass most lower bounds on tensor rank in algebraic geometry (where the the rank method is called Flattening). \nI will describe the rank method\, explain how it is used to prove lower bounds\, and then explain its limits via the new barrier result. As an example\, it shows that while the best lower bound on the tensor rank of any explicit 3-dimensional tensor of side n (which is achieved by a rank method) is 2n\, no rank method can prove a lower bound which exceeds 8n \n(despite the fact that a random such tensor has rank quadratic in n). \nNo special background knowledge is assumed. The audience is expected to come up with new lower bounds\, or else\, with new excuses for their absence. \n\n\n\n10:30-11:00am\nCoffee Break\n\n\n\n11:00-12:00pm \nVideo\nVenkat Guruswami\n\nSubspace evasion\, list decoding\, and dimension expanders \n Abstract: A subspace design is a collection of subspaces of F^n (F = finite field) most of which are disjoint from every low-dimensional subspace of F^n. This notion was put forth in the context of algebraic list decoding where it enabled the construction of optimal redundancy list-decodable codes over small alphabets as well as for error-correction in the rank-metric. Explicit subspace designs with near-optimal parameters have been constructed over large fields based on polynomials with structured roots. (Over small fields\, a construction via cyclotomic function fields with slightly worse parameters is known.) Both the analysis of the list decoding algorithm as well as the subspace designs crucially rely on the *polynomial method*. \nSubspace designs have since enabled progress on linear-algebraic analogs of Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In particular\, they yield an explicit construction of constant-degree dimension expanders over large fields. While constructions of such dimension expanders are known over any field\, they are based on a reduction to a highly non-trivial form of vertex expanders called monotone expanders. In contrast\, the subspace design approach is simpler and works entirely within the linear-algebraic realm. Further\, in recent (ongoing) work\, their combination with rank-metric codes yields dimension expanders with expansion proportional to the degree. \nThis talk will survey these developments revolving around subspace designs\, their motivation\, construction\, analysis\, and connections. \n(Based on several joint works whose co-authors include Chaoping Xing\, Swastik Kopparty\, Michael Forbes\, Nicolas Resch\, and Chen Yuan.) \n\n\n\n12:00-1:30pm\nLunch\n\n\n\n1:30-2:30pm \n \nDavid Conlon\n\nFinite reflection groups and graph norms \nAbstract: For any given graph $H$\, we may define a natural corresponding functional $\|.\|_H$. We then say that $H$ is norming if $\|.\|_H$ is a semi-norm. A similar notion $\|.\|_{r(H)}$ is defined by $\| f \|_{r(H)} := \| | f | \|_H$ and $H$ is said to be weakly norming if $\|.\|_{r(H)}$ is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction\, Hatami showed that even cycles\, complete bipartite graphs\, and hypercubes are all weakly norming. Using results from the theory of finite reflection groups\, we identify a much larger class of weakly norming graphs. This result includes all previous examples of weakly norming graphs and adds many more. We also discuss several applications of our results. In particular\, we define and compare a number of generalisations of Gowers’ octahedral norms and we prove some new instances of Sidorenko’s conjecture. Joint work with Joonkyung Lee. \n \n\n\n2:30-3:00pm\nCoffee Break\n\n\n\n3:00-4:00pm \nVideo\nLaszlo Miklós Lovasz\n\nRemoval lemmas for triangles and k-cycles. \nAbstract: Let p be a fixed prime. A k-cycle in F_p^n is an ordered k-tuple of points that sum to zero; we also call a 3-cycle a triangle. Let N=p^n\, (the size of F_p^n). Green proved an arithmetic removal lemma which says that for every k\, epsilon>0 and prime p\, there is a delta>0 such that if we have a collection of k sets in F_p^n\, and the number of k-cycles in their cross product is at most a delta fraction of all possible k-cycles in F_p^n\, then we can delete epsilon times N elements from the sets and remove all k-cycles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma\, and\, in particular\, asked whether a polynomial bound holds. Despite considerable attention\, prior to our work\, the best known bound for any k\, due to Fox\, showed that 1/delta can be taken to be an exponential tower of twos of height logarithmic in 1/epsilon (for a fixed k). \nIn this talk\, we will discuss recent work on Green’s problem. For triangles\, we prove an essentially tight bound for Green’s arithmetic triangle removal lemma in F_p^n\, using the recent breakthroughs with the polynomial method. For k-cycles\, we also prove a polynomial bound\, however\, the question of the optimal exponent is still open. \nThe triangle case is joint work with Jacob Fox\, and the k-cycle case with Jacob Fox and Lisa Sauermann. \n\n\n\n\nThursday\, Nov. 16 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n\n\n\n9:30-10:30am \nVideo\nJanos Pach\nLet’s talk about multiple crossings \nAbstract: Let k>1 be a fixed integer. It is conjectured that any graph on n vertices that can be drawn in the plane without k pairwise crossing edges has O(n) edges. Two edges of a hypergraph cross each other if neither of them contains the other\, they have a nonempty intersection\, and their union is not the whole vertex set. It is conjectured that any hypergraph on n vertices that contains no k pairwise crossing edges has at most O(n) edges. We discuss the relationship between the above conjectures and explain some partial answers\, including a recent result of Kupavskii\, Tomon\, and the speaker\, improving a 40 years old bound of Lomonosov.\n\n\n10:30-11:00am\nCoffee Break\n\n\n\n11:00-12:00pm \nVideo\nMisha Rudnev\n\nFew products\, many sums \nAbstract: This is what I like calling “weak Erd\H os-Szemer\’edi conjecture”\, still wide open over the reals and in positive characteristic. The talk will focus on some recent progress\, largely based on the ideas of I. D. Shkredov over the past 5-6 years of how to use linear algebra to get the best out of the Szemer\’edi-Trotter theorem for its sum-product applications. One of the new results is strengthening (modulo the log term hidden in the $\lesssim$ symbol) the textbook Elekes inequality \n$$ \n|A|^{10} \ll |A-A|^4|AA|^4 \n$$ \nto \n$$|A|^{10}\lesssim |A-A|^3|AA|^5.$$ \nThe other is the bound  \n$$E(H) \lesssim |H|^{2+\frac{9}{20}}$$ for additive energy of sufficiently small multiplicative subgroups in $\mathbb F_p$. \n\n\n\n12:00-1:30pm\nLunch\n\n\n\n1:30-2:30pm \nVideo\nAdam Sheffer\n\nGeometric Energies: Between Discrete Geometry and Additive Combinatorics \nAbstract: We will discuss the rise of geometric variants of the concept of Additive energy. In recent years such variants are becoming more common in the study of Discrete Geometry problems. We will survey this development and then focus on a recent work with Cosmin Pohoata. This work studies geometric variants of additive higher moment energies\, and uses those to derive new bounds for several problems in Discrete Geometry.   \n\n\n\n2:30-3:00pm\nCoffee Break\n\n\n\n3:00-4:00pm \nVideo\nBoris Bukh\n\nRanks of matrices with few distinct entries \nAbstract: Many applications of linear algebra method to combinatorics rely on the bounds on ranks of matrices with few distinct entries and constant diagonal. In this talk\, I will explain some of these application. I will also present a classification of sets L for which no low-rank matrix with entries in L exists. \n\n\n\n\nFriday\, Nov. 17 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n\n\n\n9:30-10:30am \nVideo\nBenny Sudakov\n\nSubmodular minimization and set-systems with restricted intersections \nAbstract: Submodular function minimization is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly\, there is only very little known about constraint types under which it remains efficiently solvable. The arguably most relevant non-trivial constraint class for which polynomial algorithms are known are parity constraints\, i.e.\, optimizing submodular function only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the odd-cut problem\, and they are a key tool in a recent technique to efficiently solve integer programs with a constraint matrix whose subdeter-minants are bounded by two in absolute value. \nWe show that efficient submodular function minimization is possible even for a significantly larger class than parity constraints\, i.e.\, over all sets (of any given lattice) of cardinality r mod m\, as long as m is a constant prime power. To obtain our results\, we combine tools from Combinatorial Optimization\, Combinatorics\, and Number Theory. In particular\, we establish an interesting connection between the correctness of a natural algorithm\, and the non-existence of set systems with specific intersection properties. \nJoint work with M. Nagele and R. Zenklusen \n\n\n\n10:30-11:00am\nCoffee Break\n\n\n\n11:00-12:00pm \nVideo\nRobert Kleinberg\n  \nExplicit sum-of-squares lower bounds via the polynomial method \nAbstract: The sum-of-squares (a.k.a. Positivstellensatz) proof system is a powerful method for refuting systems of multivariate polynomial inequalities\, i.e. proving that they have no solutions. These refutations themselves involve sum-of-squares (sos) polynomials\, and while any unsatisfiable system of inequalities has a sum-of-squares refutation\, the sos polynomials involved might have arbitrarily high degree. However\, if a system admits a refutation where all polynomials involved have degree at most d\, then the refutation can be found by an algorithm with running time polynomial in N^d\, where N is the combined number of variables and inequalities in the system. \nLow-degree sum-of-squares refutations appear throughout mathematics. For example\, the above proof search algorithm captures as a special case many a priori unrelated algorithms from theoretical computer science; one example is Goemans and Williamson’s algorithm to approximate the maximum cut in a graph. Specialized to extremal graph theory\, they become equivalent to flag algebras. They have also seen practical use in robotics and optimal control. \nTherefore\, it is of interest to identify “hard” systems of low-degree polynomial inequalities that have no solutions but also have no low-degree sum-of-squares refutations. Until recently\, the only known examples were either not explicit (i.e.\, known to exist by non-constructive means such as the probabilistic method) or not robust (i.e.\, a system is constructed which is not refutable by degree d sos polynomials\, but becomes refutable when perturbed by an amount tending to zero with d). We present a new family of instances derived from the cap-set problem\, and we show a super-constant lower bound on the degree of its sum-of-squares refutations. Our instances are both explicit and robust. \nThis is joint work with Sam Hopkins. \n\n\n\n12:00-1:30pm\nLunch\n\n\n\n\n  \n\n\n\nEvents\,Past Events\,Programs
URL:https://cmsa.fas.harvard.edu/event/workshop-on-algebraic-methods-in-combinatorics/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Event,Workshop
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20171002T090000
DTEND;TZID=America/New_York:20171006T160000
DTSTAMP:20260506T100301
CREATED:20230717T173144Z
LAST-MODIFIED:20250304T211134Z
UID:10000037-1506934800-1507305600@cmsa.fas.harvard.edu
SUMMARY:Workshop on Additive Combinatorics\, Oct. 2-6\, 2017
DESCRIPTION:The workshop on additive combinatorics will take place October 2-6\, 2017 at the Center of Mathematical Sciences and Applications\, located at 20 Garden Street\, Cambridge\, MA. \nAdditive combinatorics is a mathematical area bordering on number theory\, discrete mathematics\, harmonic analysis and ergodic theory. It has achieved a number of successes in pure mathematics in the last two decades in quite diverse directions\, such as: \n\nThe first sensible bounds for Szemerédi’s theorem on progressions (Gowers);\nLinear patterns in the primes (Green\, Tao\, Ziegler);\nConstruction of expanding sets in groups and expander graphs (Bourgain\, Gamburd);\nThe Kakeya Problem in Euclidean harmonic analysis (Bourgain\, Katz\, Tao).\n\nIdeas and techniques from additive combinatorics have also had an impact in theoretical computer science\, for example \n\nConstructions of pseudorandom objects (eg. extractors and expanders);\nConstructions of extremal objects (eg. BCH codes);\nProperty testing (eg. testing linearity);\nAlgebraic algorithms (eg. matrix multiplication).\n\nThe main focus of this workshop will be to bring together researchers involved in additive combinatorics\, with a particular inclination towards the links with theoretical computer science. Thus it is expected that a major focus will be additive combinatorics on the boolean cube (Z/2Z)^n \, which is the object where the exchange of ideas between pure additive combinatorics and theoretical computer science is most fruitful. Another major focus will be the study of pseudorandom phenomena in additive combinatorics\, which has been an important contributor to modern methods of generating provably good randomness through deterministic methods. Other likely topics of discussion include the status of major open problems (the polynomial Freiman-Ruzsa conjecture\, inverse theorems for the Gowers norms with bounds\, explicit correlation bounds against low degree polynomials) as well as the impact of new methods such as the introduction of algebraic techniques by Croot–Pach–Lev and Ellenberg–Gijswijt. \nConfirmed participants include: \n\nArnab Bhattacharyya (Indian Institute of Science)\nThomas Bloom (University of Bristol)\nJop Briët (Centrum Wiskunde & Informatica\, Amsterdam)\nMei-Chu Chang (University of California\, Riverside)\nNoam Elkies (Harvard University)\nAsaf Ferber (MIT)\nJacob Fox (Stanford University)\nShafi Goldwasser (MIT)\nElena Grigorescu (Purdue University)\nHamed Hatami (McGill University)\nPooya Hatami (Institute for Advanced Study)\nKaave Hosseini (University of California\, San Diego)\nGuy Kindler (Hebrew University of Jerusalem)\nVsevolod Lev (University of Haifa at Oranim)\nSean Prendiville (University of Manchester)\nRonitt Rubinfeld (MIT)\nWill Sawin (ETH Zürich)\nFernando Shao (Oxford University)\nOlof Sisask (KTH Royal Institute of Technology)\nMadhur Tulsiani (University of Chicago)\nJulia Wolf (University of Bristol)\nEmanuele Viola (Northeastern University)\nYufei Zhao (MIT)\n\nCo-organizers of this workshop include Ben Green\, Swastik Kopparty\, Ryan O’Donnell\, Tamar Ziegler. \nMonday\, October 2 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n \n\n\n9:30-10:20am\nJacob Fox\nTower-type bounds for Roth’s theorem with popular differences \nAbstract: A famous theorem of Roth states that for any $\alpha > 0$ and $n$ sufficiently large in terms of $\alpha$\, any subset of $\{1\, \dots\, n\}$ with density $\alpha$ contains a 3-term arithmetic progression. Green developed an arithmetic regularity lemma and used it to prove that not only is there one arithmetic progression\, but in fact there is some integer $d > 0$ for which the density of 3-term arithmetic progressions with common difference $d$ is at least roughly what is expected in a random set with density $\alpha$. That is\, for every $\epsilon > 0$\, there is some $n(\epsilon)$ such that for all $n > n(\epsilon)$ and any subset $A$ of $\{1\, \dots\, n\}$ with density $\alpha$\, there is some integer $d > 0$ for which the number of 3-term arithmetic progressions in $A$ with common difference $d$ is at least $(\alpha^3-\epsilon)n$. We prove that $n(\epsilon)$ grows as an exponential tower of 2’s of height on the order of $\log(1/\epsilon)$. We show that the same is true in any abelian group of odd order $n$. These results are the first applications of regularity lemmas for which the tower-type bounds are shown to be necessary. \nThe first part of the talk by Jacob Fox includes an overview and discusses the upper bound. The second part of the talk by Yufei Zhao focuses on the lower bound construction and proof. These results are all joint work with Huy Tuan Pham.\n\n\n10:20-11:00am\nCoffee Break\n \n\n\n11:00-11:50am\nYufei Zhao\nTower-type bounds for Roth’s theorem with popular differences \nAbstract:  Continuation of first talk by Jacob Fox. The first part of the talk by Jacob Fox includes an overview and discusses the upper bound. The second part of the talk by Yufei Zhao focuses on the lower bound construction and proof. These results are all joint work with Huy Tuan Pham.\n\n\n12:00-1:30pm\nLunch\n \n\n\n1:30-2:20pm\nJop Briët\nLocally decodable codes and arithmetic progressions in random settings \nAbstract: This talk is about a common feature of special types of error correcting codes\, so-called locally decodable codes (LDCs)\, and two problems on arithmetic progressions in random settings\, random differences in Szemerédi’s theorem and upper tails for arithmetic progressions in a random set in particular. It turns out that all three can be studied in terms of the Gaussian width of a set of vectors given by a collection of certain polynomials. Using a matrix version of the Khintchine inequality and a lemma that turns such polynomials into matrices\, we give an alternative proof for the best-known lower bounds on LDCs and improved versions of prior results due to Frantzikinakis et al. and Bhattacharya et al. on arithmetic progressions in the aforementioned random settings. \nJoint work with Sivakanth Gopi\n\n\n2:20-3:00pm\nCoffee Break\n \n\n\n3:00-3:50pm\nFernando Shao\n\nLarge deviations for arithmetic progressions \nAbstract: We determine the asymptotics of the log-probability that the number of k-term arithmetic progressions in a random subset of integers exceeds its expectation by a constant factor. This is the arithmetic analog of subgraph counts in a random graph. I will highlight some open problems in additive combinatorics that we encountered in our work\, namely concerning the “complexity” of the dual functions of AP-counts. \n\n\n\n4:00-6:00pm\nWelcome Reception\n\n\n\n\nTuesday\, October 3 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n\n\n\n9:30-10:20am\nEmanuele Viola\nInterleaved group products \nAuthors: Timothy Gowers and Emanuele Viola \nAbstract: Let G be the special linear group SL(2\,q). We show that if (a1\,a2) and (b1\,b2) are sampled uniformly from large subsets A and B of G^2 then their interleaved product a1 b1 a2 b2 is nearly uniform over G. This extends a result of Gowers (2008) which corresponds to the independent case where A and B are product sets. We obtain a number of other results. For example\, we show that if X is a probability distribution on G^m such that any two coordinates are uniform in G^2\, then a pointwise product of s independent copies of X is nearly uniform in G^m\, where s depends on m only. Similar statements can be made for other groups as well. \nThese results have applications in computer science\, which is the area where they were first sought by Miles and Viola (2013).\n\n\n10:20-11:00am\nCoffee Break\n\n\n\n11:00-11:50am\nVsevolod Lev\nOn Isoperimetric Stability \nAbstract: We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular\, if $A$ and $S$ are finite\, non-empty subsets of an abelian group such that $S$ is independent\, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-c)|S||A|$ with a real $c\in(0\,1]$\, then $|A|\ge4^{(1-1/d)c|S|}$\, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. \nAs a corollary\, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For groups of exponent $2$ and $3$\, our bound translates into a sharp estimate for the additive  dimension of the popular difference set. \nWe also prove\, as an auxiliary result\, the following estimate of possible independent interest: if $A\subseteq{\mathbb Z}^n$ is a finite\, non-empty downset\, then\, denoting by $w(z)$ the number of non-zero components of the vector $z\in\mathbb{Z}^n$\, we have   $$ \frac1{|A|} \sum_{a\in A} w(a) \le \frac12\\, \log_2 |A|. $$\n\n\n12:00-1:30pm\nLunch\n\n\n\n1:30-2:20pm\nElena Grigorescu\nNP-Hardness of Reed-Solomon Decoding and the Prouhet-Tarry-Escott Problem \nAbstract: I will discuss the complexity of decoding Reed-Solomon codes\, and some results establishing NP-hardness for asymptotically smaller decoding radii than the maximum likelihood decoding radius. These results follow from the study of a generalization of the classical Subset Sum problem to higher moments\, which may be of independent interest. I will further discuss a connection with the Prouhet-Tarry-Escott problem studied in Number Theory\, which turns out to capture a main barrier in extending our techniques to smaller radii. \nJoint work with Venkata Gandikota and Badih Ghazi.\n\n\n2:20-3:00pm\nCoffee Break\n\n\n\n3:00-3:50pm\nSean Prendiville\nPartition regularity of certain non-linear Diophantine equations. \nAbstract:  We survey some results in additive Ramsey theory which remain valid when variables are restricted to sparse sets of arithmetic interest\, in particular the partition regularity of a class of non-linear Diophantine equations in many variables.\n\n\n\nWednesday\, October 4 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n \n\n\n9:30-10:20am\nOlof Sisask\nBounds on capsets via properties of spectra \nAbstract: A capset in F_3^n is a subset A containing no three distinct elements x\, y\, z satisfying x+z=2y. Determining how large capsets can be has been a longstanding problem in additive combinatorics\, particularly motivated by the corresponding question for subsets of {1\,2\,…\,N}. While the problem in the former setting has seen spectacular progress recently through the polynomial method of Croot–Lev–Pach and Ellenberg–Gijswijt\, such progress has not been forthcoming in the setting of the integers. Motivated by an attempt to make progress in this setting\, we shall revisit the approach to bounding the sizes of capsets using Fourier analysis\, and in particular the properties of large spectra. This will be a two part talk\, in which many of the ideas will be outlined in the first talk\, modulo the proof of a structural result for sets with large additive energy. This structural result will be discussed in the second talk\, by Thomas Bloom\, together with ideas on how one might hope to achieve Behrend-style bounds using this method. \nJoint work with Thomas Bloom.\n\n\n10:20-11:00am\nCoffee Break\n \n\n\n11:00-11:50am\nThomas Bloom\nBounds on capsets via properties of spectra \nThis is a continuation of the previous talk by Olof Sisask.\n\n\n12:00-1:30pm\nLunch\n \n\n\n1:30-2:20pm\nHamed Hatami\nPolynomial method and graph bootstrap percolation \nAbstract: We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel\, and the latter provides an alternative and simpler proof for one of their main results. This is based on a joint work with Lianna Hambardzumyan and Yingjie Qian.\n\n\n2:20-3:00pm\nCoffee Break\n\n\n\n3:00-3:50pm\nArnab Bhattacharyya\nAlgorithmic Polynomial Decomposition \nAbstract: Fix a prime p. Given a positive integer k\, a vector of positive integers D = (D_1\, …\, D_k) and a function G: F_p^k → F_p\, we say a function P: F_p^n → F_p admits a (k\, D\, G)-decomposition if there exist polynomials P_1\, …\, P_k: F_p^n -> F_p with each deg(P_i) <= D_i such that for all x in F_p^n\, P(x) = G(P_1(x)\, …\, P_k(x)). For instance\, an n-variate polynomial of total degree d factors nontrivially exactly when it has a (2\, (d-1\, d-1)\, prod)-decomposition where prod(a\,b) = ab. \nWhen show that for any fixed k\, D\, G\, and fixed bound d\, we can decide whether a given polynomial P(x_1\, …\, x_n) of degree d admits a (k\,D\,G)-decomposition and if so\, find a witnessing decomposition\, in poly(n) time. Our approach is based on higher-order Fourier analysis. We will also discuss improved analyses and algorithms for special classes of decompositions. \nJoint work with Pooya Hatami\, Chetan Gupta and Madhur Tulsiani.\n\n\n\nThursday\, October 5 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n\n\n\n9:30-10:20am\nMadhur Tulsiani\nHigher-order Fourier analysis and approximate decoding of Reed-Muller codes \n Abstract: Decomposition theorems proved by Gowers and Wolf provide an appropriate notion of “Fourier transform” for higher-order Fourier analysis. I will discuss some questions and techniques that arise from trying to develop polynomial time algorithms for computing these decompositions. \nI will discuss constructive proofs of these decompositions based on boosting\, which reduce the problem of computing these decompositions to a certain kind of approximate decoding problem for codes. I will also discuss some earlier and recent works on this decoding problem. \nBased on joint works with Arnab Bhattacharyya\, Eli Ben-Sasson\, Pooya Hatami\, Noga Ron-Zewi and Julia Wolf.\n\n\n10:20-11:00am\nCoffee Break\n\n\n\n11:00-11:50am\nJulia Wolf\nStable arithmetic regularity \nThe arithmetic regularity lemma in the finite-field model\, proved by Green in 2005\, states that given a subset A of a finite-dimensional vector space over a prime field\, there exists a subspace H of bounded codimension such that A is Fourier-uniform with respect to almost all cosets of H. It is known that in general\, the growth of the codimension of H is required to be of tower type depending on the degree of uniformity\, and that one must allow for a small number of non-uniform cosets. \nOur main result is that\, under a natural model-theoretic assumption of stability\, the tower-type bound and non-uniform cosets in the arithmetic regularity lemma are not necessary.  Specifically\, we prove an arithmetic regularity lemma for k-stable subsets in which the bound on the codimension of the subspace is a polynomial (depending on k) in the degree of uniformity\, and in which there are no non-uniform cosets. \nThis is joint work with Caroline Terry. \n\n\n\n12:00-1:30pm\nLunch\n \n\n\n1:30-2:20pm\nWill Sawin\n\nConstructions of Additive Matchings \nAbstract: I will explain my work\, with Robert Kleinberg and David Speyer\, constructing large tri-colored sum-free sets in vector spaces over finite fields\, and how it shows that some additive combinatorics problems over finite fields are harder than corresponding problems over the integers.  \n\n\n\n2:20-3:00pm\nCoffee Break\n\n\n\n3:00-3:50pm\nMei-Chu Chang\nArithmetic progressions in multiplicative groups of finite fields \nAbstract:   Let G be a multiplicative subgroup of the prime field F_p of size |G|> p^{1-\kappa} and r an arbitrarily fixed positive integer. Assuming \kappa=\kappa(r)>0 and p large enough\, it is shown that any proportional subset A of G contains non-trivial arithmetic progressions of length r.\n\n\n\nFriday\, October 6 \n\n\n\nTime\nSpeaker\nTitle/Abstract\n\n\n9:00-9:30am\nBreakfast\n\n\n\n9:30-10:20am\nAsaf Ferber\nOn a resilience version of the Littlewood-Offord problem \nAbstract:  In this talk we consider a resilience version of the classical Littlewood-Offord problem. That is\, consider the sum X=a_1x_1+…a_nx_n\, where the a_i-s are non-zero reals and x_i-s are i.i.d. random variables with     (x_1=1)= P(x_1=-1)=1/2. Motivated by some problems from random matrices\, we consider the question: how many of the x_i-s  can we typically allow an adversary to change without making X=0? We solve this problem up to a constant factor and present a few interesting open problems. \nJoint with: Afonso Bandeira (NYU) and Matthew Kwan (ETH\, Zurich).\n\n\n10:20-11:00am\nCoffee Break\n\n\n\n11:00-11:50am\nKaave Hosseini\nProtocols for XOR functions and Entropy decrement \nAbstract: Let f:F_2^n –> {0\,1} be a function and suppose the matrix M defined by M(x\,y) = f(x+y) is partitioned into k monochromatic rectangles.  We show that F_2^n can be partitioned into affine subspaces of co-dimension polylog(k) such that f is constant on each subspace. In other words\, up to polynomial factors\, deterministic communication complexity and parity decision tree complexity are equivalent. \nThis relies on a novel technique of entropy decrement combined with Sanders’ Bogolyubov-Ruzsa lemma. \nJoint work with Hamed Hatami and Shachar Lovett\n\n\n12:00-1:30pm\nLunch\n\n\n\n1:30-2:20pm\nGuy Kindler\n\nFrom the Grassmann graph to Two-to-Two games \nAbstract: In this work we show a relation between the structure of the so called Grassmann graph over Z_2 and the Two-to-Two conjecture in computational complexity. Specifically\, we present a structural conjecture concerning the Grassmann graph (together with an observation by Barak et. al.\, one can view this as a conjecture about the structure of non-expanding sets in that graph) which turns out to imply the Two-to-Two conjecture. \nThe latter conjecture its the lesser-known and weaker sibling of the Unique-Games conjecture [Khot02]\, which states that unique games (a.k.a. one-to-one games) are hard to approximate. Indeed\, if the Grassmann-Graph conjecture its true\, it would also rule out some attempts to refute the Unique-Games conjecture\, as these attempts provide potentially efficient algorithms to solve unique games\, that would actually also solve two-to-two games if they work at all. \nThese new connections between the structural properties of the Grassmann graph and complexity theoretic conjectures highlight the Grassmann graph as an interesting and worthy object of study. We may indicate some initial results towards analyzing its structure. \nThis is joint work with Irit Dinur\, Subhash Khot\, Dror Minzer\, and Muli Safra. \n\n\n\n\n\n\n\nEvents\,Past Events
URL:https://cmsa.fas.harvard.edu/event/workshop-on-additive-combinatorics-oct-2-6-2017/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Event,Workshop
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20170418T110000
DTEND;TZID=America/New_York:20170418T120000
DTSTAMP:20260506T100301
CREATED:20240213T093734Z
LAST-MODIFIED:20240220T144719Z
UID:10002348-1492513200-1492516800@cmsa.fas.harvard.edu
SUMMARY:4-18-2017 Social Science Applications Forum
DESCRIPTION:
URL:https://cmsa.fas.harvard.edu/event/4-18-2017-social-science-applications-forum/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Seminars
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20170324T100000
DTEND;TZID=America/New_York:20170324T110000
DTSTAMP:20260506T100301
CREATED:20240213T102725Z
LAST-MODIFIED:20240220T143442Z
UID:10002428-1490349600-1490353200@cmsa.fas.harvard.edu
SUMMARY:3-24-2017 Random Matrix & Probability Theory Seminar
DESCRIPTION:
URL:https://cmsa.fas.harvard.edu/event/3-24-2017-random-matrix-probability-theory-seminar/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Random Matrix & Probability Theory Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20170308T121100
DTEND;TZID=America/New_York:20170419T121100
DTSTAMP:20260506T100301
CREATED:20230717T174006Z
LAST-MODIFIED:20250328T194543Z
UID:10000024-1488975060-1492603860@cmsa.fas.harvard.edu
SUMMARY:Special Lecture Series on Donaldson-Thomas and Gromov-Witten Theories
DESCRIPTION:From March 8 to April 19\, the Center of Mathematical Sciences and Applications will be hosting a special lecture series on Donaldson-Thomas and Gromov-Witten Theories. Artan Sheshmani (QGM Aarhus and CMSA Harvard) will give eight talks on the topic on Wednesdays and Fridays from 9:00-10:30 am\, which will be recorded and promptly available on CMSA’s Youtube Channel.
URL:https://cmsa.fas.harvard.edu/event/special-lecture-series-on-donaldson-thomas-and-gromov-witten-theories/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Event,Special Lectures
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20161203T090000
DTEND;TZID=America/New_York:20161204T170000
DTSTAMP:20260506T100301
CREATED:20230717T172404Z
LAST-MODIFIED:20250305T201523Z
UID:10000018-1480755600-1480870800@cmsa.fas.harvard.edu
SUMMARY:Mini-school on Nonlinear Equations\, December 3-4\, 2016
DESCRIPTION:The Center of Mathematical Sciences and Applications will be hosting a Mini-school on Nonlinear Equations on December 3-4\, 2016. The conference will have speakers and will be hosted at Harvard CMSA Building: Room G10 20 Garden Street\, Cambridge\, MA 02138. \nSpeakers:\n\nCliff Taubes (Harvard University)\nValentino Tosatti (Northwestern University)\nPengfei Guan (McGill University)\nJared Speck (MIT)\n\nSchedule:\n\n\n\nDecember 3rd – Day 1\n\n\n9:00am – 10:30am\nCliff Taubes\, “Compactness theorems in gauge theories”\n\n\n10:45am – 12:15pm\nValentino Tosatti\, “Complex Monge-Ampère Equations”\n\n\n\n\n\n12:15pm – 1:45pm\nLUNCH\n\n\n\n\n\n\n1:45pm – 3:15pm\nPengfei Guan\, “Monge-Ampère type equations and related geometric problems”\n\n\n3:30pm – 5:00pm\nJared Speck\, “Finite-time degeneration of hyperbolicity without blowup for solutions to quasilinear wave equations”\n\n\n\n\n\n\n\n\nDecember 4th – Day 2\n\n\n9:00am – 10:30am\nCliff Taubes\, “Compactness theorems in gauge theories”\n\n\n10:45am – 12:15pm\nValentino Tosatti\, “Complex Monge-Ampère Equations”\n\n\n\n\n\n12:15pm – 1:45pm\nLUNCH\n\n\n\n\n\n\n1:45pm – 3:15pm\nPengfei Guan\, “Monge-Ampère type equations and related geometric problems”\n\n\n3:30pm – 5:00pm\nJared Speck\, “Finite-time degeneration of hyperbolicity without blowup for solutions to quasilinear wave equations”\n\n\n\n\n  \n* This event is sponsored by National Science Foundation (NSF) and CMSA Harvard University.
URL:https://cmsa.fas.harvard.edu/event/mini-school-on-nonlinear-equations-december-3-4-2016/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Conference,Event,Workshop
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/minischool.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20160506T090000
DTEND;TZID=America/New_York:20160508T170000
DTSTAMP:20260506T100301
CREATED:20230831T035136Z
LAST-MODIFIED:20250304T171749Z
UID:10000008-1462525200-1462726800@cmsa.fas.harvard.edu
SUMMARY:The Simons Collaboration Program in Homological Mirror Symmetry
DESCRIPTION:The Simons Collaboration program in Homological Mirror Symmetry at Harvard CMSA and Brandeis University is part of the bigger Simons collaboration program on Homological mirror symmetry (https://schms.math.berkeley.edu) which brings to CMSA experts on algebraic geometry\, Symplectic geometry\, Arithmetic geometry\, Quantum topology and mathematical aspects of high energy physics\, specially string theory with the goal of proving the homological mirror symmetry conjecture (HMS) in full generality and explore its applications. Mirror symmetry\, which emerged in the late 1980s as an unexpected physical duality between quantum field theories\, has been a major source of progress in mathematics. At the 1994 ICM\, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or Landau-Ginzburg model). We are happy to announce that the Simons Foundation has agreed to renew funding for the HMS collaboration program for three additional years. \nA brief induction of the Brandeis-Harvard CMSA HMS/SYZ research agenda and team members are as follows: \n\nDirectors: \n\nShing-Tung Yau (Harvard University) \nBorn in Canton\, China\, in 1949\, S.-T. Yau grew up in Hong Kong\, and studied in the Chinese University of Hong Kong from 1966 to 1969. He did his PhD at UC Berkeley from 1969 to 1971\, as a student of S.S. Chern. He spent a year as a postdoc at the Institute for Advanced Study in Princeton\, and a year as assistant professor at SUNY at Stony Brook. He joined the faculty at Stanford in 1973. On a Sloan Fellowship\, he spent a semester at the Courant Institute in 1975. He visited UCLA the following year\, and was offered a professorship at UC Berkeley in 1977. He was there for a year\, before returning to Stanford. He was a plenary speaker at the 1978 ICM in Helsinki. The following year\, he became a faculty member at the IAS in Princeton. He moved to UCSD in 1984. Yau came to Harvard in 1987\, and was appointed the Higgins Professor of Mathematics in 1997. He has been at Harvard ever since. Yau has received numerous prestigious awards and honors throughout his career. He was named a California Scientist of the Year in 1979. In 1981\, he received a Oswald Veblen Prize in Geometry and a John J. Carty Award for the Advancement of Science\, and was elected a member of the US National Academy of Sciences. In 1982\, he received a Fields Medal for “his contributions to partial differential equations\, to the Calabi conjecture in algebraic geometry\, to the positive mass conjecture of general relativity theory\, and to real and complex MongeAmpre equations”. He was named Science Digest\, America’s 100 Brightest Scientists under 40\, in 1984. In 1991\, he received a Humboldt Research Award from the Alexander von Humboldt Foundation in Germany. He was awarded a Crafoord Prize in 1994\, a US National Medal of Science in 1997\, and a China International Scientific and Technological Cooperation Award\, for “his outstanding contribution to PRC in aspects of making progress in sciences and technology\, training researchers” in 2003. In 2010\, he received a Wolf Prize in Mathematics\, for “his work in geometric analysis and mathematical physics”. Yau has also received a number of research fellowships\, which include a Sloan Fellowship in 1975-1976\, a Guggenheim Fellowship in 1982\, and a MacArthur Fellowship in 1984-1985. Yau’s research interests include differential and algebraic geometry\, topology\, and mathematical physics. As a graduate student\, he started to work on geometry of manifolds with negative curvature. He later became interested in developing the subject of geometric analysis\, and applying the theory of nonlinear partial differential equations to solve problems in geometry\, topology\, and physics. His work in this direction include constructions of minimal submanifolds\, harmonic maps\, and canonical metrics on manifolds. The most notable\, and probably the most influential of this\, was his solution of the Calabi conjecture on Ricci flat metrics\, and the existence of Kahler-Einstein metrics. He has also succeeded in applying his theory to solve a number of outstanding conjectures in algebraic geometry\, including Chern number inequalities\, and the rigidity of complex structures of complex projective spaces. Yau’s solution to the Calabi conjecture has been remarkably influential in mathematical physics over the last 30 years\, through the creation of the theory of Calabi-Yau manifolds\, a theory central to mirror symmetry. He and a team of outstanding mathematicians trained by him\, have developed many important tools and concepts in CY geometry and mirror symmetry\, which have led to significant progress in deformation theory\, and on outstanding problems in enumerative geometry. Lian\, Yau and his postdocs have developed a systematic approach to study and compute period integrals of CY and general type manifolds. Lian\, Liu and Yau (independently by Givental) gave a proof of the counting formula of Candelas et al for worldsheet instantons on the quintic threefold. In the course of understanding mirror symmetry\, Strominger\, Yau\, and Zaslow proposed a new geometric construction of mirror symmetry\, now known as the SYZ construction. This has inspired a rapid development in CY geometry over the last two decades. In addition to CY geometry and mirror symmetry\, Yau has done influential work on nonlinear partial differential equations\, generalized geometry\, Kahler geometry\, and general relativity. His proof of positive mass conjecture is a widely regarded as a cornerstone in the classical theory of general relativity. In addition to publishing well over 350 research papers\, Yau has trained more than 60 PhD students in a broad range of fields\, and mentored dozens of postdoctoral fellows over the last 40 years. \n\nProfessor Bong Lian (Brandeis University) \nBorn in Malaysia in 1962\, Bong Lian completed his PhD in physics at Yale University under the direction of G. Zuckerman in 1991. He joined the permanent faculty at Brandeis University in 1995\, and has remained there since. Between 1995 and 2013\, he had had visiting research positions at numerous places\, including the National University of Taiwan\, Harvard University\, and Tsinghua University. Lian received a J.S. Guggenheim Fellowship in 2003. He was awarded a Chern Prize at the ICCM in Taipei in 2013\, for his “influential and fundamental contributions in mathematical physics\, in particular in the theory of vertex algebras and mirror symmetry.” He has also been co-Director\, since 2014\, of the Tsinghua Mathcamp\, a summer outreach program launched by him and Yau for mathematically talented teenagers in China. Since 2008\, Lian has been the President of the International Science Foundation of Cambridge\, a non-profit whose stated mission is “to provide financial and logistical support to scholars and universities\, to promote basic research and education in mathematical sciences\, especially in the Far East.” Over the last 20 years\, he has mentored a number of postdocs and PhD students. His research has been supported by an NSF Focused Research Grant since 2009. Published in well over 60 papers over 25 years\, Lian’s mathematical work lies in the interface between representation theory\, Calabi-Yau geometry\, and string theory. Beginning in the late 80’s\, Lian\, jointly with Zuckerman\, developed the theory of semi-infinite cohomology and applied it to problems in string theory. In 1994\, he constructed a new invariant (now known as the Lian- Zuckerman algebra) of a topological vertex algebra\, and conjectured the first example of a G algebra in vertex algebra theory. The invariant has later inspired a new construction of quantum groups by I. Frenkel and A. Zeitlin\, as semi-infinite cohomology of braided vertex algebras\, and led to a more recent discovery of new relationships between Courant algebroids\, A-algebras\, operads\, and deformation theory of BV algebras. In 2010\, he and his students Linshaw and Song developed important applications of vertex algebras in equivariant topology. Lian’s work in CY geometry and mirror symmetry began in early 90’s. Using a characteristic p version of higher order Schwarzian equations\, Lian and Yau gave an elementary proof that the instanton formula of Candelas et al implies Clemens’s divisibility conjecture for the quintic threefold\, for infinitely many degrees. In 1996\, Lian (jointly with Hosono and Yau) answered the so-called Large Complex Structure Limit problem in the affirmative in many important cases. Around the same year\, they announced their hyperplane conjecture\, which gives a general formula for period integrals for a large class of CY manifolds\, extending the formula of Candelas et al. Soon after\, Lian\, Liu and Yau (independently by Givental) gave a proof of the counting formula. In 2003\, inspired by mirror symmetry\, Lian (jointly with Hosono\, Oguiso and Yau) discovered an explicit counting formula for Fourier-Mukai partners\, and settled an old problem of Shioda on abelian and K3 surfaces. Between 2009 and 2014\, Lian (jointly with Bloch\, Chen\, Huang\, Song\, Srinivas\, Yau\, and Zhu) developed an entirely new approach to study the so-called Riemann-Hilbert problem for period integrals of CY manifolds\, and extended it to general type manifolds. The approach leads to an explicit description of differential systems for period integrals with many applications. In particular\, he answered an old question in physics on the completeness of Picard-Fuchs systems\, and constructed new differential zeros of hypergeometric functions. \n\nDenis Auroux (Harvard University) \nDenis Auroux’s research concerns symplectic geometry and its applications to mirror symmetry. While his early work primarily concerned the topology of symplectic 4-manifolds\, over the past decade Auroux has obtained pioneering results on homological mirror symmetry outside of the Calabi-Yau setting (for Fano varieties\, open Riemann surfaces\, etc.)\, and developed an extension of the SYZ approach to non-Calabi-Yau spaces.After obtaining his PhD in 1999 from Ecole Polytechnique (France)\, Auroux was employed as Chargé de Recherche at CNRS and CLE Moore Instructor at MIT\, before joining the faculty at MIT in 2002 (as Assistant Professor from 2002 to 2004\, and as Associate Professor from 2004 to 2009\, with tenure starting in 2006). He then moved to UC Berkeley as a Full Professor in 2009.\nAuroux has published over 30 peer-reviewed articles\, including several in top journals\, and given 260 invited presentations about his work. He received an Alfred P. Sloan Research Fellowship in 2005\, was an invited speaker at the 2010 International Congress of Mathematicians\, and in 2014 he was one of the two inaugural recipients of the Poincaré Chair at IHP. He has supervised 10 PhD dissertations\, won teaching awards at MIT and Berkeley\, and participated in the organization of over 20 workshops and conferences in symplectic geometry and mirror symmetry.\n\n \n\n\n\nSenior Personnel: \n\nArtan Sheshmani (Harvard CMSA) \nArtan Sheshmani’s research is focused on enumerative algebraic geometry and mathematical aspects of string theory. He is interested in applying techniques in algebraic geometry\, such as\, intersection theory\, derived category theory\, and derived algebraic geometry to construct and compute the deformation invariants of algebraic varieties\, in particular Gromov-Witten (GW) or Donaldson-Thomas (DT) invariants. In the past Professor Sheshmani has worked on proving modularity property of certain DT invariants of K3-fibered threefolds (as well as their closely related Pandharipande-Thomas (PT) invariants)\, local surface threefolds\, and general complete intersection Calabi-Yau threefolds. The modularity of DT/PT invariants in this context is predicted in a famous conjecture of  string theory called S-duality modularity conjecture\, and his joint work has provided the proof to some cases of it\, using degenerations\, virtual localizations\, as well as wallcrossing techniques. Recently\, Sheshmani has focused on proving a series of dualities relating the various enumerative invariants over threefolds\, notably the GW invariants and invariants that arise in topological gauge theory. In particular in his joint work with Gholampour\, Gukov\, Liu\, Yau he studied DT gauge theory and its reductions to D=4 and D=2 which are equivalent to local theory of surfaces in Calabi-Yau threefolds. Moreover\, in a recent joint work with Yau and Diaconescu\, he has studied the construction and computation of DT invariants of Calabi-Yau fourfolds via a suitable derived categorical reduction of the theory to the DT theory of threefolds. Currently Sheshmani is interested in a wide range of problems in enumerative geometry of CY varieties in dimensions 3\,4\,5. \nArtan has received his PhD and Master’s degrees in pure mathematics under Sheldon Katz and Thomas Nevins from the University of Illinois at Urbana Champaign (USA) in 2011 and 2008 respectively. He holds a Master’s degree in Solid Mechanics (2004) and two Bachelor’s degrees\, in Mechanical Engineering and Civil Engineering from the Sharif University of Technology\, Tehran\, Iran.  Artan has been a tenured Associate Professor of Mathematics with joint affiliation at Harvard CMSA and center for Quantum Geometry of Moduli Spaces (QGM)\, since 2016. Before that he has held visiting Associate Professor and visiting Assistant Professor positions at MIT. \nAn Huang (Brandeis University) \nThe research of An Huang since 2011 has been focused on the interplay between algebraic geometry\, the theory of special functions and mirror symmetry. With S. Bloch\, B. Lian\, V. Srinivas\, S.-T. Yau\, X. Zhu\, he has developed the theory of tautological systems\, and has applied it to settle several important problems concerning period integrals in relation to mirror symmetry. With B. Lian and X. Zhu\, he has given a precise geometric interpretation of all solutions to GKZ systems associated to Calabi-Yau hypersurfaces in smooth Fano toric varieties. With B. Lian\, S.-T. Yau\, and C.-L. Yu\, he has proved a conjecture of Vlasenko concerning an explicit formula for unit roots of the zeta functions of hypersurfaces\, and has further related these roots to p-adic interpolations of complex period integrals. Beginning in 2018\, with B. Stoica and S.-T. Yau\, he has initiated the study of p-adic strings in curved spacetime\, and showed that general relativity is a consequence of the self-consistency of quantum p-adic strings. One of the goals of this study is to understand p-adic A and B models. \nAn Huang received his PhD in Mathematics from the University of California at Berkeley in 2011. He was a postdoctoral fellow at the Harvard University Mathematics Department\, and joined Brandeis University as an Assistant Professor in Mathematics in 2016.\n\n\n\nSiu Cheong Lau (Boston University) \nThe research interest of Siu Cheong Lau lies in SYZ mirror symmetry\, symplectic and algebraic geometry.  His thesis work has successfully constructed the SYZ mirrors for all toric Calabi-Yau manifolds based on quantum corrections by open Gromov-Witten invariants and their wall-crossing phenomenon.  In collaboration with N.C. Leung\, H.H. Tseng and K. Chan\, he derived explicit formulas for the open Gromov-Witten invariants for semi-Fano toric manifolds which have an obstructed moduli theory.  It has a beautiful relation with mirror maps and Seidel representations.   Recently he works on a local-to-global approach to SYZ mirror symmetry.  In joint works with C.H. Cho and H. Hong\, he developed a noncommutative local mirror construction for immersed Lagrangians\, and a natural gluing method to construct global mirrors.  The construction has been realized in various types of geometries including orbifolds\, focus-focus singularities and pair-of-pants decompositions of Riemann surfaces. \nSiu-Cheong Lau has received the Doctoral Thesis Gold Award (2012) and the Best Paper Silver Award (2017) at the International Congress of Chinese Mathematicians.  He was awarded the Simons Collaboration Grant in 2018.  He received a Certificate of Teaching Excellence from Harvard University in 2014. \n\nAffiliates: \n\nNetanel Rubin-Blaier (Cambridge)\nKwokwai Chan (Chinese University of Hong Kong)\nMandy Cheung (Harvard University\, BP)\nChuck Doran (University of Alberta)\nHansol Hong (Yonsei University)\nShinobu Hosono (Gakushuin University\, Japan)\nConan Leung (Chinese University of Hong Kong)\nYu-Shen Lin (Boston University)\nHossein Movassati (IMPA Brazil)\nArnav Tripathhy (Harvard University\, BP)\n\n  \nPostdocs: \n\nDennis Borisov\nTsung-Ju Lee\nDingxin Zhang\nJingyu Zhao\nYang Zhou\n\n  \nTo learn about previous programming as part of the Simons Collaboration\, click here.
URL:https://cmsa.fas.harvard.edu/event/the-simons-collaboration-in-homological-mirror-symmetry/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Programs
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20140915T140000
DTEND;TZID=America/New_York:20140915T190000
DTSTAMP:20260506T100301
CREATED:20230717T180624Z
LAST-MODIFIED:20240209T214309Z
UID:10000011-1410789600-1410807600@cmsa.fas.harvard.edu
SUMMARY:Topological Insulators and Mathematical Science – Conference and Program
DESCRIPTION:The CMSA will be hosting a conference on the subject of topological insulators and mathematical science on September 15-17.  Seminars will take place each day from 2:00-7:00pm in Science Center Hall D\, 1 Oxford Street\, Cambridge\, MA.
URL:https://cmsa.fas.harvard.edu/event/topological-insulators-and-mathematical-science-conference-and-program/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Conference,Event
END:VEVENT
END:VCALENDAR